两阶段反应扩散系统的持久性

摘要

在这篇文章中，我们研究了异质环境中阶段结构种群的反应扩散模型中的扩散率如何影响该模型对种群持久性或灭绝的预测。对于没有阶段结构的种群，更快的扩散通常是有害的。与此相反，我们发现，在阶段结构的人群中，它既可能有害，也可能有益。如果成年人可以繁殖的区域与青少年可以成熟的区域相同，则通常会倾向于扩散较慢，但如果这些区域分开，则可能会倾向于更快的扩散。我们的分析主要包括对周围线性化系统主特征值的估计$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$以及它们在大小扩散率下的渐近行为的结果。我们研究的模型一般不是一个合作系统，但如果成年人只与其他成年人竞争，而青少年只与其他青少年竞争，那么它就是。在这种情况下，合作系统的一般理论暗示，当模型预测持续性时，它具有独特的正平衡。在这种情况下，我们得到了关于小扩散和大成人生殖率正平衡点渐近行为的一些结果。

1.简介

近年来，人们对扩散如何与空间异质性相互作用以影响种群动态和物种相互作用的问题进行了广泛的研究，特别是从反应扩散系统和相关模型的角度——例如[1,2,三]以及其中引用的参考文献。关于这个主题的大多数工作都假设每个种群都是由空间构成的，并且只有一种扩散模式。然而，种群通常是按年龄、阶段或其他属性构成的，个体之间的扩散速度或模式可能会有所不同。在这里，我们将研究阶段结构的存在如何影响扩散率如何影响具有两个阶段的种群的一类反应扩散模型中的种群动力学。如果种群具有逻辑增长，没有年龄或阶段结构，在时间恒定的封闭有界空间异质环境中扩散，众所周知，反应扩散模型预测，较慢的扩散速度相对于较快的扩散速度是有利的[4,5]. 中的结果[5]也适用于补丁模型。更广泛地说，出现在种群遗传学、种群动力学和相关领域的一大类模型显示了某种形式的减少原理，即扩散会导致更快的混合，通常会降低种群增长速度[6]. 然而，对于阶段结构种群，情况似乎大不相同。在[7]，作者考虑了结构化种群的离散时间斑块模型，发现在某些情况下，没有针对快速扩散的选择。本论文的目标是使用空间显式反应扩散模型来理解有利于成虫繁殖和有利于幼虫生存和生长的栖息地的空间分布如何影响快速扩散对阶段结构种群是有利还是有害。我们将看到，答案取决于有利和不利栖息地的空间分布细节。

我们将研究的反应扩散模型的类型是

$$\left\{\begin{array}{c}{\displaystyle \frac{<trans\; data-src="\partial ">\partial </trans>\mathrm{<trans\; data-src="u">u个</trans>}}{<trans\; data-src="\partial ">\partial </trans>\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>}\hfill \\ {\displaystyle \frac{<trans\; data-src="\partial ">\partial </trans>\mathrm{<trans\; data-src="v">v（v）</trans>}}{<trans\; data-src="\partial ">\partial </trans>\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>}\hfill \\ {\displaystyle <trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans><trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>}\hfill \end{array}\right.$$

(1)

$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$是中的有界域${\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="N">N个</trans>}}$、和$\mathrm{<trans\; data-src="\nu ">\nu </trans>}$外单位是否垂直于$<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$，使得系统具有Neumann边界条件，即简单扩散的无通量边界条件。在这个系统中，u个和v（v）表示同一物种中已达到生育年龄的青少年和个体（即分别为成年人）的人口密度。因此，术语$\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$代表青少年成熟为成年人的速度，由达到生育年龄的个体比例和成熟速度决定，而$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$说明了当地成年人的繁殖力$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$描述了具有密度的成年种群产生新幼仔的速度v（v）在位置x个.条款$\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$、和$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$解释了由于logistic自模拟导致的全脑死亡率和饱和因子。扩散系数${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$和${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$分别解释了青少年和成年人的传播速度。假设系数在$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}$。这是中引入的阶段结构人口的模型类型[8]. 具有不同解释的相关模型在中进行了讨论[9,10]以及那些论文中的参考文献。用方程式表示的模型(1)不是显式的年龄结构模型。它假设处于青少年阶段的个体以某种空间依赖性的速度成熟，但不跟踪每个阶段中个体的年龄。在[11,12,13]. 基于延迟反应扩散方程，在[14]. 我们这里的重点是空间异质性、扩散和舞台结构是如何相互作用的，所以我们选择了最简单的舞台结构形式。在以下情况下$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，系统是协作的，以及[15,16]方程的线性化(1)大约$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$是合作的，所以[15]适用于它；特别是，通过一些技术假设，它们意味着它具有主特征值。

我们将在本工作中解决的主要问题与理解方程中不同函数和系数的作用有关(1)物种的持久性。在本文的剩余部分中，我们将主要关注如何理解方程线性化的主特征值(1)大约$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$取决于系数以及这种依赖性在生物学上的含义。我们将看到，更快的扩散对种群的持续生存是有害的还是有益的，取决于有利于成虫繁殖和有利于幼虫生存和成熟的栖息地分布的细节。在某些情况下，较慢的扩散仍然是一个优势，但有时更快的扩散会有所帮助，甚至可能需要足够快的扩散才能持久。有利于成虫繁殖的栖息地的空间分布($\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$大）相对于那些有利于青少年发展的($\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$大型）在某些情况下非常重要。我们在这里的分析在精神上与在[三,17,18,19]. 特别是，我们将研究系统在小扩散率、大扩散率和一般扩散率下的行为。一些流行病学模型的相关结果在[20,21].

方程的线性化(1)大约$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$有一个主特征值，其符号决定了模型预测的是持久性还是灭绝性。自符号的主特征值线性化方程(1)大约$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$决定方程式的命运(1)对于它描述的人口预测，我们将详细研究以下问题：

$$\left\{\begin{array}{cc}\hfill {\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}& <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="\phi ">\phi </trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill {\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}& <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="\psi ">\psi </trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill <trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans><trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}& <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=".">.</trans>\hfill \end{array}\right.$$

(2)

2.基本属性

在本节中，我们将讨论方程的一些基本性质(1). 从现在开始，我们假设$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=")">)</trans>$,$<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$是一流的${\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}$，并且以下假设成立：

假设 1 （H1）。

$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$在里面Ω，带有 $\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="0">0</trans>}$, $\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="0">0</trans>}$ 对一些人来说 ${\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="x">x个</trans>}}_{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$.

假设 2 （H2）。

$\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$在里面Ω.

假设 三 （H3）。

$\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$,$\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$在里面$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}$.

用方程式表示的模型(1)与中讨论的模型有许多共同的数学特征[9,10]对于个体可以在两种不同运动模式之间切换的人群。一个关键特征是方程的线性部分(1)是协同的，因此它将具有决定平衡稳定性的主特征值$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$从而导致种群的持续或灭绝。方程式的另一个关键特征(1)非线性是亚均匀的。合作线性系统（如方程右侧的线性部分）的最大值原理和主特征值的存在性(1)派生于[15]. 方程等系统的一般理论(1)开发于[16]对于完全合作的情况（其中$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，例如成年人只与其他成年人竞争，青少年与其他青少年竞争），在一般情况下[8,9,10,22]. 正如所料，方程线性化的主特征值的符号(1)周围$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$为我们研究物种的持久性提供了相关信息。如果是积极的，人口将持续存在。如果是非阳性，人口将灭绝。如果系数c（c）和克在方程式中(1)零，这样系统是合作的，结果和方法[16]这意味着，如果线性部分的主特征值为正，则系统具有唯一的全局吸引平衡。如果这些系数很小[9,10]可以应用于显示方程(1)是渐近合作的，并且仍然具有唯一的全局吸引正平衡点。合并中给出的结果[8,9,10,15,16]或者直接遵循这些论文中使用的相同论点，我们有以下几点：

引理 1

用方程表示的特征值问题(2)具有唯一的主特征值${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$具有正特征向量的$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src=")">)</trans>$.

引理 2

如果${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后用方程式表示系统(1)是持久的，并且至少有一个正平衡。如果${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$在方程中是全局渐近稳定的(1).

引理 三。

如果${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$如果c和g足够小，那么方程中表示的系统(1)具有独特的全球吸引正平衡。

备注 1

在这种情况下$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，系统用等式表示(1)是合作的，因此在适当的空间上生成单调的半流。

3.案例${\mathit{<trans\; data-src="d">d日</trans>}}_{\mathbf{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathit{<trans\; data-src="d">d日</trans>}}_{\mathbf{<trans\; data-src="2">2</trans>}}$小型

遵循中的方法[23]，我们将建立方程主特征值的渐近行为(2)何时${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$规模较小，在完全合作的情况下（其中$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$)方程相应稳态系统的非负解轮廓(1).

$$\left\{\begin{array}{c}{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ {\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ {\displaystyle <trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans><trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>}\hfill \end{array}\right.$$

(3)

也。相关结果见[16]. 我们观察到，对应于方程式(1)，由给出

$$\left\{\begin{array}{c}{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="U">单位</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\hfill \\ {\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="t">t吨</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\hfill \end{array}\right.$$

(4)

对于每个$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$.

对于每个x个，此系统与方程式具有相同的属性(1)在引理1-3中给出，为了方便起见，我们进行了说明。

引理 4

设置$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$.方程（0,0）的线性化(4)具有主特征值${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$此外，

（i） 

如果${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="0">0</trans>}$，则方程具有非负初始条件的所有解(4)收敛到（0,0），当$\mathrm{<trans\; data-src="t">t吨</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$.

（ii） 

如果${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后是方程式(4)是持久的，并且至少有一个正平衡。

（iii） 

如果$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$和${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$，则公式(4)是合作的，并且承认一个唯一的正平衡点，它是所有非负非平凡解的全局吸引子。

请注意，当${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$方程（0,0）附近线性化的特征值(4)是的根$<trans\; data-src="det">det（探测）</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="I">我</trans>}<trans\; data-src=")">)</trans>$，带有

$$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left[\begin{array}{cc}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>& \mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\\ \mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>& <trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\end{array}\right]$$

(5)

通过简单的计算，我们得到最大特征值由下式给出

$$\mathsf{<trans\; data-src="\Lambda ">\Lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}\left[<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\sqrt{{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="4">4</trans>}\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}\right]<trans\; data-src=",">,</trans>$$

（6）

这是积极的，前提是$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$.我们的第一个结果是定理1.4的直接应用[23]（中的定理A1附录A)，指出这确实是具有正主特征值的必要和充分条件，当${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$和${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$都很小。

提议 1

主特征值${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$方程式的(2)满足

$${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\to ">\to </trans>\underset{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}}{<trans\; data-src="max">最大值</trans>}\mathsf{<trans\; data-src="\Lambda ">\Lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="as">作为</trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

(7)

因此，存在一个$\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样，如果

$$\underset{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}}{<trans\; data-src="min">最小值</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

(8)

方程的主特征值(2)对所有人都是积极的$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}$，而如果

$$\underset{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}}{<trans\; data-src="min">最小值</trans>}<trans\; data-src="(">(</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

(9)

主特征值为负。

因此，如果方程式(9)成立，系统的唯一非负平衡，用方程表示(三)是$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$这种平衡在全球范围内具有吸引力，无论何时${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$都很小；if公式(8)保持，系统用等式表示(三)是持久的，并且对小${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$.如果公式(8)持有和$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后通过引理3，系统用等式表示(三)具有独特的全局吸引正平衡，我们用$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src=")">)</trans>$具有$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>$.

在本节的其余部分中，我们将假设$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$在里面$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$，在这种情况下，系统表示为(1)是合作的。

下一个结果确定了$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src=")">)</trans>$到唯一的非负稳态$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$满足

$$\begin{array}{c}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="is">是</trans>}\mathrm{<trans\; data-src="positive">积极的</trans>}\mathrm{<trans\; data-src="where">哪里</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\hfill \\ <trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="where">哪里</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(10)

定理 1

假设方程式(8)持有。然后$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src=")">)</trans>$作为$\mathrm{<trans\; data-src="d">d日</trans>}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$局部一致Ω.

为了证明这个定理，我们遵循的定理1.5的证明[23]特别是他们在[23]及其假设，这些假设列在[23]如（A1）–（A4）所示，并在附录A作为（L1）-（L4），以避免与此处的方程式标签混淆。我们应该指出[23]在我们的案例中不成立，因此我们无法直接应用该结果。不同的是，我们允许在下列情况下，动力学系统用方程式表示(4)对于某些值具有正平衡$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}$但对于其他情况则不然，而条件（A2）要求所有的动力学系统都达到正平衡x个。因此，我们需要构造中参数的版本[23]那是本地的x个条件（A3）in[23]仅用于证明与等式对应的系统的非平凡次解的存在性(三)，独立于${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$在下一个引理中，我们证明了在我们的例子中需要的类似局部亚解的存在性。

引理 5

假设$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src="\in ">\in </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$，其中

$${\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="/">/</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src=".">.</trans>$$

(11)

然后就有了${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$,${\mathrm{<trans\; data-src="\rho ">\rho </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$、和函数${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$在里面$\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\subset ">\subset </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$，它是等式的次解(三)为所有人$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$.

证明。 

让$\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>$，的正特征向量$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=")">)</trans>$具有${\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$，与其主特征值相关$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$.设置$\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$而且很小。我们可以选择$\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样的话$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src="\subset ">\subset </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$和

$$\begin{array}{c}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans><trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\hfill \\ <trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}\mathrm{<trans\; data-src="and">和</trans>}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="|">|</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}\mathrm{<trans\; data-src="for">对于</trans>}\mathrm{<trans\; data-src="all">全部的</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=",">,</trans>\hfill \end{array}$$

（12）

哪里$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=")">)</trans>$,$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=")">)</trans>$,$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=")">)</trans>$、和$\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=")">)</trans>$.设置$\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$作为与$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$，的主特征值

$$\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

(13)

具有${<trans\; data-src="max">最大值</trans>}_{\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$。我们声称我们可以选择$\mathrm{<trans\; data-src="\delta ">\delta </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样的话$\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\mathrm{<trans\; data-src="p">对</trans>}$是方程式的次解(三)为所有人${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="<"><</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$为了简单起见，并保持符号与[23]，我们定义$\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$，带有

$${\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="s">秒</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="b">b条</trans>}{\mathrm{<trans\; data-src="u">u个</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="and">和</trans>}{\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="s">秒</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="f">（f）</trans>}{\mathrm{<trans\; data-src="v">v（v）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

(14)

其中我们省略了变量x个在里面$\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="f">（f）</trans>}$缩短表达式。请注意

$$\begin{array}{c}{\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="b">b条</trans>}\mathrm{<trans\; data-src="\delta ">\delta </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans>\hfill \\ {\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="f">（f）</trans>}\mathrm{<trans\; data-src="\delta ">\delta </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \end{array}$$

并使用方程式(12)，如果我们选择$\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$和一个小的$\mathrm{<trans\; data-src="\delta ">\delta </trans>}$，我们有

$$\begin{array}{c}{\displaystyle {\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="b">b条</trans>}\mathrm{<trans\; data-src="\delta ">\delta </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\frac{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}}\hfill \\ {\displaystyle {\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="\epsilon ">\epsilon </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="f">（f）</trans>}\mathrm{<trans\; data-src="\delta ">\delta </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\frac{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=".">.</trans>}\hfill \end{array}$$

(15)

因此，在方程式中替换这些不等式(三)，我们获得

$$\begin{array}{c}{\displaystyle {\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="\delta ">\delta </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\left(<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="+">+</trans>\frac{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\right)}\hfill \\ {\displaystyle {\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="\delta ">\delta </trans>}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="\eta ">\eta </trans>}\left(<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="+">+</trans>\frac{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="p">对</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\right)<trans\; data-src=";">;</trans>}\hfill \end{array}$$

因此，如果我们设置${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\frac{\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}}}{\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}$.我们获得了预期的结果。□

使用这个引理，我们可以遵循中命题5.2的证明[23]. 为了便于说明，我们将使用相同的符号。设置运算符$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="diag">诊断</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>$,$\mathcal{<trans\; data-src="L">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="diag">诊断</trans>}<trans\; data-src="(">(</trans>\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src=",">,</trans>\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}<trans\; data-src=")">)</trans>$为了证明定理1，我们将说明所需的引理，并讨论它们与[23]从而证明命题5.2。

假设方程式(8)保持，设置${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\eta ">\eta </trans>}\mathrm{<trans\; data-src="\delta ">\delta </trans>}\mathrm{<trans\; data-src="p">对</trans>}$如引理5所示，以及${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="M">M（M）</trans>}$哪里$\mathrm{<trans\; data-src="M">M（M）</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$假设（A4）中给出${\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="c">c（c）</trans>}\mathrm{<trans\; data-src="u">u个</trans>}$和${\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="c">c（c）</trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}$为所有人$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="M">M（M）</trans>}$和$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$，带有$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$固定的。设置$\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样的话$\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="+">+</trans>{<trans\; data-src="\partial ">\partial </trans>}_{\mathrm{<trans\; data-src="u">u个</trans>}}{\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$和$\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="+">+</trans>{<trans\; data-src="\partial ">\partial </trans>}_{\mathrm{<trans\; data-src="v">v（v）</trans>}}{\mathrm{<trans\; data-src="F">如果</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$为所有人$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="M">M（M）</trans>}$，我们定义$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="=">=</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$作为唯一的解决方案

$$\left\{\begin{array}{c}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="D">D类</trans>}\mathcal{<trans\; data-src="L">我</trans>}\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="K">K</trans>}\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="K">K</trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ {\displaystyle <trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>}\hfill \end{array}\right.$$

对于$\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}$.

引理 6

假设方程式(8)持有。对于每个k，我们有${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}$和作为$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$,${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$一致收敛于方程的唯一正解w(三)，满足${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$在里面Ω为所有人 $\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$.

证明。 

我们将证明这一点${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$通过归纳法。假设这是真的k个.请注意${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}$通过构造。在场景中，$\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\subset ">\subset </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$如引理5所示${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}$满足

$$<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="D">D类</trans>}\mathcal{<trans\; data-src="L">我</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="-">-</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

在里面$\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans>$.通过归纳假说${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$，从哪里$\mathrm{<trans\; data-src="K">K</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$; 因此，通过对每个分量应用强最大值原理，我们得到了${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$在里面$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}$因此，我们有

$$\left\{\begin{array}{c}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="D">D类</trans>}\mathcal{<trans\; data-src="L">我</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \\ {\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}<trans\; data-src="\partial ">\partial </trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\hfill \end{array}\right.$$

这样我们就有了${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$在里面$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans>}$证明的其余部分是一个标准的单调迭代参数，正如[23]. 我们观察到

$$\left\{\begin{array}{c}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="D">D类</trans>}\mathcal{<trans\; data-src="L">我</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="K">K</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="K">K</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ <trans\; data-src="\nabla ">\nabla </trans><trans\; data-src="[">[</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \end{array}\right.$$

因此，根据强最大值原理，我们有${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}$.

类似地，如果${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}$，然后

$$\left\{\begin{array}{c}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="D">D类</trans>}\mathcal{<trans\; data-src="L">我</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="K">K</trans>}<trans\; data-src="(">(</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\hfill \\ \mathrm{<trans\; data-src="K">K</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="K">K</trans>}{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ <trans\; data-src="\nabla ">\nabla </trans><trans\; data-src="[">[</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="-">-</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=".">.</trans>\hfill \end{array}\right.$$

通过归纳，序列$<trans\; data-src="\{">\{</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\}">\}</trans>$是递减的，其范围如下$\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\}">\}</trans>$因此，根据标准椭圆理论，它收敛于方程的非负非平凡解(三)作为$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$.由于通过引理3，方程的非平凡非负解(三)是唯一的，它与作为序列极限的构造一致$<trans\; data-src="\{">\{</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\}">\}</trans>$. □

定义${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}$和${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$.根据中的引理5.6和5.7的证明[23]，我们可以证明以下结果。

引理 7

假设方程式(8)持有。对于每个k，我们有

$${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src=".">.</trans>$$

然后，${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$局部一致收敛到${\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}$作为$\mathrm{<trans\; data-src="k">k个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$，带有

$${\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="in">在里面</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="and">和</trans>}{\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="\backslash ">\backslash </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=",">,</trans>$$

哪里${\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$由方程式给出(11).

证明。 

通过方程式观察(15)，函数${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}$是动力学系统的一种亚溶液。在中重复引理5.6的证明[23]，或者根据导致上述引理6证明单调性的参数，我们得到了序列$<trans\; data-src="\{">\{</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\}">\}</trans>$单调递减且有界于${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}$因此，${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}$逐点，满足$\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$即动力学系统的非负平衡。因此，如果$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$我们有这个$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后${\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$另一方面，${\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>{\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$在里面$\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="˜">˜</trans>}{\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src=")">)</trans><trans\; data-src="\subset ">\subset </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$对于$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$.自x个是任意的，并且序列不依赖于${\underset{<trans\; data-src="̲">̲</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="0">0</trans>}}$，我们得到${\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="V">V（V）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$，唯一的正动力学平衡，无论何时$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$特别是，${\mathrm{<trans\; data-src="W">W公司</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}$是连续的。使用的定理5.8[23]，我们得到了在任意紧集上的收敛是一致的$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$. □

引理 8

对于每个k，作为${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$，我们有${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$收敛到${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="w">周</trans>}}}^{\mathrm{<trans\; data-src="k">k个</trans>}}$在中一致$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}$.

这个结果的证明与中引理5.5的证明相同[23].

证明 属于 定理 7

使用对角线参数和引理（8），唯一正解周方程式的(三)收敛到${\mathrm{<trans\; data-src="W">W公司</trans>}}_{<trans\; data-src="\infty ">\infty </trans>}$作为${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$. □

4.案例${\mathit{<trans\; data-src="d">d日</trans>}}_{\mathbf{<trans\; data-src="1">1</trans>}}$和${\mathit{<trans\; data-src="d">d日</trans>}}_{\mathbf{<trans\; data-src="2">2</trans>}}$大型

我们将首先给出一个结果的证明，这个结果被称为“民间定理”。它的表述比具体应用所需的更具普遍性。对于$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="N">N个</trans>}$，让${\mathrm{<trans\; data-src="L">我</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}$表示运算符

$${\mathrm{<trans\; data-src="L">我</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans><trans\; data-src="\nabla ">\nabla </trans><trans\; data-src="·">·</trans>{\mathrm{<trans\; data-src="\mu ">\mu </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="[">[</trans><trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\nabla ">\nabla </trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>\mathrm{<trans\; data-src="for">对于</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$$

(16)

无流量边界条件

$$<trans\; data-src="[">[</trans><trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="\nabla ">\nabla </trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="]">]</trans><trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="for">对于</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=".">.</trans>$$

(17)

假设${\mathrm{<trans\; data-src="\mu ">\mu </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\ge ">\ge </trans>{\mathrm{<trans\; data-src="\mu ">\mu </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$在$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}$为所有人我.让$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$做一个$\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\times ">\times </trans>\mathrm{<trans\; data-src="N">N个</trans>}$不可约矩阵${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="0">0</trans>}$如果$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="\ne ">\ne </trans>\mathrm{<trans\; data-src="j">j个</trans>}$.考虑特征值问题

$${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="L">我</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="+">+</trans>\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\cdots ">\cdots </trans>\mathrm{<trans\; data-src="N">N个</trans>}$$

(18)

哪里${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$为所有人我和${\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$满足方程中表示的边界条件(17)对于每个我注意，如果我们允许${\mathsf{<trans\; data-src="\Phi ">\Phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$，然后${\mathsf{<trans\; data-src="\Phi ">\Phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$满足Neumann边界条件，使得方程中表示的系统(18)根据变量重写${\mathsf{<trans\; data-src="\Phi ">\Phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$仍然合作。由于经典边界条件的存在，从[15,23]可以应用于系统${\mathsf{<trans\; data-src="\Phi ">\Phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$值，从而使系统和方程式(18)在域上的适当条件下将具有主特征值$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$和系数。这个想法已经被用于没有年龄结构的单个种群或这种种群的竞争对的模型中，例如[2,24].

此外，我们还有${\mathrm{<trans\; data-src="L">我</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$这样${\mathrm{<trans\; data-src="L">我</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}$为零，本征函数是$\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$.让$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}$是由定义的矩阵

$${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src=":">:</trans><trans\; data-src="=">=</trans>\frac{{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}}{{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=".">.</trans>$$

（19）

表示方程的主特征值(18)作为${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="\to ">\to </trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src=")">)</trans>$哪里$\stackrel{<trans\; data-src="\to ">\to </trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src=")">)</trans>$.表示的主特征值$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}$作为$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Lambda ">\Lambda </trans>}}$.

引理 9

假设，对于某些人来说$\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}<trans\; data-src="\in ">\in </trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=")">)</trans>$，方程系数(18)满足$\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=")">)</trans>$,$\mathrm{<trans\; data-src="\mu ">\mu </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=")">)</trans>$、和${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=")">)</trans>$对于$\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\cdots ">\cdots </trans>\mathrm{<trans\; data-src="N">N个</trans>}$，以及$<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$是一流的${\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\gamma ">\gamma </trans>}}$进一步假设$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}$是不可约的。如果$\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$，然后${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="\to ">\to </trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Lambda ">\Lambda </trans>}}$.

证明。 

选择任意序列${\stackrel{<trans\; data-src="\to ">\to </trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="N">N个</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$这样的话$\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$。选择任何子序列，然后将其重新编号为${\stackrel{<trans\; data-src="\to ">\to </trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}}_{\mathrm{<trans\; data-src="n">n个</trans>}}$.让${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}$是方程的主要特征值(18)对应于${\stackrel{<trans\; data-src="\to ">\to </trans>}{\mathrm{<trans\; data-src="d">d日</trans>}}}_{\mathrm{<trans\; data-src="n">n个</trans>}}$然后让${\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$成为我特征向量的第个分量，其中特征向量通过$\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=".">.</trans>$集成我方程的第个方程(18)超过$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$并进行总结我产量

$${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>{\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\underset{\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}$$

哪里${\mathrm{<trans\; data-src="A">A类</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$是一个常量，仅取决于A类。由此可见${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}$从上方均匀限定。同样，${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}$从下方均匀边界。因此${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}$它本身有一个收敛的子序列。然后将我方程的第个方程(18)由${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}$那个${\mathrm{<trans\; data-src="L">我</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}$一致有界，并且${\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans>\mathrm{<trans\; data-src="0">0</trans>}$作为$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$.通过椭圆正则性，序列${\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}$一致有界于${\mathrm{<trans\; data-src="W">W公司</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="p">对</trans>}}<trans\; data-src="(">(</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=")">)</trans>$对于任何$\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="<"><</trans><trans\; data-src="\infty ">\infty </trans>$然后通过Sobolev嵌入，它有一个收敛于${\mathrm{<trans\; data-src="C">C类</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=")">)</trans>$在中弱收敛${\mathrm{<trans\; data-src="W">W公司</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="p">对</trans>}}<trans\; data-src="(">(</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=")">)</trans>$。任何情况下都是如此我.如果需要，再取一个子序列并重新编号，我们得到一个序列，其中${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="*">*</trans>}$对一些人来说${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="*">*</trans>}$和${\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="*">*</trans>}$为所有人我，带有${\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="*">*</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$。那么我们必须${\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="*">*</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=")">)</trans>$对于一些非负常数${\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$，和$\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="*">*</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\cdots ">\cdots </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$.积分方程(18)超过$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$并使用无流量边界条件给出

$$\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\left[{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}^{<trans\; data-src="*">*</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\right]<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="*">*</trans>}{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}^{<trans\; data-src="*">*</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\cdots ">\cdots </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=",">,</trans>$$

（20）

这样的话

$$\underset{\mathrm{<trans\; data-src="j">j个</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}}{\overset{\mathrm{<trans\; data-src="N">N个</trans>}}{<trans\; data-src="\sum ">\sum </trans>}}\left[\frac{{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}{\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}}{{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}}\right]{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}^{<trans\; data-src="*">*</trans>}{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\cdots ">\cdots </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src=".">.</trans>$$

(21)

由此可见$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans><trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="N">N个</trans>}}<trans\; data-src=")">)</trans>$必须是的非平凡非负特征向量$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}$按照$\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="a">一</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\{">\{</trans>{\mathrm{<trans\; data-src="c">c（c）</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=":">:</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="i">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\cdots ">\cdots </trans>\mathrm{<trans\; data-src="N">N个</trans>}<trans\; data-src="\}">\}</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$。这些最后的条件唯一地确定了原始子序列的子序列的极限$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="\to ">\to </trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\}">\}</trans>$.由于原始序列的每个子序列$<trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>{\stackrel{<trans\; data-src="\to ">\to </trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\}">\}</trans>$有一个子序列收敛到由方程式确定的值(21)，对于原始序列也必须如此。由于原始的值序列$<trans\; data-src="\{">\{</trans>\stackrel{<trans\; data-src="\to ">\to </trans>}{{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src="\}">\}</trans>$可以是任何接近无穷大的递增序列$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$，引理的结论如下。□

在方程式中表示的特定系统中(1)我们认为，${\mathrm{<trans\; data-src="L">我</trans>}}^{\mathrm{<trans\; data-src="i">我</trans>}}<trans\; data-src="=">=</trans>\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}$，使得${\mathrm{<trans\; data-src="\alpha ">\alpha </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$和${\mathrm{<trans\; data-src="\mu ">\mu </trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}}$是常量。在这种情况下，我们有${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}<trans\; data-src="=">=</trans>{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="A">A类</trans>}}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}$，其中${\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}$是的平均值${\mathrm{<trans\; data-src="a">一</trans>}}_{\mathrm{<trans\; data-src="i">我</trans>}\mathrm{<trans\; data-src="j">j个</trans>}}$结束$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$表示方程式中系数的平均值(1)由$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}$等类似于方程式中的计算(5)，方程式(6)，然后通过相关讨论得出以下结论：

推论 1

假设引理9的假设满足。存在一个$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$这样，如果

$$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="+">+</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

主特征值${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$方程式的(2)对所有人都是积极的${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="D">D类</trans>}$，而如果

$$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="+">+</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>$$

主特征值是非正的。

备注 2

在ODE系统中，对应于方程式(1)系数平均值为Ω，一个人可以计算 ${\mathrm{<trans\; data-src="R">R（右）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$ 作为 $\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="/">/</trans><trans\; data-src="[">[</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="+">+</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans>$ 通过以下方法[25]. 推论1中的第一个不等式等价于 ${\mathrm{<trans\; data-src="R">R（右）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="1">1</trans>}$，而第二个相当于 ${\mathrm{<trans\; data-src="R">R（右）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="1">1</trans>}$.通过书面形式 ${\mathrm{<trans\; data-src="R">R（右）</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans><trans\; data-src="[">[</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src="/">/</trans><trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="+">+</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="]">]</trans><trans\; data-src="[">[</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="/">/</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="]">]</trans>$，我们可以将持续性条件解释为成年人和青少年的增长项与损失项之比的乘积应大于1。

5.一般扩散率

案例1：所有扩散速率的持续性或消光

提议 2

如果

$${<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\sqrt{\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="s">秒</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$$

(22)

然后${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于所有正扩散率。

如果

$$\mathrm{<trans\; data-src="m">米</trans>}\mathrm{<trans\; data-src="i">我</trans>}{\mathrm{<trans\; data-src="n">n个</trans>}}_{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}}<trans\; data-src="[">[</trans>\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="]">]</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$$

(23)

然后${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$对于所有正扩散速率。

证明。 

如果我们把方程中的第一个方程分开(2)由$\mathrm{<trans\; data-src="\phi ">\phi </trans>}$和集成$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$，使用格林公式积分该项$\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="/">/</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}$，我们得到了不等式

$$<trans\; data-src="|">|</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\ge ">\ge </trans>{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\mathrm{<trans\; data-src="r">第页</trans>}\left(\frac{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}\right)\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=".">.</trans>$$

(24)

类似地，如果我们将第二个方程除以$\mathrm{<trans\; data-src="\psi ">\psi </trans>}$并将我们得到的

$$<trans\; data-src="|">|</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="|">|</trans>{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\ge ">\ge </trans>{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\mathrm{<trans\; data-src="s">秒</trans>}\left(\frac{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}\right)\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=".">.</trans>$$

(25)

如果我们添加方程式(24)和(25)除以2，我们得到

$${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\ge ">\ge </trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}<trans\; data-src="|">|</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="|">|</trans>}\left({<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\left[\mathrm{<trans\; data-src="r">第页</trans>}\left(\frac{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}\right)<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}\left(\frac{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}\right)\right]\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\right)<trans\; data-src=".">.</trans>$$

(26)

根据柯西不等式，$\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\ge ">\ge </trans>\mathrm{<trans\; data-src="2">2</trans>}\sqrt{\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="s">秒</trans>}}$为所有人$\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$，因此从方程式(26)我们获得

$${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="\ge ">\ge </trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="|">|</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="|">|</trans>}\left[{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\sqrt{\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="s">秒</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}\right]$$

(27)

因此，${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$if公式(22)成立，所以命题2的第一部分成立。朝另一个方向，如果我们乘以方程的第一个方程(2)由$\mathrm{<trans\; data-src="\phi ">\phi </trans>}$和集成，在$\mathrm{<trans\; data-src="\phi ">\phi </trans>}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\phi ">\phi </trans>}$项，并类似地将第二个方程乘以$\mathrm{<trans\; data-src="\psi ">\psi </trans>}$并进行积分，然后将结果相加，我们得到

$${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\le ">\le </trans>{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="[">[</trans><trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="]">]</trans><trans\; data-src=".">.</trans>$$

（28）

方程右侧的被积函数(28)是中的二次型$\mathrm{<trans\; data-src="\phi ">\phi </trans>}$和$\mathrm{<trans\; data-src="\psi ">\psi </trans>}$，如果

$$\mathrm{<trans\; data-src="4">4</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=">">></trans>{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src=")">)</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>$$

(29)

所以${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$if公式(23)持有，这证明了命题2的第二部分。□

备注：注意方程中的第一个积分(22)是Bhattacharyya系数公式中出现的内容[26,27]，用于比较概率分布相互匹配的程度。具体来说，如果两个概率分布P（P）和问具有概率密度函数$\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$对于$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="U">单位</trans>}<trans\; data-src="\subset ">\subset </trans>{\mathbb{<trans\; data-src="R">R（右）</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}$，巴塔查里亚系数为

$$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="Q">问</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathrm{<trans\; data-src="U">单位</trans>}}\sqrt{\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="q">q个</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=".">.</trans>$$

对于任何P（P）和问,$\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="Q">问</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="1">1</trans>}$.如果$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="Q">问</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$，然后P（P）和问都是一样的，也就是说，$\mathrm{<trans\; data-src="p">对</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="q">q个</trans>}$a.e.如果$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="P">P（P）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="Q">问</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后是对和q个是不相交的。如果我们写$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$这样的话${<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\mathrm{<trans\; data-src="\rho ">\rho </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$，然后我们可以计算${\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src="|">|</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="|">|</trans>$和${\mathrm{<trans\; data-src="s">秒</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="=">=</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="|">|</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="|">|</trans>$.我们可以治疗$\mathrm{<trans\; data-src="\rho ">\rho </trans>}$和$\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}$好像它们是分布的概率密度函数R（右）和S公司。然后我们有

$${<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\sqrt{\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="s">秒</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="=">=</trans><trans\; data-src="|">|</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="|">|</trans>\sqrt{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}}\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="R">R（右）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="S">S公司</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

(30)

最大值为$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="R">R（右）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="S">S公司</trans>}<trans\; data-src=")">)</trans>$为1，对应于以下情况第页和秒是彼此的倍数，最小值为0，对应于第页和秒是不相交的。因此$\mathrm{<trans\; data-src="\rho ">\rho </trans>}$和$\mathrm{<trans\; data-src="\sigma ">\sigma </trans>}$相互匹配对$\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}$在方程式中(27).

使用方程式(30)事实上$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="R">R（右）</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="S">S公司</trans>}<trans\; data-src=")">)</trans><trans\; data-src="\le ">\le </trans>\mathrm{<trans\; data-src="1">1</trans>}$在方程式中(22)显示了方程式(22)暗示$\mathrm{<trans\; data-src="2">2</trans>}\sqrt{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}}<trans\; data-src=">">></trans><trans\; data-src="[">[</trans><trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="+">+</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="]">]</trans>$.求两边的平方和使用柯西不等式意味着$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="+">+</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$如推论1的第一个例子。类似地，如果方程(22)保持，然后$\mathrm{<trans\; data-src="2">2</trans>}\sqrt{\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=">">></trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$对一些人来说$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$然后，它以与命题1第一种情况下的不等式相同的方式进行推导。If公式(23)成立，则公式(29)成立，然后根据柯西不等式，命题1的第二种情况成立。因此，方程式中表示的条件(22)和方程式(23)在命题2中，这意味着${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$或${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="<"><</trans>\mathrm{<trans\; data-src="0">0</trans>}$对于所有的扩散率，也意味着我们对于大扩散率或小扩散率所获得的一些相应条件。

在栖息地质量的空间分布情况下第页供成年人繁殖和秒因为青少年的存活和成熟是完全相关的，所以$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$对于某个常数${\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$，方程中表示的特征值问题(2)可以通过将方程中的第二个方程相乘而改写为加权对称特征值问题(2)由${\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$，它产生

$$\begin{array}{c}{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}\mathrm{<trans\; data-src="\phi ">\phi </trans>}\hfill \\ {\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(31)

方程的主特征值(31)具有以下变化特征${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$作为

$${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src="=">=</trans>\underset{\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="\in ">\in </trans>{\mathrm{<trans\; data-src="W">W公司</trans>}}^{\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="(">(</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=")">)</trans>}{<trans\; data-src="max">最大值</trans>}\frac{{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="(">(</trans><trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="\phi ">\phi </trans>}<trans\; data-src="|">|</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{<trans\; data-src="|">|</trans><trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="|">|</trans>}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="2">2</trans>}\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="\phi ">\phi </trans>}\mathrm{<trans\; data-src="\psi ">\psi </trans>}<trans\; data-src="-">-</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="e">e（电子）</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}}{{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}}<trans\; data-src=".">.</trans>$$

(32)

在这种情况下${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$两者都在减少${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}$和${\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}$，因此较慢的扩散是有利的。

案例2：大生育率的渐近行为

假设$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="n">n个</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$而且$\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$具有${\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\ne ">\ne </trans><trans\; data-src="\varnothing ">\varnothing </trans>$; 因此，存在一个成虫繁殖率和幼虫成熟率均为正的区域。因素n个衡量以下地区成年人的生育率${\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于任何固定的扩散率，结果表明，对于足够大的标度系数值n个，方程的主特征值(2)是正的，所以系统用等式表示(1)是持久的。我们将主要特征值的渐近行为描述为$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$.如果我们进一步假设$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$，然后用方程式表示系统(1)是合作的，所以对于n个足够大，使得方程的主特征值(2)为正，方程式(1)具有唯一的正平衡，我们将该平衡的行为描述为$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$在这种情况下也是如此。

让${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}$表示方程的主要特征值(2)带有$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="n">n个</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$观察一下，因为$\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$和${\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\ne ">\ne </trans><trans\; data-src="\varnothing ">\varnothing </trans>$，命题2意味着${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于n个足够大，事实上通过方程式(27),${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$作为$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$以下命题陈述了${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}$作为$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$.

提议 三。

如果$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="n">n个</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$和$\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$对于$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}$具有${\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="\ne ">\ne </trans><trans\; data-src="\varnothing ">\varnothing </trans>$，然后

$$\underset{\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>}{<trans\; data-src="lim">林</trans>}\frac{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}}{\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src="\to ">\to </trans>\underset{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}}{<trans\; data-src="max">最大值</trans>}<trans\; data-src="(">(</trans>\sqrt{{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

证明。 

我们首先要注意，如果${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}$，然后$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$是方程的主特征值和相应的特征函数(2)的$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="n">n个</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$; 因此，${\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="/">/</trans>\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}$,${\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}$、和${\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src="=">=</trans>\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}$是以下问题的主特征值和本征函数：

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}}{\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}<trans\; data-src="-">-</trans>\frac{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>}{\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}}\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}<trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}}& <trans\; data-src="=">=</trans>\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill {\displaystyle \frac{{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}}{\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}{\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}}\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}}& <trans\; data-src="=">=</trans>\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ \hfill <trans\; data-src="\nabla ">\nabla </trans>\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\phi ">\phi </trans>}}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans><trans\; data-src="\nabla ">\nabla </trans>\stackrel{<trans\; data-src="^">^</trans>}{\mathrm{<trans\; data-src="\psi ">\psi </trans>}}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}& <trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=".">.</trans>\hfill \end{array}\right.$$

(33)

考虑椭圆算子${\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="u">u个</trans>}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="u">u个</trans>}$和${\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="v">v（v）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="v">v（v）</trans>}$,$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="diag">诊断</trans>}\left(\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src=",">,</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}}\right)$和$\mathcal{<trans\; data-src="L">我</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="diag">诊断</trans>}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans>$，系统用等式表示(33)满足的定理1.4的假设[23]. 因此，$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$

$$\frac{{\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}^{\mathrm{<trans\; data-src="n">n个</trans>}}}{\sqrt{\mathrm{<trans\; data-src="n">n个</trans>}}}<trans\; data-src="\to ">\to </trans>\underset{\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}}{<trans\; data-src="max">最大值</trans>}\mathrm{<trans\; data-src="\lambda ">\lambda </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>$$

哪里

$$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left(\begin{array}{cc}\mathrm{<trans\; data-src="0">0</trans>}& {\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\\ \mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>& \mathrm{<trans\; data-src="0">0</trans>}\end{array}\right)<trans\; data-src=",">,</trans>$$

它有特征值$<trans\; data-src="\pm ">\pm </trans>\sqrt{{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}$结果如下。□

在以下情况下$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$这样方程(1)是合作的，引理3暗示方程(1)如果方程的主特征值为(2)是积极的。下一个结果说明了方程唯一正平衡点的渐近行为(1)的n个在那种情况下是很大的。

提议 4

假设命题3的假设得到满足，并且$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$.让$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans>$是方程的唯一正解(三). 然后${\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="\to ">\to </trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="U">单位</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src=")">)</trans>$在中一致$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}$哪里

$$\begin{array}{c}{\displaystyle {\mathrm{<trans\; data-src="U">单位</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}{\mathrm{<trans\; data-src="b">b条</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}\frac{\mathrm{<trans\; data-src="s">秒</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}{\mathrm{<trans\; data-src="f">（f）</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\frac{\mathrm{<trans\; data-src="s">秒</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}{\mathrm{<trans\; data-src="f">（f）</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}\frac{{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}{\mathrm{<trans\; data-src="b">b条</trans>}{<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}^{\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}}\mathrm{<trans\; data-src="w">周</trans>}\mathrm{<trans\; data-src="h">小时</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}}\hfill \\ {\mathrm{<trans\; data-src="U">单位</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="V">V（V）</trans>}}^{<trans\; data-src="\infty ">\infty </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="w">周</trans>}\mathrm{<trans\; data-src="h">小时</trans>}\mathrm{<trans\; data-src="e">e（电子）</trans>}\mathrm{<trans\; data-src="n">n个</trans>}\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>\hfill \end{array}$$

(34)

证明。 

为了证明这个结果，我们使用了不同的缩放比例。经过一些简单的计算，我们发现

$$<trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="w">周</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\left({\mathrm{<trans\; data-src="u">u个</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}{\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="v">v（v）</trans>}}_{\mathrm{<trans\; data-src="n">n个</trans>}}{\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}\right)<trans\; data-src=",">,</trans>$$

是标度系统的唯一正解

$$\left\{\begin{array}{c}{\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="]">]</trans><trans\; data-src="+">+</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="w">周</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ {\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src="[">[</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="]">]</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="in">在里面</trans>}\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=",">,</trans>\hfill \\ {\displaystyle <trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans><trans\; data-src="\nabla ">\nabla </trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="·">·</trans>\mathrm{<trans\; data-src="\nu ">\nu </trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}\mathrm{<trans\; data-src="on">在</trans>}<trans\; data-src="\partial ">\partial </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src=".">.</trans>}\hfill \end{array}\right.$$

(35)

我们设置操作员

$${\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="-">-</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="w">周</trans>}\mathrm{<trans\; data-src="and">和</trans>}{\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="=">=</trans>{\mathrm{<trans\; data-src="d">d日</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}\mathsf{<trans\; data-src="\Delta ">\Delta </trans>}\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src=",">,</trans>$$

$$\mathrm{<trans\; data-src="D">D类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="diag">诊断</trans>}\left({\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="2">2</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="n">n个</trans>}}^{<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="3">三</trans>}}}\right)\mathrm{<trans\; data-src="and">和</trans>}\mathcal{<trans\; data-src="L">我</trans>}<trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="1">1</trans>}}<trans\; data-src=",">,</trans>{\mathrm{<trans\; data-src="L">我</trans>}}_{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="and">和</trans>}$$

$$\mathrm{<trans\; data-src="F">如果</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans><trans\; data-src="(">(</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="z">z（z）</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="b">b条</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="w">周</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="w">周</trans>}<trans\; data-src="-">-</trans>\mathrm{<trans\; data-src="f">（f）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="z">z（z）</trans>}}^{\mathrm{<trans\; data-src="2">2</trans>}}<trans\; data-src=")">)</trans><trans\; data-src=".">.</trans>$$

为了证明这个命题，我们可以遵循与定理1的证明相同的步骤。实际上，线性化的主要特征值$<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=")">)</trans>$方程的相关动力学系统(35)是$\sqrt{\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>}$，当它为正时，动力学平衡由等式右侧给出(34). 证明到此结束。□

6.结论

从我们的分析中得出的最基本的结论是，在空间异质环境中具有阶段结构的种群的反应扩散模型不一定预测较慢的扩散有利于持久性。这与人口仅由空间位置构成的情况形成对比，在这种情况下，缩减原则的一个版本[6]适用；在具有不同扩散率的相同种群之间的竞争中，预测是“扩散速度较慢的种群获胜”[4,5]. 这一观察结果背后的机制是，在我们的结构化模型中，成年人有可能产生后代的区域可能与青少年能够存活并成熟为成年人的区域分开。我们发现暗示持久性的条件通常要求产品$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$成年人的生殖率和青少年的成熟率相对于其死亡率而言足够大。对于缓慢扩散，持久性的条件是$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="+">+</trans><trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=")">)</trans>$在某个时刻$\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="\in ">\in </trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$。对于快速扩散，它是$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src=">">></trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}<trans\; data-src="(">(</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src="+">+</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}<trans\; data-src=")">)</trans>$哪里$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src=",">,</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="e">e（电子）</trans>}}$、和$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="a">一</trans>}}$是这些量的空间平均值。如果第页和秒密切相关，在少数地方较大，但在大多数地方较小，因此最大值为$\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$很大，但平均值$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}$和$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}$较小时，可以满足缓慢扩散的持久性条件，而快速扩散的条件可能会失败。在这种类型的环境中，缓慢的扩散显然是有利的。此外，如果第页和秒在它们是彼此的倍数的意义上是完全相关的，决定低密度种群增长率的主特征值相对于扩散率是递减的，就像异质环境中的非结构化种群一样。另一方面，如果两者都是第页和秒在某些区域上较大，但在它们之外非常小，并且它们较大的区域是不相交的（即彼此分离），然后产品$\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="s">秒</trans>}$可能到处都很小，但平均值$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}$和$\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}$可能很大。在这种情况下，具有小扩散的持久性条件可能会失败，但具有快速扩散的条件可能会得到满足，从而有利于快速扩散。

我们发现，对于所有扩散速率，持久性的一个充分条件是

$${<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}\sqrt{\mathrm{<trans\; data-src="r">第页</trans>}\mathrm{<trans\; data-src="s">秒</trans>}}\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src="-">-</trans>\frac{\mathrm{<trans\; data-src="1">1</trans>}}{\mathrm{<trans\; data-src="2">2</trans>}}{<trans\; data-src="\int ">\int </trans>}_{\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="a">一</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="e">e（电子）</trans>}<trans\; data-src=")">)</trans>\mathrm{<trans\; data-src="d">d日</trans>}\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=".">.</trans>$$

第一个术语可以写成$\sqrt{\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}}<trans\; data-src="|">|</trans>\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}<trans\; data-src="|">|</trans>\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="/">/</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="/">/</trans>\stackrel{<trans\; data-src="¯">¯</trans>}{\mathrm{<trans\; data-src="s">秒</trans>}}<trans\; data-src=")">)</trans>$，其中$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}$表示Bhattacharya系数（参见[26,27])测量概率密度之间的匹配程度。对于彼此相等的分布，$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="1">1</trans>}$，但对于相互排斥的分布，正的区域不相交，$\mathrm{<trans\; data-src="B">B类</trans>}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$这一观察再次表明第页和秒在确定方程式中表示的模型预测时，相互匹配非常重要(1).

最后，我们发现，如果我们将成人生殖率定为$\mathrm{<trans\; data-src="r">第页</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="n">n个</trans>}{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>$并且在第页和秒这样的话$\mathrm{<trans\; data-src="s">秒</trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans>{\mathrm{<trans\; data-src="r">第页</trans>}}_{\mathrm{<trans\; data-src="0">0</trans>}}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="x">x个</trans>}<trans\; data-src=")">)</trans><trans\; data-src=">">></trans>\mathrm{<trans\; data-src="0">0</trans>}$关于的某些子集$\mathsf{<trans\; data-src="\Omega ">\Omega </trans>}$对于正测度，则对于任何固定的扩散速率，系统用方程表示(1)将持续存在，如果n个足够大。这意味着，只要成年人的繁殖率足够大，在成年人可以繁殖的区域和青少年可以成熟的区域之间有适度的重叠，任何扩散率的种群都可以持续存在。我们将渐近行为描述为$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$方程的主特征值(2). 在合作案例中$\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="\equiv ">\equiv </trans>\mathrm{<trans\; data-src="0">0</trans>}$，系统用等式表示(1)如果它持续存在，将有一个唯一的正平衡，在这种情况下，我们还将渐近行为描述为$\mathrm{<trans\; data-src="n">n个</trans>}<trans\; data-src="\to ">\to </trans><trans\; data-src="\infty ">\infty </trans>$达到平衡。

对于本文的一般主题，还有几个需要进一步研究的方向。采取[4]并考虑由方程等系统描述的两阶段结构种群之间的竞争(1)只在扩散速度上有所不同。这将有点困难，因为它将涉及四个方程组，但至少在合作的情况下$\mathrm{<trans\; data-src="c">c（c）</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="g">克</trans>}<trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="0">0</trans>}$单调动力系统的一般理论[28]以及[10]将适用。考虑具有明确年龄结构的模型也很有意思，如[11]并在[12,13]. 最后，考虑具有空间异质性的时间周期环境的情况是很有趣的，尽管具有挑战性。在某些情况下，仅时间变化就足以在这种环境中促进更快的扩散（参见[29])，但即使没有阶段结构，与时间相关的情况也是具有挑战性的，并且有许多悬而未决的问题。