﻿ 一类具有连续分布时滞的Nicholson飞蝇模型正周期解的全局指数稳定性 The Global Exponential Stability of the Positive Periodic Solution for a Class of Nicholson’s Blowflies Model with Continuously Distributed Delays

Vol. 08  No. 05 ( 2019 ), Article ID: 30568 , 9 pages
10.12677/AAM.2019.85115

The Global Exponential Stability of the Positive Periodic Solution for a Class of Nicholson’s Blowflies Model with Continuously Distributed Delays

Qiufeng Chen, Jianli Li

School of Mathematics and Statistics, Hunan Normal University, Changsha Hunan

Received: May 9th, 2019; accepted: May 24th, 2019; published: May 31st, 2019

ABSTRACT

In this paper, we study the existence of positive periodic solution for Nicholson’s blowflies system with continuously distributed delay. Under appropriate conditions, we obtain that the system has unique positive periodic solution and its global exponential stability.

Keywords:Nicholson’s Blowflies System, Continuous Distributed Delays, Periodic Solution, Global Exponential Stability

1. 引言

${x}^{\prime }\left(t\right)=-\frac{a\left(t\right)x\left(t\right)}{b\left(t\right)+x\left(t\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(t\right){\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)x\left(t-s\right){\text{e}}^{-x\left(t-s\right)}\text{d}s$ (1.1)

$C=C\left(\left[-r,0\right],R\right)$$\left[-r,0\right]$ 上全体连续函数的集合组成的Banach空间，赋予上确界范数 $‖\text{ }\cdot \text{ }‖$${C}^{+}=C\left(\left[-r,0\right],{R}^{+}\right)$，定义 ${x}_{t}\left(\theta \right)=x\left(t+\theta \right)$$\theta \in \left[-r,0\right]$，初始条件为 ${x}_{{t}_{0}}=\phi$$\phi \in {C}^{+}$$\phi \left(0\right)>0$

$\frac{1-k}{{\text{e}}^{k}}=\frac{1}{{\text{e}}^{2}}$ (1.2)

$\underset{x\ge k}{\mathrm{sup}}|\frac{1-x}{{\text{e}}^{x}}|=\frac{1}{{\text{e}}^{2}}$ (1.3)

$x{\text{e}}^{-x}$$\left[0,1\right]$ 上单调递增，在 $\left[1,+\infty \right)$ 上单调递减，则存在唯一的 $\stackrel{˜}{k}\in \left(1,+\infty \right)$，使得

$k{\text{e}}^{-k}=\stackrel{˜}{k}{\text{e}}^{-\stackrel{˜}{k}}$ (1.4)

$\left\{\begin{array}{l}\underset{t\in R}{\mathrm{sup}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right)\left(b\left(t\right)+K\right)}{a\left(t\right)K\text{e}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}s<1,\\ \underset{t\in R}{\mathrm{inf}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right)b\left(t\right)}{a\left(t\right){\text{e}}^{k}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}s>1,\end{array}$ (1.5)

$\begin{array}{c}{x}^{\prime }\left(t\right)=-\frac{a\left(t\right)x\left(t\right)}{b\left(t\right)+x\left(t\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(t\right){\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)x\left(t-s\right){\text{e}}^{-x\left(t-s\right)}\text{d}s\\ \ge -\frac{a\left(t\right)x\left(t\right)}{b\left(t\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(t\right){\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)x\left(t-s\right){\text{e}}^{-x\left(t-s\right)}\text{d}s\end{array}$

$\begin{array}{c}x\left(t\right)\ge {\text{e}}^{-{\int }_{{t}_{o}}^{t}\frac{a\left(s\right)}{b\left(s\right)}\text{d}s}\underset{{t}_{0}}{\overset{t}{\int }}{\text{e}}^{{\int }_{{t}_{o}}^{s}\frac{a\left(\tau \right)}{b\left(\tau \right)}\text{d}\tau }\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(s\right){\int }_{0}^{{\sigma }_{j}\left(s\right)}{K}_{j}\left(\tau \right)x\left(s-\tau \right){\text{e}}^{-x\left(s-\tau \right)}\text{d}\tau \text{d}s+x\left({t}_{0}\right){\text{e}}^{-{\int }_{{t}_{o}}^{t}\frac{a\left(s\right)}{b\left(s\right)}\text{d}s}\\ >0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall t\in \left[{t}_{0},\eta \left(\phi \right)\right)\end{array}$

$N\left(t\right)=\mathrm{max}\left\{\gamma :\gamma \le t,x\left(\gamma \right)=\underset{{t}_{0}-r\le s\le t}{\mathrm{max}}x\left(s\right)\right\}$$\forall t\in \left[{t}_{0},\eta \left(\phi \right)\right)$

$0\le {x}^{\prime }\left(N\left(t\right)\right)=-\frac{a\left(N\left(t\right)\right)x\left(N\left(t\right)\right)}{b\left(N\left(t\right)\right)+x\left(N\left(t\right)\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(N\left(t\right)\right){\int }_{0}^{{\sigma }_{j}\left(N\left(t\right)\right)}{K}_{j}\left(s\right)x\left(N\left(t\right)-s\right){\text{e}}^{-x\left(N\left(t\right)-s\right)}\text{d}s$

$\frac{a\left(N\left(t\right)\right)x\left(N\left(t\right)\right)}{b\left(N\left(t\right)\right)+x\left(N\left(t\right)\right)}\le \underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(N\left(t\right)\right){\int }_{0}^{{\sigma }_{j}\left(N\left(t\right)\right)}{K}_{j}\left(s\right)x\left(N\left(t\right)-s\right){\text{e}}^{-x\left(N\left(t\right)-s\right)}\text{d}s$

$\underset{x\ge 0}{\mathrm{sup}}x{\text{e}}^{-x}=\frac{1}{\text{e}}$，有

$\frac{a\left(N\left({t}_{n}\right)\right)x\left(N\left({t}_{n}\right)\right)}{b\left(N\left({t}_{n}\right)\right)+x\left(N\left({t}_{n}\right)\right)}\le \underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(N\left({t}_{n}\right)\right){\int }_{0}^{{\sigma }_{j}\left(N\left({t}_{n}\right)\right)}{K}_{j}\left(s\right)\text{d}s\frac{1}{\text{e}}$

$\begin{array}{c}\frac{x\left(N\left({t}_{n}\right)\right)}{b\left(N\left({t}_{n}\right)\right)+x\left(N\left({t}_{n}\right)\right)}\le \underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(N\left({t}_{n}\right)\right)}{a\left(N\left({t}_{n}\right)\right)\text{e}}{\int }_{0}^{{\sigma }_{j}\left(N\left({t}_{n}\right)\right)}{K}_{j}\left(s\right)\text{d}s\\ \le \underset{t\in R}{\mathrm{sup}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right)}{a\left(t\right)\text{e}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}s\end{array}$

$n\to +\infty$，有

$1\le \underset{t\in R}{\mathrm{sup}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right)}{a\left(t\right)\text{e}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}s$

$\begin{array}{c}0\le {x}^{\prime }\left({t}_{2}\right)=-\frac{a\left({t}_{2}\right)x\left({t}_{2}\right)}{b\left({t}_{2}\right)+x\left({t}_{2}\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left({t}_{2}\right){\int }_{0}^{{\sigma }_{j}\left({t}_{2}\right)}{K}_{j}\left(s\right)x\left({t}_{2}-s\right){\text{e}}^{-x\left({t}_{2}-s\right)}\text{d}s\\ \le -\frac{a\left({t}_{2}\right)K}{b\left({t}_{2}\right)+K}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left({t}_{2}\right){\int }_{0}^{{\sigma }_{j}\left({t}_{2}\right)}{K}_{j}\left(s\right)\text{d}s\frac{1}{\text{e}}\end{array}$

$1\le \underset{t\in R}{\mathrm{sup}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left({t}_{2}\right)\left(b\left({t}_{2}\right)+K\right)}{a\left({t}_{2}\right)K\text{e}}{\int }_{0}^{{\sigma }_{j}\left({t}_{2}\right)}{K}_{j}\left(s\right)\text{d}s\le \underset{t\in R}{\mathrm{sup}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right)\left(b\left(t\right)+K\right)}{a\left(t\right)K\text{e}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}s$

$h\left(t\right)=\mathrm{max}\left\{\gamma :\gamma \le t,x\left(\gamma \right)=\underset{{t}_{0}\le s\le t}{\mathrm{min}}x\left(s\right)\right\}$

$\underset{t\to +\infty }{\mathrm{lim}}h\left(t\right)=+\infty$$\underset{t\to +\infty }{\mathrm{lim}}x\left(h\left(t\right)\right)=0$ (1.6)

$h\left(t\right)$ 的定义，有 $x\left(h\left(t\right)\right)=\underset{{t}_{0}\le s\le t}{\mathrm{min}}x\left(s\right)$${x}^{\prime }\left(h\left(t\right)\right)\le 0$$h\left(t\right)>{t}_{0}$。因此，

$\begin{array}{c}0\ge {x}^{\prime }\left(h\left(t\right)\right)=-\frac{a\left(h\left(t\right)\right)x\left(h\left(t\right)\right)}{b\left(h\left(t\right)\right)+x\left(h\left(t\right)\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(h\left(t\right)\right){\int }_{0}^{{\sigma }_{j}\left(h\left(t\right)\right)}{K}_{j}\left(s\right)x\left(h\left(t\right)-s\right){\text{e}}^{-x\left(h\left(t\right)-s\right)}\text{d}s\\ \ge -\frac{a\left(h\left(t\right)\right)x\left(h\left(t\right)\right)}{b\left(h\left(t\right)\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(h\left(t\right)\right){\int }_{0}^{{\sigma }_{j}\left(h\left(t\right)\right)}{K}_{j}\left(s\right)x\left(h\left(t\right)-s\right){\text{e}}^{-x\left(h\left(t\right)-s\right)}\text{d}s\end{array}$

$\begin{array}{c}\frac{a\left(h\left(t\right)\right)x\left(h\left(t\right)\right)}{b\left(h\left(t\right)\right)+x\left(h\left(t\right)\right)}\ge \underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(h\left(t\right)\right){\int }_{0}^{{\sigma }_{j}\left(h\left(t\right)\right)}{K}_{j}\left(s\right)x\left(h\left(t\right)-s\right){\text{e}}^{-x\left(h\left(t\right)-s\right)}\text{d}s\\ =\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(h\left(t\right)\right){\int }_{0}^{{\sigma }_{j}\left(h\left(t\right)\right)}{K}_{j}\left(s\right)\text{d}sx\left(h\left(t\right)-{\tau }_{j}\left(h\left(t\right)\right)\right){\text{e}}^{-x\left(h\left(t\right)-{\tau }_{j}\left(h\left(t\right)\right)\right)}\end{array}$

$\left\{\begin{array}{l}{t}_{n}\to +\infty ,x\left(h\left({t}_{n}\right)\right)\to 0,a\left(h\left({t}_{n}\right)\right)\to {a}^{\ast }\\ b\left(h\left({t}_{n}\right)\right)\to {b}^{\ast },{\beta }_{j}\left(h\left({t}_{n}\right)\right)\to {\beta }_{j}^{\ast },\\ {\sigma }_{j}\left(h\left({t}_{n}\right)\right)\to {\sigma }_{j}^{\ast },{\tau }_{j}\left(h\left({t}_{n}\right)\right)\to {\tau }_{j}^{\ast }\end{array}$

$\begin{array}{c}\frac{a\left(h\left({t}_{n}\right)\right)}{b\left(h\left({t}_{n}\right)\right)+x\left(h\left({t}_{n}\right)\right)}\ge \underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(h\left({t}_{n}\right)\right){\int }_{0}^{{\sigma }_{j}\left(h\left({t}_{n}\right)\right)}{K}_{j}\left(s\right)\text{d}s\frac{x\left(h\left({t}_{n}\right)-{\tau }_{j}\left(h\left({t}_{n}\right)\right)\right)}{x\left(h\left({t}_{n}\right)\right)}{\text{e}}^{-x\left(h\left({t}_{n}\right)-{\tau }_{j}\left(h\left({t}_{n}\right)\right)\right)}\\ \ge \underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(h\left(t\right)\right){\int }_{0}^{{\sigma }_{j}\left(h\left(t\right)\right)}{K}_{j}\left(s\right)\text{d}s\text{ }{\text{e}}^{-x\left(h\left({t}_{n}\right)-{\tau }_{j}\left(h\left({t}_{n}\right)\right)\right)}\end{array}$

$1\ge \underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(h\left({t}_{n}\right)\right)\left(b\left(h\left({t}_{n}\right)\right)+x\left(h\left({t}_{n}\right)\right)\right)}{a\left(h\left({t}_{n}\right)\right)}{\int }_{0}^{{\sigma }_{j}\left(h\left({t}_{n}\right)\right)}{K}_{j}\left(s\right)\text{d}s\text{ }{\text{e}}^{-x\left(h\left({t}_{n}\right)-{\tau }_{j}\left(h\left({t}_{n}\right)\right)\right)}$

$n\to +\infty$，有

$\begin{array}{c}1\ge \underset{n\to +\infty }{\mathrm{lim}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(h\left({t}_{n}\right)\right)\left(b\left(h\left({t}_{n}\right)\right)+x\left(h\left({t}_{n}\right)\right)\right)}{a\left(h\left({t}_{n}\right)\right)}{\int }_{0}^{{\sigma }_{j}\left(h\left({t}_{n}\right)\right)}{K}_{j}\left(s\right)\text{d}s\text{ }{\text{e}}^{-x\left(h\left({t}_{n}\right)-{\tau }_{j}\left(h\left({t}_{n}\right)\right)\right)}\\ \ge \underset{t\in R}{\mathrm{inf}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right)b\left(t\right)}{a\left(t\right)}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}s\end{array}$

${x}^{\prime }\left(t\right)=-\frac{a\left(t\right)x\left(t\right)}{b\left(t\right)+x\left(t\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(t\right){\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}sx\left(t-{\tau }_{j}\left(t\right)\right){\text{e}}^{-x\left(t-{\tau }_{j}\left(t\right)\right)}$

${x}^{\prime }\left({t}_{i}\right)=-\frac{a\left({t}_{i}\right)x\left({t}_{i}\right)}{b\left({t}_{i}\right)+x\left({t}_{i}\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left({t}_{i}\right){\int }_{0}^{{\sigma }_{j}\left({t}_{i}\right)}{K}_{j}\left(s\right)\text{d}sx\left({t}_{i}-{\tau }_{j}\left({t}_{i}\right)\right){\text{e}}^{-x\left({t}_{i}-{\tau }_{j}\left({t}_{i}\right)\right)}$

${t}_{i}\to +\infty$ 时，有

$\begin{array}{c}0=-\frac{{a}^{\ast }c}{{b}^{\ast }+c}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}^{\ast }{\int }_{0}^{{\sigma }_{j}^{\ast }}{K}_{j}\left(s\right)\text{d}s\text{ }{\phi }^{\ast }\left(-{\tau }_{j}^{\ast }\right){\text{e}}^{-{\phi }^{\ast }\left(-{\tau }_{j}^{\ast }\right)}\\ \ge -\frac{{a}^{\ast }c}{{b}^{\ast }}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}^{\ast }{\int }_{0}^{{\sigma }_{j}^{\ast }}{K}_{j}\left(s\right)\text{d}s\text{ }c{\text{e}}^{-c}\end{array}$

$\underset{t\in R}{\mathrm{sup}}\left\{\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right){\left(b\left(t\right)+K\right)}^{2}}{a\left(t\right)b\left(t\right){\text{e}}^{2}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right){\text{e}}^{s}\text{d}s\right\}<1$ (1.7)

$x\left(t\right)=x\left(t;{t}_{0},\phi \right)$${x}^{\ast }\left(t\right)=x\left(t;{t}_{0},{\phi }^{\ast }\right)$，则存在正常数 $\lambda$${t}^{\ast }$ 使得

$|x\left(t;{t}_{0},\phi \right)-x\left(t;{t}_{0},{\phi }^{\ast }\right)|\le {L}_{\phi ,{\phi }^{\ast }}{\text{e}}^{-\lambda t}$$t\ge {t}^{\ast }$

${L}_{\phi ,{\phi }^{\ast }}={\text{e}}^{\lambda {t}^{\ast }}\left(\underset{t\in \left[{t}_{0}-r,{t}^{\ast }\right]}{\mathrm{max}}|x\left(t\right)-{x}^{\ast }\left(t\right)|+1\right)$

$\begin{array}{c}{y}^{\prime }\left(t\right)=-\left[\frac{a\left(t\right)x\left(t\right)}{b\left(t\right)+x\left(t\right)}-\frac{a\left(t\right){x}^{\ast }\left(t\right)}{b\left(t\right)+{x}^{\ast }\left(t\right)}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(t\right){\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\left[x\left(t-s\right){\text{e}}^{-x\left(t-s\right)}-{x}^{\ast }\left(t-s\right){\text{e}}^{-{x}^{\ast }\left(t-s\right)}\right]\text{d}s\end{array}$

$\begin{array}{c}{D}^{-}\left(V\left(t\right)\right)=\lambda |y\left(t\right)|{\text{e}}^{\lambda t}-{\text{e}}^{\lambda t}\frac{a\left(t\right)x\left(t\right)}{\left(b\left(t\right)+x\left(t\right)\right)\left(b\left(t\right)+{x}^{\ast }\left(t\right)\right)}|x\left(t\right)-{x}^{\ast }\left(t\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\mathrm{sgn}\left(x\left(t\right)-{x}^{\ast }\left(t\right)\right){\text{e}}^{\lambda t}\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(t\right){\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\left[x\left(t-s\right){\text{e}}^{-x\left(t-s\right)}-{x}^{\ast }\left(t-s\right){\text{e}}^{-{x}^{\ast }\left(t-s\right)}\right]\text{d}s\end{array}$

$\underset{t\in R}{\mathrm{sup}}\left\{\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right){\left(b\left(t\right)+K\right)}^{2}}{\left(a\left(t\right)b\left(t\right)-\lambda {\left(b\left(t\right)+K\right)}^{2}\right){\text{e}}^{2}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right){\text{e}}^{s}\text{d}s\right\}<\mu <1$

$\begin{array}{c}0\le {D}^{-}\left(V\left({t}^{\ast }\right)\right)=\lambda V\left({t}^{\ast }\right)-\frac{a\left({t}^{\ast }\right)b\left({t}^{\ast }\right)}{\left(b\left({t}^{\ast }\right)+x\left({t}^{\ast }\right)\right)\left(b\left({t}^{\ast }\right)+{x}^{\ast }\left({t}^{\ast }\right)\right)}V\left({t}^{\ast }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\mathrm{sgn}\left(x\left({t}^{\ast }\right)-{x}^{\ast }\left({t}^{\ast }\right)\right){\text{e}}^{\lambda {t}^{\ast }}\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left({t}^{\ast }\right){\int }_{0}^{{\sigma }_{j}\left({t}^{\ast }\right)}{K}_{j}\left(s\right)\left[x\left({t}^{\ast }-s\right){\text{e}}^{-x\left({t}^{\ast }-s\right)}-{x}^{\ast }\left({t}^{\ast }-s\right){\text{e}}^{-{x}^{\ast }\left({t}^{\ast }-s\right)}\right]\text{d}s\end{array}$

$\begin{array}{l}\left(\frac{a\left({t}^{\ast }\right)b\left({t}^{\ast }\right)}{\left(b\left({t}^{\ast }\right)+x\left({t}^{\ast }\right)\right)\left(b\left({t}^{\ast }\right)+{x}^{\ast }\left({t}^{\ast }\right)\right)}-\lambda \right){L}_{\phi ,{\phi }^{\ast }}\\ \le \underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left({t}^{\ast }\right){\int }_{0}^{{\sigma }_{j}\left({t}^{\ast }\right)}{K}_{j}\left(s\right){\text{e}}^{\lambda s}V\left({t}^{\ast }-s\right)\text{d}s\frac{1}{{\text{e}}^{2}}\\ \le \underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left({t}^{\ast }\right){\int }_{0}^{{\sigma }_{j}\left({t}^{\ast }\right)}{K}_{j}\left(s\right){\text{e}}^{s}\text{d}s\frac{1}{{\text{e}}^{2}}{L}_{\phi ,{\phi }^{\ast }}\end{array}$

$1\le \underset{t\in R}{\mathrm{sup}}\left\{\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right){\left(b\left(t\right)+K\right)}^{2}}{\left(a\left(t\right)b\left(t\right)-\lambda {\left(b\left(t\right)+K\right)}^{2}\right){\text{e}}^{2}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right){\text{e}}^{s}\text{d}s\right\}$

2. 周期解及全局指数稳定性

$\begin{array}{l}{\left[x\left(t+\left(q+1\right)T\right)\right]}^{\prime }=\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(t\right){\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)x\left(t+\left(q+1\right)T-s\right){\text{e}}^{-x\left(t+\left(q+1\right)T-s\right)}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{a\left(t\right)x\left(t+\left(q+1\right)T\right)}{b\left(t\right)+x\left(t+\left(q+1\right)T\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t+\left(q+1\right)T\in \left[{t}_{0},+\infty \right)\end{array}$

$\begin{array}{l}|x\left(t+\left(q+1\right)T;{t}_{0},\phi \right)-x\left(t+qT;{t}_{0},\phi \right)|\\ =|x\left(t+qT;{t}_{0},\psi \right)-x\left(t+qT;{t}_{0},\phi \right)|\\ \le {L}_{\phi ,\psi }{\text{e}}^{-\lambda \left(t+qT\right)}\end{array}$

${L}_{\phi ,\psi }={\text{e}}^{\lambda Q}\left(\underset{t\in \left[{t}_{0}-r,Q\right]}{\mathrm{max}}|x\left(t\right)-{x}^{\ast }\left(t\right)|+1\right)$

$\left[a,b\right]\subset R$ 是R中的任意区间，选择非负整数 ${p}_{0}$，使得当 $t\in \left[a,b\right]$ 时，有 $t+{p}_{0}T\ge Q$，则对 $\forall t\in \left[a,b\right]$$p>{p}_{0}$，有

$x\left(t+pT\right)=x\left(t+{p}_{0}T\right)+\underset{q={p}_{0}}{\overset{p-1}{\sum }}\left[x\left(t+\left(q+1\right)T\right)-x\left(t+qT\right)\right]$

$\left[a,b\right]$ 的任意性可知， ${\left\{x\left(t+pT\right)\right\}}_{p}$ 在R上内闭一致收敛到 ${x}^{\ast }\left(t\right)$，且 $k\le {x}^{\ast }\left(t\right)\le K$$\forall t\in R$。取极限有

$\begin{array}{c}{x}^{\ast }\left(t+T\right)=\underset{p\to +\infty }{\mathrm{lim}}x\left(\left(t+T\right)+pT\right)\\ =\underset{p+1\to +\infty }{\mathrm{lim}}x\left(t+\left(p+1\right)T\right)\\ ={x}^{\ast }\left(t\right)\end{array}$

$\begin{array}{l}x\left(t+pT\right)-x\left({t}_{0}+pT\right)\\ ={\int }_{{t}_{0}}^{t}\left[-\frac{a\left(s\right)x\left(s+pT\right)}{b\left(s\right)+x\left(s+pT\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(s\right){\int }_{0}^{{\sigma }_{j}\left(s\right)}{K}_{j}\left(u\right)x\left(s+pT-u\right){\text{e}}^{-x\left(s+pT-u\right)}\text{d}u\right]\text{d}s\end{array}$

$p\to +\infty$，有

$\begin{array}{l}{x}^{\ast }\left(t\right)-{x}^{\ast }\left({t}_{0}\right)\\ ={\int }_{{t}_{0}}^{t}\left[-\frac{a\left(s\right){x}^{\ast }\left(s\right)}{b\left(s\right)+{x}^{\ast }\left(s\right)}+\underset{j=1}{\overset{m}{\sum }}{\beta }_{j}\left(s\right){\int }_{0}^{{\sigma }_{j}\left(s\right)}{K}_{j}\left(u\right){x}^{\ast }\left(s-u\right){\text{e}}^{-{x}^{\ast }\left(s-u\right)}\text{d}u\right]\text{d}s\end{array}$

3. 举例应用

$\begin{array}{c}{x}^{\prime }\left(t\right)=-\frac{\left(16+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\right)x\left(t\right)}{12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)+x\left(t\right)}+\frac{80+{\mathrm{cos}}^{2}\left(t\right)}{100}{\int }_{0}^{1}{\text{e}}^{s}x\left(t-s\right){\text{e}}^{-x\left(t-s\right)}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{80+{\mathrm{cos}}^{2}\left(2t\right)}{100}{\int }_{0}^{1}{\text{e}}^{s}x\left(t-s\right){\text{e}}^{-x\left(t-s\right)}\text{d}s\end{array}$

$a\left(t\right)=16+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)$$b\left(t\right)=12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)$${\beta }_{1}\left(t\right)=\frac{80+{\mathrm{cos}}^{2}\left(t\right)}{100}$${\beta }_{2}\left(t\right)=\frac{80+{\mathrm{cos}}^{2}\left(2t\right)}{100}$${\sigma }_{1}\left(t\right)={\sigma }_{2}\left(t\right)=1$${K}_{1}\left(s\right)={K}_{2}\left(s\right)={e}^{s}$

$r=1$${a}^{+}=16.01$${a}^{-}=16$${b}^{+}=12.01$${b}^{-}=12$${\beta }_{1}^{+}={\beta }_{2}^{+}=\frac{4}{5}$${\beta }_{1}^{-}={\beta }_{2}^{-}=\frac{81}{100}$

$K=0.9$$\lambda =0.4$$\mu =0.94$，由计算可知， $k\approx 0.72154$$\stackrel{˜}{k}\approx 1.34228$

$\begin{array}{l}\underset{t\in R}{\mathrm{sup}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right)\left(b\left(t\right)+K\right)}{a\left(t\right)K\text{e}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}s\\ \text{=}\underset{t\in R}{\mathrm{sup}}\left\{\frac{\frac{80+{\mathrm{cos}}^{2}\left(t\right)}{100}\left(12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)+\frac{9}{10}\right)}{16+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\frac{9}{10}\text{e}}{\int }_{0}^{1}{\text{e}}^{s}\text{d}s+\frac{\frac{80+{\mathrm{cos}}^{2}\left(2t\right)}{100}\left(12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)+\frac{9}{10}\right)}{16+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\frac{9}{10}\text{e}}{\int }_{0}^{1}{\text{e}}^{s}\text{d}s\right\}\\ <0.54<1\end{array}$

$\begin{array}{l}\underset{t\in R}{\mathrm{inf}}\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right)b\left(t\right)}{a\left(t\right){\text{e}}^{k}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right)\text{d}s\\ =\underset{t\in R}{\mathrm{inf}}\left\{\frac{\frac{80+{\mathrm{cos}}^{2}\left(t\right)}{100}\left(12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\right)}{\left(16+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\right){\text{e}}^{k}}{\int }_{0}^{1}{\text{e}}^{s}\text{d}s+\frac{\frac{80+{\mathrm{cos}}^{2}\left(2t\right)}{100}\left(12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\right)}{\left(16+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\right){\text{e}}^{k}}{\int }_{0}^{1}{\text{e}}^{s}\text{d}s\right\}\\ >1.07>1\end{array}$

$\begin{array}{l}\underset{t\in R}{\mathrm{sup}}\left\{\underset{j=1}{\overset{m}{\sum }}\frac{{\beta }_{j}\left(t\right){\left(b\left(t\right)+K\right)}^{2}}{a\left(t\right)b\left(t\right){\text{e}}^{2}}{\int }_{0}^{{\sigma }_{j}\left(t\right)}{K}_{j}\left(s\right){\text{e}}^{s}\text{d}s\right\}\\ =\underset{t\in R}{\mathrm{sup}}\left\{\frac{\left(\frac{80+{\mathrm{cos}}^{2}\left(t\right)}{100}+\frac{80+{\mathrm{cos}}^{2}\left(2t\right)}{100}\right){\left(12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)+\frac{9}{10}\right)}^{2}}{\left(\left(16+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\right)\left(12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)\right)-\frac{2}{5}{\left(12+\frac{1}{100}{\mathrm{sin}}^{2}\left(t\right)+\frac{9}{10}\right)}^{2}\right){\text{e}}^{2}}{\int }_{0}^{1}{\text{e}}^{2s}\text{d}s\right\}\\ <0.94<1\end{array}$

The Global Exponential Stability of the Positive Periodic Solution for a Class of Nicholson’s Blowflies Model with Continuously Distributed Delays[J]. 应用数学进展, 2019, 08(05): 1007-1015. https://doi.org/10.12677/AAM.2019.85115

1. 1. Wang, L.J. (2013) Almost Periodic Solution for Nicholson’s Blowflies Model with Patch Structure and Linear Har-vesting Terms. Applied Mathematical Modeling, 37, 2153-2165. https://doi.org/10.1016/j.apm.2012.05.009

2. 2. (2012) Positive Almost Periodic Solution for a Class of Nichol-son’s Blowflies Model with Linear Harvesting Term. Nonlinear Analysis: Real World Applications, 13, 686-693. https://doi.org/10.1016/j.nonrwa.2011.08.009

3. 3. Yao, Z.J. (2015) Almost Periodic Solution of Nicholson’s Blowflies Model with Linear Harvesting Term and Impulsive Effects. World Scientific, 3, 1-18. https://doi.org/10.19029/mca-2015-001

4. 4. 刘炳文, 田雪梅, 杨孪山, 黄创霞. 具有非线性死亡密度和连续分布时滞的Nicholson飞蝇模型的周期解[J]. 应用数学学报, 2018, 41(1): 98-109.

5. 5. Tang, Y. and Xie, S.L. (2018) Global Attractivity of Asymptotically Almost Periodic Nicholson’s Blowflies Model with a Nonlinear Densi-ty-Dependent Mortality Term. World Scientific, 6, 1-15.

6. 6. Liu, B.W. (2014) Almost Periodic Solutions for a Delayed Nicholson’s Blowflies Model with a Noninear Density-Dependent Mortality Term. Advances in Difference Equations, 2014, 72. https://doi.org/10.1186/1687-1847-2014-72

7. 7. Liu, P.Y., Zhang, L., Liu, S.T. and Zheng, L.F. (2017) Global Exponential Stability of Almost Periodic Solutions for Nicholson’s Blowflies System with Noninear Density-Dependent Mortality Terms and Patch Structure. Mathematical Modelling and Analysis, 22, 484-502. https://doi.org/10.3846/13926292.2017.1329171

8. 8. Yao, L.G. (2018) Global Attractivity of a Delayed Nichol-son-Type System Involving Nonlinear Density-Dependent Mortality Terms. Mathematical Methods in the Applied Sciences, 41, 2379-2391. https://doi.org/10.1002/mma.4747

9. 9. Chen, W. (2012) Permanence for Nichol-son’s-Type Delay Systems with Patch Structure and Nonlinear Density-Dependent Mortality Terms. Electronic Journal of Qualitative Theory of Differential Equations, 2012, 1-14. https://doi.org/10.14232/ejqtde.2012.1.73

10. 10. Smith, H.L. (2011) An Introduction to Delay Differential Equations to the Life Sciences. Springer, New York. https://doi.org/10.1007/978-1-4419-7646-8_1

11. 11. Hale, J.K. and Verduyn Lunel, S.M. (1993) Introduction to Functional Differential Equations. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4612-4342-7