﻿ 污染环境下具有尺度结构的捕食种群模型解的存在唯一性 Existence and Uniqueness of Solution for Predator-Prey Population Model with Size-Structured in Polluted Environment

Vol.07 No.07(2018), Article ID:25844,8 pages
10.12677/AAM.2018.77091

Existence and Uniqueness of Solution for Predator-Prey Population Model with Size-Structured in Polluted Environment

Xuejing Cao

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou Gansu

Received: Jun. 17th, 2018; accepted: Jul. 5th, 2018; published: Jul. 12th, 2018

ABSTRACT

In recent years, people pay more and more attention to the problem of environmental pollution, and it is very important to study the law of population development in polluted environment. In this paper, a predator-prey population model with size-structured in polluted environment is proposed. The formal solution of the model is obtained by establishing the corresponding assumptions and using the characteristic line method. Then the existence and uniqueness of the solution are proved by inequality estimation and Banach fixed point theorem.

Keywords:Size-Structured, Existence and Uniqueness, Environmental Pollution, Banach Fixed Point Theorem

1. 建立模型

$\left\{\begin{array}{l}\frac{\partial p\left(s,t\right)}{\partial t}+\frac{\partial \left[g\left(s\right)p\left(s,t\right)\right]}{\partial s}=-\left[{\mu }_{1}\left(s,{c}_{10}\left(t\right)\right)+{\lambda }_{1}\left(t\right)q\left(t\right)\right]p\left(s,t\right),\text{}\left(s,t\right)\in Q,\\ \frac{\text{d}q\left(t\right)}{\text{d}t}=-\left[{\mu }_{2}\left({c}_{20}\left(t\right)\right)+{\lambda }_{2}\left(t\right)P\left(t\right)\right]q\left(t\right),\text{}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{10}\left(t\right)}{\text{d}t}=k{c}_{e}\left(t\right)-l{c}_{10}\left(t\right)-n{c}_{10}\left(t\right),\text{}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{20}\left(t\right)}{\text{d}t}=k{c}_{e}\left(t\right)-l{c}_{20}\left(t\right)-n{c}_{20}\left(t\right),\text{}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{e}\left(t\right)}{\text{d}t}=-{k}_{1}{c}_{e}\left(t\right)\left[P\left(t\right)+q\left(t\right)\right]+{l}_{1}\left[{c}_{10}\left(t\right)P\left(t\right)+{c}_{20}\left(t\right)q\left(t\right)\right]-h{c}_{e}\left(t\right)+v\left(t\right),\\ g\left(0\right)p\left(0,t\right)={\lambda }_{3}\left(t\right)q\left(t\right){\int }_{0}^{m}{\beta }_{1}\left(s,{c}_{10}\left(t\right)\right)p\left(s,t\right)\text{d}s,\\ p\left(s,0\right)={p}_{0}\left(s\right),P\left(t\right)={\int }_{0}^{m}p\left(s,t\right)\text{d}s,q\left(0\right)={q}_{0},\text{}s\in \left(0,m\right),\\ 0\le {c}_{0}\left(0\right)\le 1,0\le {c}_{e}\left(0\right)\le 1.\end{array}$ (1)

${p}_{1}\left(s,t\right)$ ：t时刻尺度为s的捕食者种群个体密度；

$q\left(t\right)$ ：t时刻尺度为s的食饵种群个体密度；

$g\left(s\right)$ ：捕食者种群个体的尺度增长函数；

${c}_{10}\left(t\right),{c}_{20}\left(t\right)$ ：分别表示t时刻捕食者种群个体、食饵种群个体体内的毒素浓度；

${c}_{e}\left(t\right)$ ：t时刻环境中的毒素浓度；

$v\left(t\right)$ ：t时刻外界向环境中输入的毒素浓度；

${\lambda }_{1}\left(t\right),{\lambda }_{2}\left(t\right)$ ：t时刻捕食者、食饵的相互作用因子；

${\beta }_{1}\left(s,{c}_{10}\left(t\right)\right),{\mu }_{1}\left(s,{c}_{10}\left(t\right)\right)$ ：分别表示尺度为s，体内毒素浓度为 ${c}_{10}\left(t\right)$ 的捕食者种群的出生率和死亡率；

${\mu }_{2}\left({c}_{20}\left(t\right)\right)$ ：体内毒素浓度为 ${c}_{20}\left(t\right)$ 的食饵种群的死亡率；

(H1) ${\mu }_{1}\left(s,{c}_{10}\left(t\right)\right)\in {L}_{\text{loc}}^{1}\left(Q\right)$$0<{\mu }_{1}\left(s,{c}_{10}\left(t\right)\right)\le {\mu }^{0}$${\int }_{0}^{m}{\mu }_{1}\left(s,{c}_{10}\left(t\right)\right)\text{d}s=+\infty$

(H2) ${\beta }_{1}\left(s,{c}_{10}\left(t\right)\right)\in {L}_{\text{loc}}^{1}\left(Q\right)$$0<{\beta }_{1}\left(s,{c}_{10}\left(t\right)\right)\le {\beta }^{0}$

(H3) ${\lambda }_{i}\left(t\right)\in {L}^{\infty }\left(0,T\right)$$0<{\lambda }_{i}\left(t\right)<{\lambda }_{i}^{0}$$i=1,2,3$

(H4) $g\in {C}^{1}\left(0,m\right)$$0$s\in \left(0,m\right)$

(H5) $v\left(\cdot \right)\in {L}^{2}\left(0,T\right)$$0

(H6) $|{\beta }_{1}\left(s,{x}_{1}\right)-{\beta }_{1}\left(s,{x}_{2}\right)|\le {L}_{{\beta }_{1}}|{x}_{1}-{x}_{2}|$$|{\mu }_{1}\left(s,{x}_{1}\right)-{\mu }_{1}\left(s,{x}_{2}\right)|\le {L}_{{\mu }_{1}}|{x}_{1}-{x}_{2}|$$|{\beta }_{2}\left({x}_{1}\right)-{\beta }_{2}\left({x}_{2}\right)|\le {L}_{{\beta }_{2}}|{x}_{1}-{x}_{2}|$$|{\mu }_{2}\left({x}_{1}\right)-{\mu }_{2}\left({x}_{2}\right)|\le {L}_{{\mu }_{2}}|{x}_{1}-{x}_{2}|$

(H7) $0$0

(H8) $l${v}^{0} [6] 。

2. 模型解的存在唯一性

$\left\{\begin{array}{l}\frac{\partial p\left(s,t\right)}{\partial t}+\frac{\partial \left[g\left(s\right)p\left(s,t\right)\right]}{\partial s}=-\left[{\mu }_{1}\left(s,{c}_{10}\left(t\right)\right)+{\lambda }_{1}\left(t\right)q\left(t\right)\right]p\left(s,t\right),\text{a}\text{.e}\text{.}\left(s,t\right)\in Q,\\ \frac{\text{d}q\left(t\right)}{\text{d}t}=-\left[{\mu }_{2}\left({c}_{20}\left(t\right)\right)+{\lambda }_{2}\left(t\right)P\left(t\right)+{u}_{2}\left(t\right)\right]q\left(t\right),\text{a}\text{.e}\text{.}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{10}\left(t\right)}{\text{d}t}=k{c}_{e}\left(t\right)-l{c}_{10}\left(t\right)-n{c}_{10}\left(t\right),\text{a}\text{.e}\text{.}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{20}\left(t\right)}{\text{d}t}=k{c}_{e}\left(t\right)-l{c}_{20}\left(t\right)-n{c}_{20}\left(t\right),\text{a}\text{.e}\text{.}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{e}\left(t\right)}{\text{d}t}=-{k}_{1}{c}_{e}\left(t\right)\left[P\left(t\right)+q\left(t\right)\right]+{l}_{1}\left[{c}_{10}\left(t\right)P\left(t\right)+{c}_{20}\left(t\right)q\left(t\right)\right]-h{c}_{e}\left(t\right)+v\left(t\right),\\ \underset{\epsilon \to {0}^{+}}{\mathrm{lim}}g\left(0\right)p\left({\Gamma }^{-1}\left(\epsilon \right),t+\epsilon \right)={\lambda }_{3}\left(t\right)q\left(t\right){\int }_{0}^{m}{\beta }_{1}\left(s,{c}_{10}\left(t\right)\right)p\left(s,t\right)\text{d}s,\\ \underset{\epsilon \to {0}^{+}}{\mathrm{lim}}p\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\epsilon \right),\epsilon \right)={p}_{0}\left(s\right),\text{}P\left(t\right)={\int }_{0}^{m}p\left(s,t\right)\text{d}s,\text{}q\left(0\right)={q}_{0},\text{a}\text{.e}\text{.}s\in \left(0,m\right),\\ 0\le {c}_{10}\left(0\right)\le 1,0\le {c}_{20}\left(0\right)\le 1,0\le {c}_{e}\left(0\right)\le 1.\end{array}$ (2)

$\begin{array}{l}X=\left\{\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)\in {L}^{\infty }\left(0,T;{L}^{1}\left(0,m\right)\right)×{\left[{L}^{\infty }\left(0,T\right)\right]}^{4}|p\left(s,t\right)>0,q\left(t\right)>0,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le {\int }_{0}^{m}p\left(s,t\right)\text{d}s\le M,\text{a}\text{.e}\text{.}\left(s,t\right)\in Q\right\}.\end{array}$

$p\left(s,t\right)=\left\{\begin{array}{l}\mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),{c}_{10}\left(\tau \right)\right)+{g}_{s}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)\\ {\text{ }}_{\text{ }}^{\text{ }}+{\lambda }_{1}\left(\tau \right)q\left(\tau \right)\right]\text{d}\tau \right\}{p}_{0}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right),\text{}\Gamma \left(s\right)>t,\text{}\\ \mathrm{exp}\left\{-{\int }_{0}^{\Gamma \left(s\right)}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}\left(t+\tau -\Gamma \left(s\right)\right)\right)+{\lambda }_{2}\left(t+\tau -\Gamma \left(s\right)\right)P\left(t+\tau -\Gamma \left(s\right)\right)\\ \text{}+{g}_{s}\left({\Gamma }^{-1}\left(\tau \right)\right)\right]\text{d}\tau \right\}B\left(t-\Gamma \left(s\right)\right),\text{}\Gamma \left(s\right)

$q\left(t\right)={q}_{0}\cdot \mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{2}\left({c}_{20}\left(\tau \right)\right)+{\lambda }_{2}\left(\tau \right)P\left(\tau \right)\right]\text{d}\tau \right\}$

${c}_{i0}\left(t\right)={c}_{i0}\left(0\right)\mathrm{exp}\left\{-\left(l+n\right)t\right\}+k{\int }_{0}^{t}{c}_{e}\left(\tau \right)\mathrm{exp}\left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau$$i=1,2$

$\begin{array}{l}{c}_{e}\left(t\right)={\int }_{0}^{t}\left[{l}_{1}{c}_{10}\left(\tau \right)P\left(\tau \right)+{l}_{1}{c}_{20}\left(\tau \right)q\left(\tau \right)+v\left(\tau \right)\right]\mathrm{exp}\left\{{\int }_{t}^{\tau }\left[{k}_{1}P\left(r\right)+{k}_{1}q\left(r\right)+h\right]\text{d}r\right\}\text{d}\tau \\ \text{}+{c}_{e}\left(0\right)\mathrm{exp}\left\{-{\int }_{0}^{t}\left({k}_{1}P\left(r\right)+{k}_{1}q\left(r\right)+h\right)\text{d}r\right\}.\end{array}$

$\left(p\left(s,t\right),q\left(t\right),{c}_{10}\left(t\right),{c}_{20}\left(t\right),{c}_{e}\left(t\right)\right)\in X$

$F\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)=\left({F}_{1}\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right),{F}_{2}\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right),\cdot \cdot \cdot ,{F}_{5}\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)\right),$

${F}_{1}\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)=\left\{\begin{array}{l}\mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),{c}_{10}\left(\tau \right)\right)+{g}_{s}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)\\ {\text{ }}_{\text{ }}^{\text{ }}+{\lambda }_{1}\left(\tau \right)q\left(\tau \right)\right]\text{d}\tau \right\}{p}_{0}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right),\text{}\Gamma \left(s\right)>t,\\ \mathrm{exp}\left\{-{\int }_{0}^{\Gamma \left(s\right)}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}\left(t+\tau -\Gamma \left(s\right)\right)\right)+{\lambda }_{2}\left(t+\tau -\Gamma \left(s\right)\right)P\left(t+\tau -\Gamma \left(s\right)\right)\\ \text{}+{g}_{s}\left({\Gamma }^{-1}\left(\tau \right)\right)\right]\text{d}\tau \right\}B\left(t-\Gamma \left(s\right)\right),\text{}\Gamma \left(s\right)

${F}_{2}\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)={q}_{0}\cdot \mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{2}\left({c}_{20}\left(\tau \right)\right)+{\lambda }_{2}\left(\tau \right)P\left(\tau \right)\right]\text{d}\tau \right\},$

${F}_{3}\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)={c}_{10}\left(0\right)\mathrm{exp}\left\{-\left(l+n\right)t\right\}+k{\int }_{0}^{t}{c}_{e}\left(\tau \right)\mathrm{exp}\left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau ,$

${F}_{4}\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)={c}_{20}\left(0\right)\mathrm{exp}\left\{-\left(l+n\right)t\right\}+k{\int }_{0}^{t}{c}_{e}\left(\tau \right)\mathrm{exp}\left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau ,$

$\begin{array}{c}{F}_{5}\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)={\int }_{0}^{t}\left[{l}_{1}{c}_{10}\left(\tau \right)P\left(\tau \right)+{l}_{1}{c}_{20}\left(\tau \right)q\left(\tau \right)+v\left(\tau \right)\right]\mathrm{exp}\left\{{\int }_{t}^{\tau }\left[{k}_{1}P\left(r\right)+{k}_{1}q\left(r\right)+h\right]\text{d}r\right\}\text{d}\tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{c}_{e}\left(0\right)\mathrm{exp}\left\{-{\int }_{0}^{t}\left({k}_{1}P\left(r\right)+{k}_{1}q\left(r\right)+h\right)\text{d}r\right\}.\end{array}$

$\begin{array}{l}{\int }_{0}^{m}p\left(s,t\right)\text{d}s={\int }_{0}^{{\Gamma }^{-1}\left(t\right)}p\left(s,t\right)\text{d}s+{\int }_{{\Gamma }^{-1}\left(t\right)}^{m}p\left(s,t\right)\text{d}s\\ ={\int }_{0}^{{\Gamma }^{-1}\left(t\right)}\mathrm{exp}\left\{-{\int }_{0}^{\Gamma \left(s\right)}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}\left(t+\tau -\Gamma \left(s\right)\right)\right)+{\lambda }_{2}\left(t+\tau -\Gamma \left(s\right)\right)P\left(t+\tau -\Gamma \left(s\right)\right)\\ \text{}+{g}_{s}\left({\Gamma }^{-1}\left(\tau \right)\right)\right]\text{d}\tau \right\}B\left(t-\Gamma \left(s\right)\right)\text{d}s+{\int }_{{\Gamma }^{-1}\left(t\right)}^{m}\mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),{c}_{10}\left(\tau \right)\right)\\ \text{}+{g}_{s}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)+{\lambda }_{1}\left(\tau \right)q\left(\tau \right)\right]\text{d}\tau \right\}{p}_{0}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\text{d}s\\ \le {\int }_{0}^{{\Gamma }^{-1}\left(t\right)}B\left(t-\Gamma \left(s\right)\right)\text{d}s+{\int }_{{\Gamma }^{-1}\left(t\right)}^{m}{p}_{0}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\text{d}s\\ \le {g}^{-1}\left(0\right){\lambda }_{3}^{0}{q}^{\ast }{\beta }^{0}{\int }_{0}^{t}{\int }_{0}^{m}p\left(s,\tau \right)\text{d}s\text{d}\tau +m{p}^{\ast }\end{array}$

$\begin{array}{l}{\int }_{0}^{m}p\left(s,t\right)\text{d}s\\ \le m{p}^{\ast }\cdot \mathrm{exp}\left\{{\int }_{0}^{t}{g}^{-1}\left(0\right){\lambda }_{3}^{0}{q}^{\ast }{\beta }^{0}\text{d}t\right\}\\ =M.\end{array}$

${x}^{i}=\left({p}^{i},{q}^{i},{c}_{10}^{i},{c}_{20}^{i},{c}_{e}^{i}\right)$$i=1,\text{2}$ 。当 $\Gamma \left(s\right) 时，有如下不等式成立：

$\begin{array}{l}{\int }_{0}^{m}|{p}^{1}\left(s,t\right)-{p}^{2}\left(s,t\right)|\text{d}s\\ ={\int }_{0}^{{\Gamma }^{-1}\left(t\right)}|{p}^{1}\left(s,t\right)-{p}^{2}\left(s,t\right)|\text{d}s+{\int }_{{\Gamma }^{-1}\left(t\right)}^{m}|{p}^{1}\left(s,t\right)-{p}^{2}\left(s,t\right)|\text{d}s\\ ={\int }_{0}^{{\Gamma }^{-1}\left(t\right)}|{B}^{1}\mathrm{exp}\left\{-{\int }_{0}^{\Gamma \left(s\right)}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}^{1}\left(t+\tau -\Gamma \left(s\right)\right)\right)\\ \text{}+{\lambda }_{2}\left(t+\tau -\Gamma \left(s\right)\right){P}^{1}\left(t+\tau -\Gamma \left(s\right)\right)+{g}_{s}\left({\Gamma }^{-1}\left(\tau \right)\right)\right]\text{d}\tau \right\}\\ \text{}-{B}^{2}\mathrm{exp}\left\{-{\int }_{0}^{\Gamma \left(s\right)}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}^{2}\left(t+\tau -\Gamma \left(s\right)\right)\right)\end{array}$

$\begin{array}{l}\text{ }+{g}_{s}\left({\Gamma }^{-1}\left(\tau \right)\right)+{\lambda }_{2}\left(t+\tau -\Gamma \left(s\right)\right){P}^{2}\left(t+\tau -\Gamma \left(s\right)\right)\right]\text{d}\tau \right\}|\text{d}s\\ \text{ }+{\int }_{{\Gamma }^{-1}\left(t\right)}^{m}|\mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),{c}_{10}^{1}\left(\tau \right)\right)+{g}_{s}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)\\ \text{ }+{\lambda }_{1}\left(\tau \right){q}^{1}\left(\tau \right)\right]\text{d}\tau \right\}-\mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{1}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),{c}_{10}^{2}\left(\tau \right)\right)\\ \text{ }+{g}_{s}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)+{\lambda }_{1}\left(\tau \right){q}^{2}\left(\tau \right)\right]\text{d}\tau \right\}|{p}_{0}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\text{d}s\end{array}$

$\begin{array}{l}\le {\int }_{0}^{{\Gamma }^{-1}\left(t\right)}|{B}^{1}-{B}^{2}|\text{d}s+{B}^{2}\cdot {\int }_{0}^{{\Gamma }^{-1}\left(t\right)}{\int }_{0}^{\Gamma \left(s\right)}|{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}^{1}\left(t+\tau -\Gamma \left(s\right)\right)\right)\\ \text{}-{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}^{2}\left(t+\tau -\Gamma \left(s\right)\right)\right)|\text{d}\tau \text{d}s+{B}^{2}\cdot {\int }_{0}^{{\Gamma }^{-1}\left(t\right)}{\int }_{0}^{\Gamma \left(s\right)}{\lambda }_{2}\left(t+\tau -\Gamma \left(s\right)\right)\\ \text{}\cdot |{P}^{1}\left(t+\tau -\Gamma \left(s\right)\right)-{P}^{2}\left(t+\tau -\Gamma \left(s\right)\right)|\text{d}\tau \text{d}s\\ \text{}+{\int }_{{\Gamma }^{-1}\left(t\right)}^{m}{p}_{0}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\cdot {\int }_{0}^{t}|{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}^{1}\left(t+\tau -\Gamma \left(s\right)\right)\right)\\ \text{}-{\mu }_{1}\left({\Gamma }^{-1}\left(\tau \right),{c}_{10}^{2}\left(t+\tau -\Gamma \left(s\right)\right)\right)|\text{d}\tau \text{d}s\end{array}$

$\begin{array}{l}\text{}+{\int }_{{\Gamma }^{-1}\left(t\right)}^{m}{p}_{0}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\cdot {\int }_{0}^{t}{\lambda }_{1}\left(\tau \right)|{q}^{1}\left(\tau \right)-{q}^{2}\left(\tau \right)|\text{d}\tau \text{d}s\\ \le \left(\frac{1}{{g}^{*}}{\lambda }_{3}^{0}\cdot {\beta }^{0}\cdot {q}^{\ast }+{\lambda }_{2}^{0}\cdot {\lambda }_{3}^{0}\cdot {\beta }^{0}\cdot {q}^{\ast }\cdot MT\right){\int }_{0}^{t}{\int }_{0}^{m}|{p}^{1}\left(s,\tau \right)-{p}^{2}\left(s,\tau \right)|\text{d}s\text{d}\tau \\ \text{}+\left(\frac{1}{{g}^{*}}{\lambda }_{3}^{0}\cdot {\beta }^{0}\cdot M+m\cdot {\lambda }_{1}^{0}{p}^{*}\right){\int }_{0}^{t}|{q}^{1}\left(\tau \right)-{q}^{2}\left(\tau \right)|\text{d}\tau \\ \text{}+\left(\frac{1}{{g}^{*}}{\lambda }_{3}^{0}\cdot {q}^{\ast }\cdot M{L}_{{\beta }_{1}}+T\cdot {\lambda }_{3}^{0}\cdot {q}^{\ast }\cdot M{L}_{{\mu }_{1}}+m\cdot {p}^{*}\cdot {L}_{{\mu }_{1}}\right){\int }_{0}^{t}|{c}_{10}^{1}\left(\tau \right)-{c}_{10}^{2}\left(\tau \right)|\text{d}\tau .\end{array}$

${\int }_{0}^{m}|{F}_{1}\left({x}^{1}\right)-{F}_{1}\left({x}^{2}\right)|\text{d}s\le {M}_{1}\left({\int }_{0}^{t}{\int }_{0}^{m}|{p}^{1}\left(s,\tau \right)-{p}^{2}\left(s,\tau \right)|\text{d}s\text{d}\tau +{\int }_{0}^{t}|{q}^{1}\left(\tau \right)-{q}^{2}\left(\tau \right)|\text{d}\tau +{\int }_{0}^{t}|{c}_{10}^{1}\left(\tau \right)-{c}_{10}^{2}\left(\tau \right)|\text{d}\tau \right),$ (3)

$\begin{array}{l}{M}_{1}=\mathrm{max}\left\{\frac{1}{{g}^{*}}{\lambda }_{3}^{0}\cdot {\beta }^{0}\cdot {q}^{\ast }+{\lambda }_{2}^{0}\cdot {\lambda }_{3}^{0}\cdot {\beta }^{0}\cdot {q}^{\ast }\cdot MT,\frac{1}{{g}^{*}}{\lambda }_{3}^{0}\cdot {\beta }^{0}\cdot M+m\cdot {\lambda }_{1}^{0}{p}^{*},\\ \text{}\frac{1}{{g}^{*}}{\lambda }_{3}^{0}\cdot {q}^{\ast }\cdot M{L}_{{\beta }_{1}}+T\cdot {\lambda }_{3}^{0}\cdot {q}^{\ast }\cdot M{L}_{{\mu }_{1}}+m\cdot {p}^{*}\cdot {L}_{{\mu }_{1}}\right\}.\end{array}$

$\Gamma \left(s\right)>t$ 时同理可得上述不等式成立。

$\begin{array}{c}|{q}^{1}-{q}^{2}|={q}_{0}\cdot |\mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{2}\left({c}_{20}^{1}\left(\tau \right)\right)+{\lambda }_{2}\left(\tau \right){P}^{1}\left(\tau \right)\right]\text{d}\tau \right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\mathrm{exp}\left\{-{\int }_{0}^{t}\left[{\mu }_{2}\left({c}_{20}^{2}\left(\tau \right)\right)+{\lambda }_{2}\left(\tau \right){P}^{2}\left(\tau \right)\right]\text{d}\tau \right\}|\\ \le {q}_{0}\cdot {\int }_{0}^{t}|{\mu }_{2}\left({c}_{20}^{1}\left(\tau \right)\right)-{\mu }_{2}\left({c}_{20}^{2}\left(\tau \right)\right)|\text{d}\tau +{q}_{0}\cdot {\int }_{0}^{t}{\lambda }_{2}\left(\tau \right)|{P}^{1}\left(\tau \right)-{P}^{2}\left(\tau \right)|\text{d}\tau \\ \le {q}_{0}{L}_{{\mu }_{2}}\cdot {\int }_{0}^{t}|{c}_{20}^{1}\left(\tau \right)-{c}_{20}^{2}\left(\tau \right)|\text{d}\tau +{q}_{0}{\lambda }_{2}^{0}{\int }_{0}^{t}{\int }_{0}^{m}|{p}^{1}\left(s,\tau \right)-{p}^{2}\left(s,\tau \right)|\text{d}s\text{d}\tau \\ \le {M}_{2}\left({\int }_{0}^{t}{\int }_{0}^{m}|{p}^{1}\left(s,\tau \right)-{p}^{2}\left(s,\tau \right)|\text{d}s\text{d}\tau +{\int }_{0}^{t}|{c}_{20}^{1}\left(\tau \right)-{c}_{20}^{2}\left(\tau \right)|\text{d}\tau \right)\end{array}$ (4)

$\begin{array}{l}|{c}_{i0}^{1}-{c}_{i0}^{2}|\\ =|{c}_{i0}\left(0\right)\mathrm{exp}\left\{-\left(l+n\right)t\right\}+k{\int }_{0}^{t}{c}_{e}^{1}\left(\tau \right)\mathrm{exp}\left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau \\ \text{}-{c}_{i0}\left(0\right)\mathrm{exp}\left\{-\left(l+n\right)t\right\}+k{\int }_{0}^{t}{c}_{e}^{2}\left(\tau \right)\mathrm{exp}\left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau |\\ =k{\int }_{0}^{t}|{c}_{e}^{1}\left(\tau \right)-{c}_{e}^{2}\left(\tau \right)|\mathrm{exp}\left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau \\ \le {M}_{3}{\int }_{0}^{t}|{c}_{e}^{1}\left(\tau \right)-{c}_{e}^{2}\left(\tau \right)|\text{d}\tau \text{}\left({M}_{3}=k\right)\end{array}$ (5)

$\begin{array}{l}|{c}_{e}^{1}-{c}_{e}^{2}|\\ =|{\int }_{0}^{t}\left[{l}_{1}{c}_{10}^{1}\left(\tau \right){P}^{1}\left(\tau \right)+{l}_{1}{c}_{20}^{1}\left(\tau \right){q}^{1}\left(\tau \right)+v\left(\tau \right)\right]\mathrm{exp}\left\{{\int }_{t}^{\tau }\left[{k}_{1}{P}^{1}\left(r\right)+{k}_{1}{q}^{1}\left(r\right)+h\right]\text{d}r\right\}\text{d}\tau \\ \text{}+{c}_{e}\left(0\right)\mathrm{exp}\left\{-{\int }_{0}^{t}\left({k}_{1}{P}^{1}\left(r\right)+{k}_{1}{q}^{1}\left(r\right)+h\right)\text{d}r\right\}\\ \text{}-{\int }_{0}^{t}\left[{l}_{1}{c}_{10}^{2}\left(\tau \right){P}^{2}\left(\tau \right)+{l}_{1}{c}_{20}^{2}\left(\tau \right){q}^{2}\left(\tau \right)+v\left(\tau \right)\right]\mathrm{exp}\left\{{\int }_{t}^{\tau }\left[{k}_{1}{P}^{2}\left(r\right)+{k}_{1}{q}^{2}\left(r\right)+h\right]\text{d}r\right\}\text{d}\tau \\ \text{}+{c}_{e}\left(0\right)\mathrm{exp}\left\{-{\int }_{0}^{t}\left({k}_{1}{P}^{2}\left(r\right)+{k}_{1}{q}^{2}\left(r\right)+h\right)\text{d}r\right\}|\end{array}$

$\begin{array}{l}\le {c}_{e}\left(0\right)\cdot {\int }_{0}^{t}|{k}_{1}\left({P}^{1}\left(r\right)-{P}^{2}\left(r\right)\right)+{k}_{1}\left({q}^{1}\left(r\right)-{q}^{2}\left(r\right)\right)|\text{d}r\\ \text{}+{\int }_{0}^{t}|{l}_{1}\left({c}_{10}^{1}\left(\tau \right){P}^{1}\left(\tau \right)-{c}_{10}^{2}\left(\tau \right){P}^{2}\left(\tau \right)\right)+{l}_{1}\left({c}_{20}^{1}\left(\tau \right){q}^{1}\left(\tau \right)-{c}_{20}^{2}\left(\tau \right){q}^{2}\left(\tau \right)\right)|\text{d}\tau \\ \text{}+{\int }_{0}^{t}\left[{l}_{1}{c}_{10}^{2}\left(\tau \right){P}^{2}\left(\tau \right)+{l}_{1}{c}_{20}^{2}\left(\tau \right){q}^{2}\left(\tau \right)+v\left(\tau \right)\right]\cdot |\mathrm{exp}\left\{{\int }_{t}^{\tau }\left[{k}_{1}{P}^{1}\left(r\right)+{k}_{1}{q}^{1}\left(r\right)+h\right]\text{d}r\right\}\\ \text{}-\mathrm{exp}\left\{{\int }_{t}^{\tau }\left[{k}_{1}{P}^{2}\left(r\right)+{k}_{1}{q}^{2}\left(r\right)+h\right]\text{d}r\right\}|\text{d}\tau \end{array}$

$\begin{array}{l}\le \left({k}_{1}+{l}_{1}+TM{k}_{1}{l}_{1}+T{q}^{\ast }{k}_{1}{l}_{1}+T{k}_{1}{v}^{0}\right){\int }_{0}^{t}{\int }_{0}^{m}|{p}^{1}\left(s,\tau \right)-{p}^{2}\left(s,\tau \right)|\text{d}s\text{d}\tau \\ \text{}+\left({k}_{1}+{l}_{1}+TM{k}_{1}{l}_{1}+T{q}^{\ast }{k}_{1}{l}_{1}+T{k}_{1}{v}^{0}\right){\int }_{0}^{t}|{q}^{1}\left(\tau \right)-{q}^{2}\left(\tau \right)|\text{d}\tau \\ \text{}+M{l}_{1}\cdot {\int }_{0}^{t}|{c}_{10}^{1}\left(\tau \right)-{c}_{10}^{2}\left(\tau \right)|\text{d}\tau +{q}^{\ast }{l}_{1}\cdot {\int }_{0}^{t}|{c}_{20}^{1}\left(\tau \right)-{c}_{20}^{2}\left(\tau \right)|\text{d}\tau \\ \le {M}_{4}\left({\int }_{0}^{t}{\int }_{0}^{m}|{p}^{1}\left(s,\tau \right)-{p}^{2}\left(s,\tau \right)|\text{d}s\text{d}\tau +{\int }_{0}^{t}|{q}^{1}\left(\tau \right)-{q}^{2}\left(\tau \right)|\text{d}\tau +\underset{i=1}{\overset{2}{\sum }}{\int }_{0}^{t}|{c}_{20}^{1}\left(\tau \right)-{c}_{20}^{2}\left(\tau \right)|\text{d}\tau \right)\end{array}$ (6)

${M}_{4}=\mathrm{max}\left\{{k}_{1}+{l}_{1}+TM{k}_{1}{l}_{1}+T{q}^{\ast }{k}_{1}{l}_{1}+T{k}_{1}{v}^{0},M{l}_{1},{q}^{\ast }{l}_{1}\right\}$

${‖\left(p,q,{c}_{10},{c}_{20},{c}_{e}\right)‖}_{X}=Ess\underset{t\in \left(0,T\right)}{\mathrm{sup}}{\text{e}}^{-\lambda t}\left\{{\int }_{0}^{m}|p\left(s,t\right)|\text{d}s+|q\left(t\right)|+\underset{i=1}{\overset{2}{\sum }}|{c}_{i0}\left(t\right)|+|{c}_{e}\left(t\right)|\right\},$

$\lambda >0$ 足够大。

$\begin{array}{l}{‖F\left({x}^{1}\right)-F\left({x}^{2}\right)‖}_{X}={‖{F}_{1}\left({x}^{1}\right)-{F}_{1}\left({x}^{2}\right),{F}_{2}\left({x}^{1}\right)-{F}_{2}\left({x}^{2}\right),\cdot \cdot \cdot ,{F}_{5}\left({x}^{1}\right)-{F}_{5}\left({x}^{2}\right)‖}_{X}\\ \le {M}_{5}Ess\underset{t\in \left(0,T\right)}{\mathrm{sup}}{\text{e}}^{-\lambda t}{\int }_{0}^{t}\left\{{\int }_{0}^{m}|{p}^{1}\left(s,\tau \right)-{p}^{2}\left(s,\tau \right)|\text{d}s+|{q}^{1}\left(\tau \right)-{q}^{2}\left(\tau \right)|+\underset{i=1}{\overset{2}{\sum }}|{c}_{i0}^{1}\left(\tau \right)-{c}_{i0}^{2}\left(\tau \right)|+|{c}_{e}^{1}\left(\tau \right)-{c}_{e}^{2}\left(\tau \right)|\right\}\\ ={M}_{5}Ess\underset{t\in \left(0,T\right)}{\mathrm{sup}}{\text{e}}^{-\lambda t}{\int }_{0}^{t}{\text{e}}^{\lambda \tau }\left\{{\text{e}}^{-\lambda \tau }\left[{\int }_{0}^{m}|{p}^{1}\left(s,\tau \right)-{p}^{2}\left(s,\tau \right)|\text{d}s+|{q}^{1}\left(\tau \right)-{q}^{2}\left(\tau \right)|\begin{array}{c}\text{ }\\ \text{ }\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{2}{\sum }}|{c}_{i0}^{1}\left(\tau \right)-{c}_{i0}^{2}\left(\tau \right)|+|{c}_{e}^{1}\left(\tau \right)-{c}_{e}^{2}\left(\tau \right)|\right]\right\}\text{d}\tau \\ \le {M}_{5}{‖{x}^{1}-{x}^{2}‖}_{X}Ess\underset{t\in \left(0,T\right)}{\mathrm{sup}}\left\{{\text{e}}^{-\lambda t}{\int }_{0}^{t}{\text{e}}^{\lambda \tau }\text{d}\tau \right\}\le \frac{{M}_{5}}{\lambda }{‖{x}^{1}-{x}^{2}‖}_{X}\end{array}$

3. 结论

Existence and Uniqueness of Solution for Predator-Prey Population Model with Size-Structured in Polluted Environment[J]. 应用数学进展, 2018, 07(07): 758-765. https://doi.org/10.12677/AAM.2018.77091

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