Vol.3 No.03(2014), Article ID:14020,5 pages
DOI:10.12677/AAM.2014.33022

Stability in a Predator-Prey Model with Discrete and Distributed Delays

Huantao Zhu, Zhongde Zhang, Wuli Chen

Hunan College of Information, Changsha

Email: zhu-huan-tao@163.com

Received: May 28th, 2014; revised: Jun. 27th, 2014; accepted: Jul. 8th, 2014

ABSTRACT

The stability in a predator-prey model with discrete and distributed delays is investigated. By using linearized methods for the positive equilibrium and analyzing the corresponding characteristic equations, sufficient conditions for asymptotic stability of the positive equilibrium and the Hopf bifurcation occurring are derived.

Keywords:Delays, Predator Model, Stability, Hopf Bifurcation

Email: zhu-huan-tao@163.com

1. 引言

Song和Yuan研究了如下一类具有离散和分布时滞的捕食—食饵系统[3]

(1)

2. 引理

(2)

(3)

3. 主要结论

(4)

1) 系统(4)在平凡平衡点的特征根为，故是不稳定的；

2) 系统(4)在边界平衡点的特征根为，其它特征值由方程

(H1)

(5)

(6)

(7)

，我们有

(8)

，则方程(7)可化为

(9)

(H2) 方程(9)至少有一个正实根

(H3)

1) 当时，系统(4)的正平衡点 (即系统(1)的正平衡点)是局部渐进稳定的；

2) 系统(4)的正平衡点 (即系统(1)的正平衡点)在时经历Hopf分支。

4. 举例

(10)

1. [1]   Volterra, V. (1931) Lecons sur la theorie mathematique de la lutte pour la vie. Gauthier-Villars, Pairs.

2. [2]   Brelot, M. (1931) Sur le probleme biologique hereditaiar de deux especes devorante et devore. Annali di Matematica Pura ed Applicata, 9, 58-74.

3. [3]   Song, Y.L. and Yuan, S.L. (2006) Bifurcation analysis in a predator-prey system with time delay. Nonlinear Analysis: Real World Applications, 7, 265-284.

4. [4]   廖晓昕 (2001) 稳定性的数学理论及应用. 第二版, 华中师范大学出版社, 武汉.

5. [5]   马知恩, 周义仓 (2001) 常微分方程定性与稳定性方法. 科学出版社, 北京.

6. [6]   Ruan, S.G. and Wei, J.J. (2003) On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynamics of Continuous, Discrete and Impulsive Systems, Ser. A: Mathematical Analysis, 10, 863-874.

7. [7]   Hale, J.K. and Lunel, S.M.V. (1993) Introduction to Functional Differential Equations. Springer-Verlag, Berlin.