﻿ 不完全CI下Wolbachia传播的离散竞争模型 Modeling Wolbachia Propagation underIncomplete Cytoplasmic Incompatibility by Discrete Competition Model

Vol. 09  No. 02 ( 2020 ), Article ID: 34189 , 13 pages
10.12677/AAM.2020.92018

Modeling Wolbachia Propagation under Incomplete Cytoplasmic Incompatibility by Discrete Competition Model

Yijie Li, Zhiming Guo*

School of Mathematics and Information Science, Guangzhou University, Guangzhou Guangdong

Received: Jan. 25th, 2020; accepted: Feb. 7th, 2020; published: Feb. 14th, 2020

ABSTRACT

Dengue fever is one of the most serious mosquito-borne infectious diseases. Using Wolbachia infection mosquitoes to control those diseases is an effective strategy. In this paper, a discrete competition model is established to study the dynamic of Wolbachia propagation under incomplete cytoplasmic incompatibility (CI). We systematically analyze the existing conditions of the equilibrium and global asymptotic behaviors of solutions to this model, then we give the conditions for successful diffusion and the influence of CI strength on the Wolbachia diffusion. Finally, we verify our findings by numerical simulations.

Keywords:Wolbachia, Incomplete CI, Competition, Discrete Model, Stability

1. 引言

2. 模型的建立及其分析

$\left\{\begin{array}{l}{x}_{t+1}={b}_{1}\frac{1}{1+{c}_{11}{x}_{t}+{c}_{12}{y}_{t}}{x}_{t},\\ {y}_{t+1}={b}_{2}\frac{1}{1+{c}_{21}{x}_{t}+{c}_{22}{y}_{t}}{y}_{t}.\end{array}$ (1)

$\left\{\begin{array}{l}{x}_{t+1}=\frac{{b}_{1}{x}_{t}}{1+\alpha \left({x}_{t}+{y}_{t}\right)}+\left(1-{d}_{1}\right){x}_{t},\\ {y}_{t+1}=\frac{{b}_{2}{y}_{t}}{1+\alpha \left({x}_{t}+{y}_{t}\right)}\left(1-q\frac{{x}_{t}}{{x}_{t}+{y}_{t}}\right)+\left(1-{d}_{2}\right){y}_{t}.\end{array}$ (2)

$\begin{array}{l}{x}_{t+1}\le \frac{{b}_{1}}{\alpha }+\left(1-{d}_{1}\right){x}_{t},\\ {y}_{t+1}\le \frac{{b}_{2}}{\alpha }+\left(1-{d}_{2}\right){y}_{t},\end{array}$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}{x}_{t}\le \frac{{b}_{1}}{\alpha {d}_{1}},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}{y}_{t}\le \frac{{b}_{2}}{\alpha {d}_{2}}.$

$\Gamma =\left\{\left(x,y\right)\in {R}^{2}|0\le x\le \frac{{b}_{1}}{\alpha {d}_{1}},0\le y\le \frac{{b}_{2}}{\alpha {d}_{2}}\right\}.$

$f\left(x,y\right)=\left\{\begin{array}{l}\frac{{b}_{2}y}{1+\alpha \left(x+y\right)}\left(1-q\frac{x}{x+y}\right)+\left(1-{d}_{2}\right)y,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(x,y\right)\ne \left(0,0\right),\\ 0\text{ },\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\left(x,y\right)=\left(0,0\right).\end{array}$

$\left\{\begin{array}{l}{x}_{t+1}=\frac{{b}_{1}{x}_{t}}{1+\alpha \left({x}_{t}+{y}_{t}\right)}+\left(1-{d}_{1}\right){x}_{t},\\ {y}_{t+1}=f\left({x}_{t},{y}_{t}\right).\end{array}$

$\left\{\begin{array}{l}x=\frac{{b}_{1}x}{1+\alpha \left(x+y\right)}+\left(1-{d}_{1}\right)x,\\ y=\frac{{b}_{2}y}{1+\alpha \left(x+y\right)}\left(1-q\frac{x}{x+y}\right)+\left(1-{d}_{2}\right)y.\end{array}$ (3)

${E}_{3}\left(\frac{1}{\alpha q}\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right),\frac{1}{\alpha }\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\right)$

a) 若 ${b}_{1}\le {d}_{1},{b}_{2}\le {d}_{2}$，则 ${E}_{0}$ 是全局渐近稳定的；否则， ${E}_{0}$ 是不稳定的。

b) 若 ${b}_{1}>{d}_{1}$，则当 $\frac{{b}_{1}}{{d}_{1}}>\frac{{b}_{2}}{{d}_{2}}\left(1-q\right)$ 时， ${E}_{1}$ 是局部渐近稳定的；当 $\frac{{b}_{1}}{{d}_{1}}<\frac{{b}_{2}}{{d}_{2}}\left(1-q\right)$ 时， ${E}_{1}$ 是一个鞍点。进一步，若 ${E}_{1}$ 是局部渐近稳定的且 ${b}_{2}\le {d}_{2}$，则 ${E}_{1}$ 是全局渐近稳定的。

c) 若 ${b}_{2}>{d}_{2}$，则当 $\frac{{b}_{2}}{{d}_{2}}>\frac{{b}_{1}}{{d}_{1}}$ 时， ${E}_{2}$ 是局部渐近稳定的；当 $\frac{{b}_{2}}{{d}_{2}}<\frac{{b}_{1}}{{d}_{1}}$ 时， ${E}_{2}$ 是一个鞍点。进一步，若 ${E}_{2}$ 是局部渐近稳定的且 ${b}_{1}\le {d}_{1}$，则 ${E}_{2}$ 是全局渐近稳定的。

$J=\left(\begin{array}{cc}\frac{{b}_{1}\left(1+\alpha y\right)}{{\left[1+\alpha \left(x+y\right)\right]}^{2}}+1-{d}_{1}& \frac{-\alpha {b}_{1}x}{{\left[1+\alpha \left(x+y\right)\right]}^{2}}\\ \frac{q{b}_{2}y\left[\alpha \left({x}^{2}-{y}^{2}\right)-y\right]-\alpha {b}_{2}y{\left(x+y\right)}^{2}}{{\left[1+\alpha \left(x+y\right)\right]}^{2}{\left(x+y\right)}^{2}}& \frac{q{b}_{2}x\left[\alpha \left({y}^{2}-{x}^{2}\right)-x\right]+{b}_{2}\left(1+\alpha x\right){\left(x+y\right)}^{2}}{{\left[1+\alpha \left(x+y\right)\right]}^{2}{\left(x+y\right)}^{2}}+1-{d}_{2}\end{array}\right).$ (4)

${J}_{1}={\left(\begin{array}{cc}\frac{{d}_{1}^{2}}{{b}_{1}}+1-{d}_{1}& \frac{{d}_{1}^{2}}{{b}_{1}}-{d}_{1}\\ 0& \frac{{b}_{2}{d}_{1}\left(1-q\right)}{{b}_{1}}+1-{d}_{2}\end{array}\right)}_{,}$ (5)

${J}_{2}={\left(\begin{array}{cc}\frac{{b}_{1}{d}_{2}}{{b}_{2}}+1-{d}_{1}& 0\\ \frac{{d}_{2}^{2}}{{b}_{2}}-{d}_{2}\left(1+q\right)& \frac{{d}_{2}^{2}}{{b}_{2}}+1-{d}_{2}\end{array}\right)}_{,}$ (6)

a) 若 ${b}_{1}\le {d}_{1},{b}_{2}\le {d}_{2}$，则系统只有一个平凡平衡点 ${E}_{0}$。当 ${b}_{1}<{d}_{1},{b}_{2}<{d}_{2}$ 时，由模型(2)可得

$\begin{array}{l}0\le {x}_{t+1}\le \left({b}_{1}-{d}_{1}+1\right){x}_{t},\\ 0\le {y}_{t+1}\le \left({b}_{2}-{d}_{2}+1\right){y}_{t},\end{array}$

${x}_{t+1}=\left[\frac{{b}_{1}}{1+\alpha \left({x}_{t}+{y}_{t}\right)}+1-{d}_{1}\right]{x}_{t},$

${y}_{t+1}=\left[\frac{{b}_{2}}{1+\alpha \left({x}_{t}+{y}_{t}\right)}\left(1-q\frac{{x}_{t}}{{x}_{t}+{y}_{t}}\right)+1-{d}_{2}\right]{y}_{t},$

${x}_{t+1}=\left(\frac{{b}_{1}}{1+\alpha {x}_{t}}+1-{d}_{1}\right){x}_{t},$

b) ${b}_{1}>{d}_{1}$，则当 $\frac{{b}_{1}}{{d}_{1}}>\frac{{b}_{2}}{{d}_{2}}\left(1-q\right)$ 时，由 ${J}_{1}$ 的所有特征值的绝对值都小于1可知 ${E}_{1}$ 是局部渐近稳定的；而当 $\frac{{b}_{1}}{{d}_{1}}<\frac{{b}_{2}}{{d}_{2}}\left(1-q\right)$ 时，由 ${J}_{1}$ 的一个特征值的绝对值小于1，另一个特征值大于1，可知 ${E}_{1}$ 是一个鞍点。进一步，若 ${E}_{1}$ 是局部渐近稳定的且 ${b}_{2}\le {d}_{2}$，模型只存在两个平衡点 ${E}_{0}$${E}_{1}$，由a)知此时 ${E}_{0}$ 不稳定，则根据文献 [12] 中的定理，所有的有界轨线将最终趋于一个平衡点，因此所有轨线只能趋于 ${E}_{1}$，可得 ${E}_{1}$ 是全局渐近稳定的。

c) 这里的证明与b)类似，所以省略。

$y=-x+\frac{1}{\alpha }\left(\frac{{b}_{1}}{{d}_{1}}-1\right),$

${L}_{1}$ 表示，这是一条斜率为−1，截距为 $\frac{1}{\alpha }\left(\frac{{b}_{1}}{{d}_{1}}-1\right)$ 的直线段。由模型(2)的第二个式子可得水平等倾线

$\alpha {x}^{2}+2\alpha xy+\alpha {y}^{2}+\left[1+\frac{{b}_{2}}{{d}_{2}}\left(q-1\right)\right]x+\left(1-\frac{{b}_{2}}{{d}_{2}}\right)y=0,$

${L}_{2}$ 表示，这是经过三个不动点 $\left(0,0\right),\left(0,\frac{1}{\alpha }\left(\frac{{b}_{2}}{{d}_{2}}-1\right)\right),\left(\frac{1}{\alpha }\left[\frac{{b}_{2}}{{d}_{2}}\left(1-q\right)-1\right],0\right)$ 的抛物线，见图1

Figure 1. The three possible cases of L1and L2

Case A的条件为

${b}_{1}>{d}_{1},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{b}_{2}>{d}_{2},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\frac{{b}_{1}}{{d}_{1}}>\frac{{b}_{2}}{{d}_{2}}.$

Case B的条件为

${b}_{1}>{d}_{1},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{b}_{2}>{d}_{2},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\frac{{b}_{2}}{{d}_{2}}>\frac{{b}_{1}}{{d}_{1}}>\frac{{b}_{2}}{{d}_{2}}\left(1-q\right).$

Case C的条件为

${b}_{1}>{d}_{1},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{b}_{2}>{d}_{2},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\frac{{b}_{1}}{{d}_{1}}<\frac{{b}_{2}}{{d}_{2}}\left(1-q\right).$

a) 若 $\frac{{b}_{1}}{{d}_{1}}>\frac{{b}_{2}}{{d}_{2}}$，则 ${E}_{1}$ 是全局渐近稳定的， ${E}_{2}$ 是一个鞍点。

b) 若 $\frac{{b}_{2}}{{d}_{2}}>\frac{{b}_{1}}{{d}_{1}}>\frac{{b}_{2}}{{d}_{2}}\left(1-q\right)$，则 ${E}_{1}$${E}_{2}$ 都是局部渐近稳定的， ${E}_{3}$ 是一个鞍点。

c) 若 $\frac{{b}_{1}}{{d}_{1}}<\frac{{b}_{2}}{{d}_{2}}\left(1-q\right)$，则 ${E}_{2}$ 是全局渐近稳定的， ${E}_{1}$ 是一个鞍点。

b) 在此条件下(即Case B)，由 ${J}_{1}$${J}_{2}$ 可得 ${E}_{1}$${E}_{2}$ 的所有特征值的绝对值都小于1，所以 ${E}_{1}$${E}_{2}$ 都是局部渐近稳定的。下面讨论 ${E}_{3}$ 的稳定性，为了表达方便，记

${E}_{3}=\left({x}^{*},{y}^{*}\right)=\left(\frac{1}{\alpha q}\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right),\frac{1}{\alpha }\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\right).$

${J}_{3}={\left(\begin{array}{cc}\frac{{d}_{1}^{2}\left(1+\alpha {y}^{*}\right)}{{b}_{1}}+1-{d}_{1}& -\frac{\alpha {d}_{1}^{2}{x}^{*}}{{b}_{1}}\\ \frac{q{b}_{2}{y}^{*}\left[\left(\frac{{b}_{1}}{{d}_{1}}-1\right){x}^{*}-\frac{{b}_{1}}{{d}_{1}}{y}^{*}\right]-\alpha {b}_{2}{y}^{*}{\left({x}^{*}+{y}^{*}\right)}^{2}}{{\left[\frac{{b}_{1}}{{d}_{1}}\left({x}^{*}+{y}^{*}\right)\right]}^{2}}& \frac{q{b}_{2}{x}^{*}\left[\left(\frac{{b}_{1}}{{d}_{1}}-1\right){y}^{*}-\frac{{b}_{1}}{{d}_{1}}{x}^{*}\right]+{b}_{2}\left(1+\alpha {x}^{*}\right){\left({x}^{*}+{y}^{*}\right)}^{2}}{{\left[\frac{{b}_{1}}{{d}_{1}}\left({x}^{*}+{y}^{*}\right)\right]}^{2}}+1-{d}_{2}\end{array}\right)}_{·}$

(7)

$|tr{J}_{3}|<1+\mathrm{det}{J}_{3}<2$

$\left(1\right).\text{}\text{ }\text{ }\text{ }1+\mathrm{det}{J}_{3}<2;\text{}\left(2\right)\text{ }\text{.}\text{ }\text{ }-1-\mathrm{det}{J}_{3}

$\begin{array}{l}\mathrm{det}{J}_{3}=\left(\frac{{d}_{1}^{2}}{{b}_{1}}+1-{d}_{1}\right)\frac{q{b}_{2}{x}^{*}\left[\left(\frac{{b}_{1}}{{d}_{1}}-1\right){y}^{*}-\frac{{b}_{1}}{{d}_{1}}{x}^{*}\right]+{b}_{2}\left(1+\alpha {x}^{*}\right){\left({x}^{*}+{y}^{*}\right)}^{2}}{{\left[\frac{{b}_{1}}{{d}_{1}}\left({x}^{*}+{y}^{*}\right)\right]}^{2}}\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+\left(1-{d}_{2}\right)\frac{{d}_{1}^{2}\left(1+\alpha {y}^{*}\right)}{{b}_{1}}+\frac{{d}_{1}^{4}\left[\alpha {b}_{2}{y}^{*}\left({x}^{*}+{y}^{*}\right)-\alpha q{b}_{2}{x}^{*}{y}^{*}\right]}{{b}_{1}^{3}\left({x}^{*}+{y}^{*}\right)}+\left(1-{d}_{1}\right)\left(1-{d}_{2}\right),\end{array}$

$tr{J}_{3}=\frac{q{b}_{2}{x}^{*}\left[\left(\frac{{b}_{1}}{{d}_{1}}-1\right){y}^{*}-\frac{{b}_{1}}{{d}_{1}}{x}^{*}\right]+{b}_{2}\left(1+\alpha {x}^{*}\right){\left({x}^{*}+{y}^{*}\right)}^{2}}{{\left[\frac{{b}_{1}}{{d}_{1}}\left({x}^{*}+{y}^{*}\right)\right]}^{2}}+\frac{{d}_{1}^{2}\left(1+\alpha {y}^{*}\right)}{{b}_{1}}+2-{d}_{1}-{d}_{2}.$

$\begin{array}{l}\left(\frac{{d}_{1}^{2}}{{b}_{1}}-{d}_{1}\right)\frac{q{b}_{2}{x}^{*}\left[\left(\frac{{b}_{1}}{{d}_{1}}-1\right){y}^{*}-\frac{{b}_{1}}{{d}_{1}}{x}^{*}\right]+{b}_{2}\left(1+\alpha {x}^{*}\right){\left({x}^{*}+{y}^{*}\right)}^{2}}{{\left[\frac{{b}_{1}}{{d}_{1}}\left({x}^{*}+{y}^{*}\right)\right]}^{2}}\\ -\frac{{d}_{1}^{2}{d}_{2}\left(1+\alpha {y}^{*}\right)}{{b}_{1}}+\frac{{d}_{1}^{4}\left[\alpha {b}_{2}{y}^{*}\left({x}^{*}+{y}^{*}\right)-\alpha q{b}_{2}{x}^{*}{y}^{*}\right]}{{b}_{1}^{3}\left({x}^{*}+{y}^{*}\right)}+{d}_{1}{d}_{2}>0.\end{array}$

$\begin{array}{l}\frac{q{b}_{2}\left(\frac{{d}_{1}}{{b}_{1}}-1\right){x}^{*}\left[\left(\frac{{b}_{1}}{{d}_{1}}-1\right){y}^{*}-\frac{{b}_{1}}{{d}_{1}}{x}^{*}\right]}{{\left[\frac{{b}_{1}}{{d}_{1}}\left({x}^{*}+{y}^{*}\right)\right]}^{2}}+\frac{{d}_{1}^{2}{b}_{2}}{{b}_{1}{}^{2}}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\left(1+\alpha {x}^{*}\right)\\ +{d}_{2}\left[1-\frac{{d}_{1}}{{b}_{1}}\left(1+\alpha {y}^{*}\right)\right]+\frac{{d}_{1}^{3}\left[\alpha {b}_{2}{y}^{*}\left({x}^{*}+{y}^{*}\right)-\alpha q{b}_{2}{x}^{*}{y}^{*}\right]}{{b}_{1}^{3}\left({x}^{*}+{y}^{*}\right)}>0,\end{array}$

${E}_{3}=\left({x}^{*},{y}^{*}\right)$ 代入上式，有

$\begin{array}{l}\frac{{d}_{1}^{2}{b}_{2}}{{b}_{1}^{2}}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\left[\frac{{b}_{1}}{{d}_{1}}-1-\frac{1}{q}\left(2\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]+\frac{{d}_{1}^{2}{b}_{2}}{{b}_{1}^{2}}\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\\ +\frac{{d}_{1}^{2}{b}_{2}}{{b}_{1}^{2}}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\left[1+\frac{1}{q}\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]+{d}_{2}\left(1-\frac{{d}_{1}}{{b}_{1}}\left(1+\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\right)\right)\\ =\frac{{d}_{1}{b}_{2}}{{b}_{1}}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\left(\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]+\frac{{d}_{2}}{{b}_{2}}\left[1+\frac{1}{q}\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\right)+{d}_{2}\end{array}$

$\begin{array}{l}-\frac{{d}_{1}{d}_{2}}{{b}_{1}}\left(1+\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\right)+\frac{{d}_{1}^{2}{d}_{2}}{{b}_{1}{}^{2}}\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\\ =\left[\frac{{d}_{1}{b}_{2}}{{b}_{1}}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)+\frac{{d}_{1}{d}_{2}}{{b}_{1}}\left(1-\frac{{d}_{1}}{{b}_{1}}\right)\right]\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]+\frac{{d}_{1}{d}_{2}}{{b}_{1}}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }+{d}_{2}\left(1-\frac{{d}_{1}}{{b}_{1}}\right)+\frac{{d}_{1}{d}_{2}}{q{b}_{1}}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\\ =\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\left[\frac{{d}_{1}{b}_{2}}{{b}_{1}}\left(1-\frac{{d}_{1}}{{b}_{1}}\right)+{d}_{2}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\right]+\left[\frac{{d}_{1}{b}_{2}}{{b}_{1}}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)+{d}_{2}\left(1-\frac{{d}_{1}}{{b}_{1}}\right)\right]\\ =\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)-1>0.\end{array}$

$\begin{array}{l}4-2{d}_{2}+{d}_{1}\left(2-{d}_{2}\right)\left[\frac{{d}_{1}\left(1+\alpha {y}^{*}\right)}{{b}_{1}}-1\right]+\frac{{d}_{1}^{4}\left[\alpha {b}_{2}{y}^{*}\left({x}^{*}+{y}^{*}\right)-\alpha q{b}_{2}{x}^{*}{y}^{*}\right]}{{b}_{1}^{3}\left({x}^{*}+{y}^{*}\right)}\\ +\left(\frac{{d}_{1}^{2}}{{b}_{1}}+2-{d}_{1}\right)\frac{q{b}_{2}{x}^{*}\left[\left(\frac{{b}_{1}}{{d}_{1}}-1\right){y}^{*}-\frac{{b}_{1}}{{d}_{1}}{x}^{*}\right]+{b}_{2}\left(1+\alpha {x}^{*}\right){\left({x}^{*}+{y}^{*}\right)}^{2}}{{\left[\frac{{b}_{1}}{{d}_{1}}\left({x}^{*}+{y}^{*}\right)\right]}^{2}}>0.\end{array}$

$\begin{array}{l}4-2{d}_{2}+{d}_{1}\left(2-{d}_{2}\right)\left(\frac{{d}_{1}}{{b}_{1}}\left(1+\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\right)-1\right)+\frac{{d}_{1}^{3}{d}_{2}}{{b}_{1}^{2}}\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\\ +\frac{{d}_{1}^{2}{b}_{2}}{{b}_{1}^{2}}\left(\frac{{d}_{1}^{2}}{{b}_{1}}+2-{d}_{1}\right)\left[1+\left(1+\frac{1}{q}\right)\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)-\frac{1}{q}\left(2\frac{{b}_{1}}{{d}_{1}}-1\right){\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)}^{2}\right]>0,\end{array}$

$\begin{array}{l}4-2{d}_{2}+{d}_{1}\left(2-{d}_{2}\right)\left(\frac{{d}_{1}}{{b}_{1}}\left(1+\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left[1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\right]\right)-1\right)\ge 4-2{d}_{2}+{d}_{1}\left(2-{d}_{2}\right)\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\\ =\left(2-{d}_{2}\right)\left[2+{d}_{1}\left(\frac{{d}_{1}}{{b}_{1}}-1\right)\right]>0\end{array}$

$\begin{array}{l}1+\left(1+\frac{1}{q}\right)\left(\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)-\frac{1}{q}\left(2\frac{{b}_{1}}{{d}_{1}}-1\right){\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)}^{2}\\ \ge 1+\left[\frac{{b}_{1}}{{d}_{1}}\left(1+\frac{1}{q}\right)-\left(1+\frac{1}{q}\right)\right]\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)-\left(2\frac{{b}_{1}}{{d}_{1}}-1\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\\ \ge 1-\left(1+\frac{1}{q}\right)\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)+\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)\\ =1-\frac{1}{q}\left(1-\frac{{b}_{1}{d}_{2}}{{b}_{2}{d}_{1}}\right)>0\end{array}$

$p\left(\lambda \right)={\lambda }^{2}-\left(tr{J}_{3}\right)\lambda +\mathrm{det}{J}_{3}.$

c) 在此条件下(即Case C)，证明类似于a)，这里我们不再赘述。定理2.2证毕。

3. 数值模拟

Figure 2. The change in the solution of the system in Case A

Figure 3. The change in the solution of the system in Case B

Figure 4. The change in the solution of the system in Case C

Figure 5. The influence of CI strength changes in Case B

3. 总结

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