﻿ 一类具有时滞效应与脉冲控制的捕食–食饵系统的动力学研究 Dynamic Analysis of a Predator-Prey System with Time Delay and Pulse Control

Vol.07 No.08(2018), Article ID:26534,13 pages
10.12677/AAM.2018.78116

Dynamic Analysis of a Predator-Prey System with Time Delay and Pulse Control

Jing Yang

College of Mathematics Physics and Electronic Information Engineering, Wenzhou University, Wenzhou Zhejiang

Received: Aug. 1st, 2018; accepted: Aug. 15th, 2018; published: Aug. 22nd, 2018

ABSTRACT

Based on theory of ecological dynamics and ecological control modeling idea, a predator-prey dynamic system with time-delay and pulse control is constructed. The sufficient criterion of local asymptotic stability and global attraction for the semi-trivial periodic solution of the system is established. The permanence of the system is proved. Further, the dynamic behavior of the model was simulated numerically, which verifies the validity and feasibility of theoretical analysis.

Keywords:Pulse Control, Time Delay, Stability, Global Attraction, Persistence

1. 引言

$\left\{\begin{array}{l}{\stackrel{˙}{N}}_{i}\left(t\right)=B\left(t\right)-{D}_{i}\left(t\right)-W\left(t\right)\\ {\stackrel{˙}{N}}_{m}\left(t\right)=\alpha W\left(t\right)-{D}_{m}\left(t\right)\end{array}$ (1)

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\beta y\left(t\right)-rx\left(t\right)-\beta {\text{e}}^{-r\tau }y\left(t-\tau \right)\\ \stackrel{˙}{y}\left(t\right)=\beta {\text{e}}^{-r\tau }y\left(t-\tau \right)-{\eta }_{2}{y}^{2}\left(t\right)\end{array}$ (2)

$\stackrel{˙}{N}\left(t\right)=N\left(t\right)\left(r-aN\left(t\right)\right)$ (3)

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\left[r-a\left(x\left(t\right)+y\left(t\right)\right)\right]x\left(t\right)-r{\text{e}}^{-{d}_{1}\tau }y\left(t-\tau \right)-{d}_{1}x\left(t\right)\\ \stackrel{˙}{y}\left(t\right)=r{\text{e}}^{-{d}_{1}\tau }y\left(t-\tau \right)+\left[r-{d}_{2}-a\left(x\left(t\right)+y\left(t\right)\right)\right]y\left(t\right)-by\left(t\right)z\left(t\right)\\ \stackrel{˙}{z}\left(t\right)=kby\left(t\right)z\left(t\right)-{d}_{3}z\left(t\right)\end{array}$ (4)

$\left\{\begin{array}{l}\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\left[r-a\left(x\left(t\right)+y\left(t\right)\right)\right]x\left(t\right)-r{\text{e}}^{-{d}_{1}\tau }y\left(t-\tau \right)-{d}_{1}x\left(t\right)\\ \stackrel{˙}{y}\left(t\right)=r{\text{e}}^{-{d}_{1}\tau }y\left(t-\tau \right)+\left[r-{d}_{2}-a\left(x\left(t\right)+y\left(t\right)\right)\right]y\left(t\right)-by\left(t\right)z\left(t\right)\\ \stackrel{˙}{z}\left(t\right)=kby\left(t\right)z\left(t\right)-{d}_{3}z\left(t\right)\end{array}\right\}t\ne nT\\ \begin{array}{l}\Delta x\left(t\right)=0\\ \Delta y\left(t\right)=0\\ \Delta z\left(t\right)=R\end{array}\right\}t=nT\end{array}$ (5)

$\left\{\begin{array}{l}\left({\varphi }_{1},{\varphi }_{2},\psi \right)\in \left(\left[-\tau ,0\right],{R}_{3}^{+}\right),\\ x\left(t\right)={\varphi }_{1}\left(t\right),y\left(t\right)={\varphi }_{2}\left(t\right),z\left(t\right)=\psi \left(t\right),\\ {R}_{3}^{+}=\left\{x\in {R}^{3}:x\ge 0\right\}.\end{array}$ (6)

$D=\left\{\left(x\left(t\right),y\left(t\right),z\left(t\right)\right):x\left(t\right)>0,y\left(t\right)>0,z\left(t\right)>0\right\}$ (7)

2. 定理

2) V对于X满足局部的Lipschitz条件。

$\left\{\begin{array}{l}\stackrel{˙}{w}\left(t\right)\le p\left(t\right)w\left(t\right)+q\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ne {\tau }_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0\\ w\left({t}^{+}\right)\le {p}_{i}w\left(t\right)+{q}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t=n{\tau }_{i}>0\\ u\left({0}^{+}\right)\le {w}_{0},\end{array}$ (8)

$\begin{array}{l}w\left(t\right)\le w\left(0\right)\frac{\Pi }{0<{\tau }_{i}

$\left\{\begin{array}{l}\stackrel{˙}{h}\left(t\right)=-{d}_{3}h\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ne nT\\ h\left({t}^{+}\right)=h\left(t\right)+R\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t=nT\\ h\left({0}^{+}\right)={h}_{0}\end{array}$ (9)

${h}^{*}\left(t\right)=\frac{R{e}^{-{d}_{3}\left(t-nT\right)}}{1-{\text{e}}^{-{d}_{3}T}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left(nT,\left(n+1\right)T\right]$ .

$\frac{\text{d}u}{\text{d}t}=au\left(t-\tau \right)-b\left(t\right)u\left(t\right)$ , (10)

1) 若 $a>b$ ，则 $\underset{t\to \infty }{\mathrm{lim}}u\left(t\right)=+\infty$

2) 若 $a\le b$ ，则 $\underset{t\to \infty }{\mathrm{lim}}u\left(t\right)=0$

$\frac{\text{d}u}{\text{d}t}=au\left(t-\tau \right)-bu\left(t\right)-c{u}^{2}\left(t\right)$ , (11)

1) 若 $a>b$ ，则 $\underset{t\to \infty }{\mathrm{lim}}u\left(t\right)=\frac{a-b}{c}$

2) 若 $a\le b$ ，则 $\underset{t\to \infty }{\mathrm{lim}}u\left(t\right)=0$

3. 理论分析

3.1. 系统(5)任意解的有界性

$t\ne nT$ 时，

$\begin{array}{l}{D}^{+}V\left(t\right)=k\left[r-a\left(x\left(t\right)+y\left(t\right)\right)\right]x\left(t\right)-kr{\text{e}}^{-{d}_{1}\tau }y\left(t-\tau \right)-k{d}_{1}x\left(t\right)+kr{\text{e}}^{-{d}_{1}\tau }y\left(t-\tau \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+k\left[r-{d}_{2}-a\left(x\left(t\right)+y\left(t\right)\right)\right]y\left(t\right)-kby\left(t\right)z\left(t\right)+kby\left(t\right)z\left(t\right)-{d}_{3}z\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=k\left[-a{x}^{2}\left(t\right)+\left(r-{d}_{1}+{d}_{3}\right)x\left(t\right)\right]-2kax\left(t\right)y\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+k\left[-a{y}^{2}\left(t\right)+\left(r-{d}_{2}+{d}_{3}\right)y\left(t\right)\right]-{d}_{3}V\left(t\right)\end{array}$

${D}^{+}V\left(t\right)+{d}_{3}V\left(t\right)\le \frac{k{\left(r-{d}_{1}+{d}_{3}\right)}^{2}}{4a}+\frac{k{\left(r-{d}_{2}+{d}_{3}\right)}^{2}}{4a}=\frac{k\left[{\left(r-{d}_{1}+{d}_{3}\right)}^{2}+{\left(r-{d}_{2}+{d}_{3}\right)}^{2}\right]}{4a}={M}_{0}$

$\left\{\begin{array}{l}{D}^{+}V\left(t\right)\le -{d}_{3}V\left(t\right)+{M}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ne nT\\ V\left({t}^{+}\right)=V\left(t\right)+R\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t=nT\end{array}$ (12)

$\begin{array}{l}V\left(t\right)\le \left[V\left({0}^{+}\right)-\frac{{M}_{0}}{{d}_{3}}\right]{\text{e}}^{-{d}_{3}t}+\frac{R{e}^{-{d}_{3}\left(t-nT\right)}\left(1-{\text{e}}^{-{d}_{3}nT}\right)}{1-{\text{e}}^{-{d}_{3}T}}+\frac{{M}_{0}}{{d}_{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left(nT,\left(n+1\right)T\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\le \frac{R{e}^{{d}_{3}T}}{1-{\text{e}}^{-{d}_{3}T}}+\frac{{M}_{0}}{{d}_{3}}\triangleq M\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(t\to \infty \right)\end{array}$ (13)

3.2. 系统(5)半平凡周期解的局部渐近稳定性

$\left\{\begin{array}{l}\begin{array}{l}\stackrel{˙}{H}\left(t\right)=\left(r-{d}_{1}\right)H\left(t\right)-r{\text{e}}^{-{d}_{1}\tau }P\left(t\right)\\ \stackrel{˙}{P}\left(t\right)=\left[r-{d}_{2}-b{z}^{*}\left(t\right)+r{\text{e}}^{-{d}_{1}\tau }\right]P\left(t\right)\\ \stackrel{˙}{W}\left(t\right)=kb{z}^{*}\left(t\right)P\left(t\right)-{d}_{3}W\left(t\right)\end{array}\right\}t\ne nT\\ \begin{array}{l}H\left({t}^{+}\right)=H\left(t\right)\\ P\left({t}^{+}\right)=P\left(t\right)\\ W\left({t}^{+}\right)=W\left(t\right)+R\end{array}\right\}t=nT\end{array}$ (14)

$\frac{\text{d}\varphi \left(t\right)}{\text{d}t}=A\left(t\right)\varphi \left(t\right)$ (15)

$t=nT$ 时， $\left[\begin{array}{c}H\left(n{T}^{+}\right)\\ P\left(n{T}^{+}\right)\\ W\left(n{T}^{+}\right)\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}H\left(nT\right)\\ P\left(nT\right)\\ W\left(nT\right)\end{array}\right]$

$\Phi \left(T\right)=\varphi \left(0\right){\text{e}}^{\underset{0}{\overset{T}{\int }}A\left(s\right)\text{d}s}\triangleq \varphi \left(0\right)\mathrm{exp}\left(\stackrel{˜}{A}\right),\text{\hspace{0.17em}}\stackrel{˜}{A}=\underset{0}{\overset{T}{\int }}A\left(s\right)\text{d}s$ (16)

${\lambda }_{1}={\text{e}}^{\underset{0}{\overset{T}{\int }}\left(r-{d}_{1}\right)\text{d}s},\text{\hspace{0.17em}}{\lambda }_{2}={\text{e}}^{\underset{0}{\overset{T}{\int }}\left[r-{d}_{2}-b{z}^{*}\left(s\right)+r{\text{e}}^{-{d}_{1}\tau }\right]\text{d}s},\text{\hspace{0.17em}}{\lambda }_{3}={\text{e}}^{\underset{0}{\overset{T}{\int }}\left(-{d}_{3}\right)\text{d}s}={\text{e}}^{-{d}_{3}T}<1$ (17)

$z\left(t\right)=\left[{z}_{0}-\frac{R}{1-{\text{e}}^{-{d}_{3}T}}\right]{\text{e}}^{-{d}_{3}t}+{z}^{*}\left(t\right),\text{\hspace{0.17em}}t\in \left(nT,\left(n+1\right)T\right]$ . (18)

3.3. 系统(5)半平凡周期解的全局吸引性

(19)

(20)

. (21)

(22)

(23)

(24)

(25)

(26)

(27)

3.4. 系统(5)的持久生存性

，知对任意的，存在，使得

(28)

(29)

(H1)：假设对上面的，对任意的，当时，有不恒成立，则可以假设存在，使得当时有成立。

(30)

(31)

(32)

(33)

(34)

(35)

(H2)：根据第一个假设，只需要考虑两种情况，

1)对所有的充分大的t，都有成立；

2)当t充分大时，关于左右波动。

(36)

(37)

4. 数值模拟

5. 结论

Figure 1. Time diagram and phase diagram of system (5) when

Figure 2. Time diagram and phase diagram of system (5) when

Dynamic Analysis of a Predator-Prey System with Time Delay and Pulse Control[J]. 应用数学进展, 2018, 07(08): 987-999. https://doi.org/10.12677/AAM.2018.78116

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