﻿ 一类集合生态系统的一致持久性

# 一类集合生态系统的一致持久性The Uniform Persistence of a Meta-Ecosystem

Abstract: In this paper we study the dynamic behavior of bi-directional resource exchange within a meta-ecosystem. By using the comparison principle, some sufficient conditions are determined that guarantee the uniform persistence of the model.

1. 引言

$\left\{\begin{array}{l}\frac{\text{d}P}{\text{d}t}={r}_{p}P\left[1-\frac{P}{{K}_{p}+{a}_{p}Q}\right]-{b}_{p}QP\\ \frac{\text{d}Q}{\text{d}t}={r}_{q}Q\left[1-\frac{Q}{{K}_{q}+{a}_{q}P}\right]-{b}_{q}QP\end{array}$

${\beta }_{i}=\frac{{b}_{i}}{{r}_{i}}{K}_{j}<1\text{\hspace{0.17em}}\left(i\ne j\right)$

$\left\{\begin{array}{l}\frac{\text{d}x}{\text{d}t}=x\left(t\right)\left[{r}_{1}\left(t\right)-\frac{{r}_{1}\left(t\right)x\left(t-{\tau }_{1}\left(t\right)\right)}{{K}_{1}\left(t\right)+{a}_{1}\left(t\right)y\left(t\right)}-{b}_{1}\left(t\right)y\left(t\right)\right]\\ \frac{\text{d}y}{\text{d}t}=y\left(t\right)\left[{r}_{2}\left(t\right)-\frac{{r}_{2}\left(t\right)y\left(t-{\tau }_{2}\left(t\right)\right)}{{K}_{2}\left(t\right)+{a}_{2}\left(t\right)x\left(t\right)}-{b}_{2}\left(t\right)x\left(t\right)\right]\end{array}$, (1)

${g}^{u}=\underset{t\in R}{\mathrm{sup}}g\left(t\right),\text{\hspace{0.17em}}{g}^{l}=\underset{t\in R}{\mathrm{inf}}g\left(t\right),\text{\hspace{0.17em}}{g}^{\ast }=\underset{x\to \infty }{\mathrm{lim}\mathrm{sup}}g\left(t\right),\text{\hspace{0.17em}}{g}_{\ast }=\underset{x\to \infty }{\mathrm{lim}\mathrm{inf}}g\left(t\right),\text{\hspace{0.17em}}\tau =\mathrm{max}\left({\tau }_{1}^{u},{\tau }_{2}^{u}\right)$

$\forall \phi =\left({\phi }_{1},{\phi }_{2}\right)\in {C}^{+}=\left\{\left({\phi }_{1},{\phi }_{2}\right)\in C\left[-\tau ,0\right],{R}^{2}|{\phi }_{i}\left(\theta \right)\ge 0,{\phi }_{i}\left(0\right)>0,i=1,2\right\}$，由 [12] 定理2.1知，方程有唯一满足初始条件 ${x}_{0}=\phi$ 的解

$x\left(t,0,\phi \right)=\left({x}_{1}\left(t,0,{\phi }_{1},{\phi }_{2}\right),{x}_{2}\left(t,0,{\phi }_{1},{\phi }_{2}\right)\right)$

${x}_{i}\left(t,0,{\phi }_{1},{\phi }_{2}\right)>0\left(\forall t>0,i=1,2\right)$

2. 一致持久性

${m}_{1}\le \underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}x\left(t,0,\phi \right)\le \underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}x\left(t,0,\phi \right)\le {M}_{1}$,

${m}_{2}\le \underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}y\left(t,0,\phi \right)\le \underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}y\left(t,0,\phi \right)\le {M}_{2}$.

${x}^{\prime }\le \frac{x\left(m-nx\right)}{K+ay}$,(2)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}x\le \frac{m}{n}$,

$x\le {\left[{\text{e}}^{-{\int }_{T}^{t}\frac{m}{g\left(s\right)}\text{d}s}\left({\int }_{T}^{t}\frac{n}{g\left(u\right)}{\text{e}}^{{\int }_{T}^{u}\frac{m}{g\left(s\right)}\text{d}s}\text{d}u+\stackrel{˜}{c}\right)\right]}^{-1}$,

${x}^{\prime }\ge \frac{x\left(m-nx\right)}{K+ay}$, (3)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}x\ge \frac{m}{n}$,

${M}_{1}=\left\{\begin{array}{l}\frac{{r}_{1}^{\ast }{K}_{1}^{\ast }\mathrm{exp}\left({r}_{1}^{\ast }{\tau }_{1}^{u}\right)}{{r}_{1}{}_{\ast }},{r}_{1}^{\ast }{a}_{1}^{\ast }<{K}_{1}^{\ast }{b}_{1}{}_{\ast }\\ \frac{{\left({r}_{1}^{\ast }{a}_{1}^{\ast }+{K}_{1}^{\ast }{b}_{1}{}_{\ast }\right)}^{2}\mathrm{exp}\left({r}_{1}^{\ast }{\tau }_{1}^{u}\right)}{4{a}_{1}^{\ast }{b}_{1}{}_{\ast }{r}_{1}{}_{\ast }},{r}_{1}^{\ast }{a}_{1}^{\ast }\ge {K}_{1}^{\ast }{b}_{1}{}_{\ast }\end{array}$,

${M}_{2}=\left\{\begin{array}{l}\frac{{r}_{2}^{\ast }{K}_{2}^{\ast }\mathrm{exp}\left({r}_{2}^{\ast }{\tau }_{2}^{u}\right)}{{r}_{2}{}_{\ast }},{r}_{2}^{\ast }{a}_{2}^{\ast }<{K}_{2}^{\ast }{b}_{2}{}_{\ast }\\ \frac{{\left({r}_{2}^{\ast }{a}_{2}^{\ast }+{K}_{2}^{\ast }{b}_{2}{}_{\ast }\right)}^{2}\mathrm{exp}\left({r}_{2}^{\ast }{\tau }_{2}^{u}\right)}{4{a}_{2}^{\ast }{b}_{2}{}_{\ast }{r}_{2}{}_{\ast }},{r}_{2}^{\ast }{a}_{2}^{\ast }\ge {K}_{2}^{\ast }{b}_{2}{}_{\ast }\end{array}$.

$\left({r}_{1}^{\ast }+\epsilon \right)\left({a}_{1}^{\ast }+\epsilon \right)<\left({K}_{1}^{\ast }+\epsilon \right)\left({b}_{1}{}_{*}-\epsilon \right)$. (4)

$\forall \epsilon \in \left(0,{\epsilon }_{0}\right),\exists {T}_{1}>0$ 使得当 $t>{T}_{1}$ 时：

${r}_{1}{}_{\ast }-\epsilon <{r}_{1}\left(t\right)<{r}_{1}^{\ast }+\epsilon ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{K}_{1}{}_{\ast }-\epsilon <{K}_{1}\left(t\right)<{K}_{1}^{\ast }+\epsilon$ (5)

${a}_{1}{}_{\ast }-\epsilon <{a}_{1}\left(t\right)<{a}_{1}^{\ast }+\epsilon ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{1}{}_{\ast }-\epsilon <{b}_{1}\left(t\right)<{b}_{1}^{\ast }+\epsilon$ (6)

$t>{T}_{1}$ 时，由(1)和(5)知

${x}^{\prime }\left(t\right)\le \left({r}_{1}^{\ast }+\epsilon \right)x\left(t\right)$.

$x\left(t\right)\le x\left(t-{\tau }_{1}\left(t\right)\right)\mathrm{exp}\left(\left({r}_{1}^{\ast }+\epsilon \right){\tau }_{1}^{u}\right)$.

$\begin{array}{c}\frac{\text{d}x}{\text{d}t}\le x\left(t\right)\left[\left({r}_{1}^{\ast }+\epsilon \right)-\frac{\left({r}_{1}^{\ast }-\epsilon \right)x\left(t\right)\mathrm{exp}\left(-\left({r}_{1}^{\ast }+\epsilon \right){\tau }_{1}{}^{u}\right)}{\left({K}_{1}^{\ast }+\epsilon \right)+\left({a}_{1}^{\ast }+\epsilon \right)y\left(t\right)}-\left({b}_{1}{}_{*}-\epsilon \right)y\left(t\right)\right]\\ =x\left(t\right)\frac{{g}_{1}\left(\epsilon \right)+{g}_{2}\left(\epsilon \right)y\left(t\right)-{g}_{3}\left(\epsilon \right){y}^{2}\left(t\right)-{g}_{4}\left(\epsilon \right)x\left(t\right)}{\left({K}_{1}^{\ast }+\epsilon \right)+\left({a}_{1}^{\ast }+\epsilon \right)y\left( t \right)}\end{array}$

$\begin{array}{l}{g}_{1}\left(\epsilon \right)=\left({r}_{1}^{\ast }+\epsilon \right)\left({K}_{1}^{\ast }+\epsilon \right),\\ {g}_{2}\left(\epsilon \right)=\left({r}_{1}^{\ast }+\epsilon \right)\left({a}_{1}^{\ast }+\epsilon \right)-\left({K}_{1}^{\ast }+\epsilon \right)\left({b}_{1}{}_{*}-\epsilon \right),\\ {g}_{3}\left(\epsilon \right)=\left({a}_{1}^{\ast }+\epsilon \right)\left({b}_{1}{}_{*}-\epsilon \right),\\ {g}_{4}\left(\epsilon \right)=\left({r}_{1}{}_{*}-\epsilon \right)\mathrm{exp}\left(-\left({r}_{1}^{\ast }+\epsilon \right){\tau }_{1}^{u}\right)\end{array}$

${x}^{\prime }\le x\left(t\right)\frac{{g}_{1}\left(\epsilon \right)-{g}_{4}\left(\epsilon \right)x\left(t\right)}{\left({K}_{1}^{\ast }+\epsilon \right)+\left({a}_{1}^{\ast }+\epsilon \right)y\left(t\right)}$.

$\epsilon$ 的任意性及引理1，

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}x\left(t\right)\le \frac{{r}_{1}^{\ast }{K}_{1}^{\ast }\mathrm{exp}\left({r}_{1}^{\ast }{\tau }_{1}^{u}\right)}{{r}_{1}{}_{*}}={M}_{1}$.

$\begin{array}{c}{x}^{\prime }\le x\left(t\right)\frac{{g}_{1}\left(\epsilon \right)+{g}_{2}\left(\epsilon \right)y\left(t\right)-{g}_{3}\left(\epsilon \right){y}^{2}\left(t\right)-{g}_{4}\left(\epsilon \right)x\left(t\right)}{\left({K}_{1}^{\ast }+\epsilon \right)+\left({a}_{1}^{\ast }+\epsilon \right)y\left(t\right)}\\ =x\left(t\right)\frac{{g}_{1}\left(\epsilon \right)-{g}_{3}\left(\epsilon \right){\left(y\left(t\right)-\frac{{g}_{2}\left(\epsilon \right)}{2{g}_{3}\left(\epsilon \right)}\right)}^{2}+\frac{{g}_{2}^{2}\left(\epsilon \right)}{4{g}_{3}\left(\epsilon \right)}-{g}_{4}\left(\epsilon \right)x\left(t\right)}{\left({K}_{1}^{\ast }+\epsilon \right)+\left({a}_{1}^{\ast }+\epsilon \right)y\left(t\right)}\\ \le x\left(t\right)\frac{{g}_{1}\left(\epsilon \right)+\frac{{g}_{2}^{2}\left(\epsilon \right)}{4{g}_{3}\left(\epsilon \right)}-{g}_{4}\left(\epsilon \right)x\left(t\right)}{\left({K}_{1}^{\ast }+\epsilon \right)+\left({a}_{1}^{\ast }+\epsilon \right)y\left( t \right)}\end{array}$

$\epsilon$ 的任意性及引理1，

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}x\left(t\right)\le \frac{{\left({r}_{1}^{\ast }{a}_{1}^{\ast }+{K}_{1}^{\ast }{b}_{1}{}_{*}\right)}^{2}\mathrm{exp}\left({r}_{1}^{\ast }{\tau }_{1}^{u}\right)}{4{a}_{1}^{\ast }{b}_{1}{}_{*}{r}_{1}{}_{*}}={M}_{1}$.

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}x\left(t\right)\le {M}_{1}=\left\{\begin{array}{l}\frac{{r}_{1}^{\ast }{K}_{1}^{\ast }\mathrm{exp}\left({r}_{1}^{\ast }{\tau }_{1}^{u}\right)}{{r}_{1}{}_{*}},{r}_{1}^{\ast }{a}_{1}^{\ast }<{K}_{1}^{\ast }{b}_{1}{}_{*}\\ \frac{{\left({r}_{1}^{\ast }{a}_{1}^{\ast }+{K}_{1}^{\ast }{b}_{1}{}_{*}\right)}^{2}\mathrm{exp}\left({r}_{1}^{\ast }{\tau }_{1}^{u}\right)}{4{a}_{1}^{\ast }{b}_{1}{}_{*}{r}_{1}{}_{*}},{r}_{1}^{\ast }{a}_{1}^{\ast }\ge {K}_{1}^{\ast }{b}_{1}{}_{*}\end{array}$.

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}y\left(t\right)\le {M}_{2}=\left\{\begin{array}{l}\frac{{r}_{2}^{\ast }{K}_{2}^{\ast }\mathrm{exp}\left({r}_{2}^{\ast }{\tau }_{2}^{u}\right)}{{r}_{2}{}_{*}},{r}_{2}^{\ast }{a}_{2}^{\ast }<{K}_{2}^{\ast }{b}_{2}{}_{*}\\ \frac{{\left({r}_{2}^{\ast }{a}_{2}^{\ast }+{K}_{2}^{\ast }{b}_{2}{}_{*}\right)}^{2}\mathrm{exp}\left({r}_{2}^{\ast }{\tau }_{2}^{u}\right)}{4{a}_{2}^{\ast }{b}_{2}{}_{*}{r}_{2}{}_{*}},{r}_{2}^{\ast }{a}_{2}^{\ast }\ge {K}_{2}^{\ast }{b}_{2}{}_{*}\end{array}$.

${r}_{i}{}_{*}-{b}_{i}^{\ast }{s}_{i}>0\text{\hspace{0.17em}}\left(i=1,2\right)$ (7)

$\frac{\left({r}_{1}{}_{*}-{b}_{1}^{\ast }{s}_{1}\left({\eta }_{1}\right)\right){K}_{1}{}_{*}\mathrm{exp}\left({r}_{1}{}_{*}-\frac{{r}_{1}^{\ast }{M}_{1}}{{K}_{1}{}_{*}}-{b}_{1}^{\ast }{s}_{1}\left({\eta }_{1}\right)\right){\tau }_{1}^{l}}{{r}_{1}^{\ast }}>{\eta }_{1}$. (8)

$\exists \phi \in {C}^{+}$，使得 $\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}x\left(t,0,\phi \right)<{\eta }_{1}$。记 $x\left(t\right)=x\left(t,0,\phi \right),y\left(t\right)=y\left(t,0,\phi \right)$。则存在 ${\epsilon }_{0}>0$，使得当 $\epsilon \in \left(0,{\epsilon }_{0}\right)$ 时， $\exists {T}_{3}>0$，当 ${T}_{3}>0$ 时有(5)与(6)成立且 $x\left(t\right)<{\eta }_{1}+\epsilon$

$\frac{\text{d}y}{\text{d}t}\le y\left(t\right)\left({r}_{2}^{\ast }+\epsilon \right)$,

$y\left(t\right)\le y\left(t-{\tau }_{2}\left(t\right)\right)\mathrm{exp}\left(\left({r}_{2}^{\ast }+\epsilon \right){\tau }_{2}^{u}\right)$. (9)

$\frac{\text{d}y}{\text{d}t}\le y\left(t\right)\left(\left({r}_{2}^{\ast }+\epsilon \right)-\frac{\left({r}_{2}{}_{*}-\epsilon \right)\mathrm{exp}\left(-\left({r}_{2}^{\ast }+\epsilon \right){\tau }_{2}^{u}\right)y\left(t\right)}{\left({K}_{2}^{\ast }+\epsilon \right)+\left({a}_{2}^{\ast }+\epsilon \right)\left({\eta }_{1}+\epsilon \right)}\right)$.

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}y\left(t\right)\le \frac{{r}_{2}^{\ast }\left({K}_{2}^{\ast }+{a}_{2}^{\ast }{\eta }_{1}\right)\mathrm{exp}\left({r}_{2}^{\ast }{\tau }_{2}^{u}\right)}{{r}_{2}{}_{*}}={s}_{1}\left({\eta }_{1}\right)$. (10)

$\frac{\text{d}x}{\text{d}t}\ge x\left(t\right)\left(\left({r}_{1}{}_{*}-\epsilon \right)-\frac{\left({r}_{1}^{\ast }+\epsilon \right){M}_{1}}{\left({K}_{1}{}_{*}-\epsilon \right)}-\left({b}_{1}^{\ast }+\epsilon \right)\left({s}_{1}\left({\eta }_{1}\right)+\epsilon \right)\right)$,

$x\left(t\right)\ge x\left(t-{\tau }_{1}\left(t\right)\right)\mathrm{exp}\left(\left({r}_{1}{}_{*}-\epsilon \right)-\frac{\left({r}_{1}^{\ast }+\epsilon \right)\left({M}_{1}+\epsilon \right)}{\left({K}_{1}{}_{*}-\epsilon \right)}-\left({b}_{1}^{\ast }+\epsilon \right)\left({s}_{1}\left({\eta }_{1}\right)+\epsilon \right)\right){\tau }_{1}^{l}$. (11)

$\frac{\text{d}x}{\text{d}t}\ge x\left(t\right)\left(\left({r}_{1}{}_{*}-\epsilon \right)-\frac{h\left(\epsilon \right)x\left(t\right)}{{K}_{1}{}_{*}-\epsilon }-\left({b}_{1}^{\ast }+\epsilon \right)\left({s}_{1}\left({\eta }_{1}\right)+\epsilon \right)\right)$,

$h\left(\epsilon \right)=\left({r}_{1}^{\ast }+\epsilon \right)\mathrm{exp}\left(-\left(\left({r}_{1}{}_{*}-\epsilon \right)-\frac{\left({r}_{1}^{\ast }+\epsilon \right)\left({M}_{1}+\epsilon \right)}{{K}_{1}{}_{*}-\epsilon }-\left({b}_{1}^{\ast }+\epsilon \right)\left({s}_{1}\left({\eta }_{1}\right)+\epsilon \right)\right){\tau }_{1}^{l}\right)$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}x\left(t\right)\ge \frac{\left({r}_{1}{}_{*}-{b}_{1}^{\ast }{s}_{1}\left({\eta }_{1}\right)\right){K}_{1}{}_{*}\mathrm{exp}\left({r}_{1}{}_{*}-\frac{{r}_{1}^{\ast }{M}_{1}}{{K}_{1}{}_{*}}-{b}_{1}^{\ast }{s}_{1}\left({\eta }_{1}\right)\right){\tau }_{1}^{l}}{{r}_{1}^{\ast }}>{\eta }_{1}$.

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}x\left(t,0,\phi \right)>{m}_{1}$. (12)

$\left({r}_{2}{}_{*}-\epsilon \right)-\frac{\left({r}_{1}^{\ast }+\epsilon \right)\epsilon }{\left({K}_{1}{}_{*}-\epsilon \right)}-\left({b}_{2}^{\ast }+\epsilon \right){\left(\frac{{r}_{1}{}_{*}-\epsilon }{\left({r}_{1}^{\ast }+\epsilon \right)\left({K}_{1}^{\ast }+\epsilon \right)\mathrm{exp}\left(\left({r}_{1}^{\ast }+\epsilon \right)\left({\tau }_{1}^{u}+\epsilon \right)\right)}\right)}^{-1}>0$. (13)

$x\left({\tau }_{k}^{\left(n\right)},0,{\phi }_{n}\right)=\frac{1}{n},\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\left({t}_{k}^{\left(n\right)},0,{\phi }_{n}\right)=\frac{1}{{n}^{2}}$. (14)

$\frac{1}{{n}^{2}}。由引理3，存在 $T\left({\phi }_{n}\right)>0$ 使得当 $t>T\left({\phi }_{n}\right)$$x\left(t,0,{\phi }_{n}\right)<2{M}_{1},y\left(t,0,{\phi }_{n}\right)<2{M}_{2}$。从而当 $t>T\left({\phi }_{n}\right)+\tau$ 时有

${x}^{\prime }\left(t,0,{\phi }_{n}\right)>x\left(t,0,{\phi }_{n}\right)\left(-\frac{2{r}_{1}^{u}{M}_{1}}{{K}_{1}^{l}}-2{b}_{1}^{u}{M}_{2}\right)\triangleq -{\beta }_{1}x\left(t,0,{\phi }_{n}\right)$. (15)

${t}_{k}^{\left(n\right)}-{\tau }_{k}^{\left(n\right)}>\frac{\mathrm{ln}n}{{\beta }_{1}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}k>{K}_{1}^{\left(n\right)}$.

${y}^{\prime }\left(t,0,{\phi }_{n}\right)\le y\left(t,0,{\phi }_{n}\right)\left(\left({r}_{2}^{*}+\epsilon \right)-\frac{\left({r}_{2}{}_{*}-\epsilon \right)\mathrm{exp}-\left({r}_{2}^{*}+\epsilon \right)\left({\tau }_{2}^{*}+\epsilon \right)}{\left({K}_{2}^{*}+\epsilon \right)+\left({a}_{2}^{*}+\epsilon \right)\frac{1}{n}}y\left(t,0,{\phi }_{n}\right)\right)$.

$\begin{array}{c}y\left(t,0,{\phi }_{n}\right)\le \left(\left(\frac{1}{y\left({\tau }_{k}^{\left(n\right)}\right)}-\frac{\left({r}_{2}{}_{*}-\epsilon \right)\mathrm{exp}-\left({r}_{2}^{*}+\epsilon \right)\left({\tau }_{2}^{*}+\epsilon \right)}{\left({r}_{2}^{*}+\epsilon \right)\left(\left({K}_{2}^{*}+\epsilon \right)+\left({a}_{2}^{*}+\epsilon \right)\frac{1}{n}\right)}\right)\mathrm{exp}-\left({r}_{2}^{*}+\epsilon \right)\left(t-{\tau }_{k}^{\left(n\right)}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\frac{\left({r}_{2}{}_{*}-\epsilon \right)\mathrm{exp}-\left({r}_{2}^{*}+\epsilon \right)\left({\tau }_{2}^{*}+\epsilon \right)}{\left({r}_{2}^{*}+\epsilon \right)\left(\left({K}_{2}^{*}+\epsilon \right)+\left({a}_{2}^{*}+\epsilon \right)\frac{1}{n}\right)}\right)}^{-1}\end{array}$ (16)

$k>{K}_{1}^{\left(n\right)}$$t-{\tau }_{k}^{\left(n\right)}>\frac{{t}_{k}^{\left(n\right)}-{\tau }_{k}^{\left(n\right)}}{2}>\frac{\mathrm{ln}n}{2{\beta }_{1}}$ 时，由于

$\underset{n\to \infty }{\mathrm{lim}}\left(\frac{1}{2{M}_{2}}-\frac{\left({r}_{2}{}_{*}-\epsilon \right)\mathrm{exp}-\left({r}_{2}^{\ast }+\epsilon \right)\left({\tau }_{2}^{\ast }+\epsilon \right)}{\left({r}_{2}^{\ast }+\epsilon \right)\left(\left({K}_{2}^{\ast }+\epsilon \right)+\left({a}_{2}^{\ast }+\epsilon \right)\frac{1}{n}\right)}\right)\frac{1}{{n}^{\frac{{r}_{2}^{*}+\epsilon }{2\beta }}}=0$.

$|\left(\frac{1}{2{M}_{2}}-\frac{\left({r}_{2}{}_{*}-\epsilon \right)\mathrm{exp}-\left({r}_{2}^{\ast }+\epsilon \right)\left({\tau }_{2}^{\ast }+\epsilon \right)}{\left({r}_{2}^{\ast }+\epsilon \right)\left(\left({K}_{2}^{\ast }+\epsilon \right)+\left({a}_{2}^{\ast }+\epsilon \right)\frac{1}{n}\right)}\right)\frac{1}{{n}^{\frac{{r}_{2}^{*}+\epsilon }{2\beta }}}|<\epsilon$. (17)

$y\left(t,0,{\phi }_{n}\right)<{\left(\frac{\left({r}_{2}{}_{*}-\epsilon \right)\mathrm{exp}-\left({r}_{2}^{\ast }+\epsilon \right)\left({\tau }_{2}^{\ast }+\epsilon \right)}{\left({r}_{2}^{\ast }+\epsilon \right)\left(\left({K}_{2}^{\ast }+\epsilon \right)+\left({a}_{2}^{\ast }+\epsilon \right)\frac{1}{n}\right)}-\epsilon \right)}^{-1}$.

${x}^{\prime }\left(t,0,{\phi }_{n}\right)>x\left(t,0,{\phi }_{n}\right)\left({r}_{1}{}_{*}-\epsilon -\frac{\left({r}_{1}^{\ast }+\epsilon \right)\epsilon }{{K}_{1}{}_{*}-\epsilon }-\left({b}_{1}^{\ast }+\epsilon \right){\left(\frac{\left({r}_{2}{}_{*}-\epsilon \right)\mathrm{exp}-\left({r}_{2}^{\ast }+\epsilon \right)\left({\tau }_{2}^{\ast }+\epsilon \right)}{\left({r}_{2}^{\ast }+\epsilon \right)\left(\left({K}_{2}^{\ast }+\epsilon \right)+\left({a}_{2}^{\ast }+\epsilon \right)\epsilon \right)}-\epsilon \right)}^{-1}\right)>0$

${r}_{1}{}_{*}-{b}_{1}^{\ast }\frac{{r}_{2}^{\ast }{K}_{2}^{\ast }\mathrm{exp}\left({r}_{2}^{\ast }{\tau }_{2}^{u}\right)}{{r}_{2}{}_{*}}>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{2}{}_{*}-{b}_{2}^{\ast }\frac{{r}_{1}^{\ast }{K}_{1}^{\ast }\mathrm{exp}\left({r}_{1}^{\ast }{\tau }_{1}^{u}\right)}{{r}_{1}{}_{*}}>0$

${b}_{1}{K}_{2}-{r}_{1}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{2}{K}_{1}-{r}_{2}<0$.

$\left\{\begin{array}{l}\frac{\text{d}x}{\text{d}t}=x\left(t\right)\left[\left(0.17+0.01\mathrm{sin}t\right)-\frac{\left(0.25+0.01\mathrm{sin}t\right)x\left(t-{\text{e}}^{-5\pi +\pi \mathrm{sin}t}\right)}{0.9+0.1\mathrm{cos}t+\left(0.4+0.1\mathrm{sin}t\right)y\left(t\right)}-\left(0.11+0.01\mathrm{sin}t\right)y\left(t\right)\right]\\ \frac{\text{d}y}{\text{d}t}=y\left(t\right)\left[\left(0.22+0.01\mathrm{sin}t\right)-\frac{\left(0.22+0.01\mathrm{sin}t\right)y\left(t-{\text{e}}^{-6\pi +\pi \mathrm{cos}t}\right)}{1+0.1\mathrm{cos}t+\left(0.4+0.1\mathrm{sin}t\right)x\left(t\right)}-\left(0.15+0.01\mathrm{cos}t\right)x\left(t\right)\right]\end{array}$

${r}_{1}^{*}=0.18,{r}_{1}{}_{*}=0.16,{r}_{2}^{*}=0.23,{r}_{2}{}_{*}=0.21，$

${b}_{1}^{*}=0.12,{b}_{2}^{*}=0.16,{K}_{1}^{*}=1,{K}_{2}^{*}=1.1,{\tau }_{1}^{u}={\text{e}}^{-4\pi },{\tau }_{2}^{u}={\text{e}}^{-5\pi }.$

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