﻿ 具收获项和周期系数的广义捕食-被捕食模型的正周期解题 Positive Periodic Solutions for a Generalized Prey–Predator Model

Vol.06 No.03(2017), Article ID:20781,9 pages
10.12677/AAM.2017.63036

Positive Periodic Solutions for a Generalized Prey–Predator Model

Xianglai Zhuo1, Fengxue Zhang2

1College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao Shandong

2College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao Shandong

Received: May 7th, 2017; accepted: May 23rd, 2017; published: May 27th, 2017

ABSTRACT

The existence of positive periodic solutions for a generalized prey-predator model with harvesting term was studied by using Mawhin’s continuation theorem of coincidence degree theory. Some sufficient conditions were obtained to ensure the existence of positive periodic solutions. The results obtained in this paper generalized the known results.

Keywords:Coincidence Degree, Prey-Predator Model, Harvesting Term, Positive Periodic Solution

1山东科技大学数学与系统科学学院，山东 青岛

2山东科技大学矿业与安全工程学院，山东 青岛

1. 引言

(1)

(2)

(3)

(1)对任意，对算子方程的任一解满足

(2)当时，

2. 主要结果

，则系统(3)成为

(4)

。于是是指标为零的Fredholm算子。

(5)

(6)

(7)

(8)

(i)对于固定的关于单调减少。

(ii)对任意的，当时，。对于固定的，当时，关于上升，而且当时，关于上升。

(iii)对，当时，

(iv)对，当时，

(v)对，当时，

(9)

(10)

，由引理2(iii)，当时，。于是

，由(10)的第一个方程得

，这与引理2(iv)矛盾；同样当时候，，(10)的第二个方程不成立。故当时，，故上是同伦映射，因此

(11)

(3)当时，

(4)当时，

，则

(12)

。事实上，若，由引理2(iii)和引理4得

(13)

，由引理4得

(14)

(13)和(14)与(12)矛盾。

(15)

。事实上，若，由引理2(iv)和引理4得

(16)

，由引理2(v)和引理4得

(17)

(16)和(17)与(15)矛盾。于是当时，。引理5证毕。于是

(18)

. (19)

，由(18)，(19)得。于是由。这里，所以(18)成立，引理6证毕。

Positive Periodic Solutions for a Generalized Prey–Predator Model[J]. 应用数学进展, 2017, 06(03): 308-316. http://dx.doi.org/10.12677/AAM.2017.63036

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