﻿ 具有Beddington-DeAngelis功能反应和密度制约的非自治捕食–食饵系统的灭绝性 Extinctionof the Density Dependent Nonautonomous Predator-Prey System with Beddington-DeAngelis Functional Response

Vol. 09  No. 03 ( 2020 ), Article ID: 34686 , 8 pages
10.12677/AAM.2020.93048

Extinction of the Density Dependent Nonautonomous Predator-Prey System with Beddington-DeAngelis Functional Response

Wenrui Zeng, Dingyong Bai, Jinshui Li

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

Received: Mar. 1st, 2020; accepted: Mar. 16th, 2020; published: Mar. 24th, 2020

ABSTRACT

In this paper, the density dependent nonautonomous predator-prey system with Beddington-DeAngelis functional response is studied. The global asymptotic stability of boundary periodic solution is obtained by using the comparison theorem, differential inequality and the dynamics of Logistic equation. Our main result indicates that, even though there is no intraspecific competition, over a periodic interval if the average predation benefit is less than the average death number of predator species, the predator species will go extinction. In addition, some numerical simulations are performed to illustrate the theoretical results.

Keywords:Beddington-DeAngelis Functional Response, Nonautonomous Predator-Prey System, Global Asymptotic Stability, Boundary Periodic Solution, Extinction

1. 引言

Fan和Kuang [5] 研究了具有Beddington-DeAngelis型功能反应的非自治捕食–食饵系统

(1)

(2)

(3)

(4)

2. 主要结果

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

1) 如果，定理2.4仍然成立，条件与文 [6] 给出的一致，但文 [6] 要求都有正的下界，而本文定理2.4允许，且允许改变符号。在本文的假设条件下无法应用文 [6] 采用的方法来证明边界周期解的全局稳定性。

2) 生物解释：定理2.4表明，即使没有种内竞争(即)，但如果在一个周期区间上捕食者种群的平均收益小于平均死亡数(即)，就会导致该种群灭绝。

3. 数值模拟

。显然，除了，其余参数均非负，。容易计算

Figure 1. Global asymptotical stability of periodic boundary solution of system (2)

。显然，是变号的周期函数，但。因此，系统(2)存在边界周期解。由于有下面形式

Figure 2. Global asymptotical stability of periodic boundary solution of system (2)

Extinctionof the Density Dependent Nonautonomous Predator-Prey System with Beddington-DeAngelis Functional Response[J]. 应用数学进展, 2020, 09(03): 400-407. https://doi.org/10.12677/AAM.2020.93048

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