Bifurcation analysis of a predator-prey system with sex-structure and sexual favoritism

Advances in Difference Equations

Zuoliang Xiong¹  Shunyi Li² 

[1]Department of Mathematics, Nanchang University, Nanchang, China

[2]Department of Mathematics, Qiannan Normal College for Nationalities, Duyun, China

关键词: predator-prey system;    time delay;    bifurcation;    chaos;    sex-structure;    sexual favoritism;   

DOI  :  10.1186/1687-1847-2013-219

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, a predator-prey system with sex-structure and sexual favoritism is considered. Firstly, the impact of the sexual favoritism coefficient on the stability of the ordinary differential equation (ODE) model is studied. By choosing sexual favoritism coefficient as a bifurcation parameter, it is shown that a Hopf bifurcation can occur as it passes some critical value, and the stability of the bifurcation is also considered by using an analytical method. Secondly, the impact of the time delay on the stability of the delay differential equation (DDE) model is investigated, where time delay is regarded as a bifurcation parameter. It is found that a Hopf bifurcation can occur as the time delay passes some critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are derived. Numerical simulations are performed to support theoretical results and some complex dynamic behaviors are observed, including period-halving bifurcations, period-doubling bifurcations, high-order periodic oscillations, chaotic oscillation, fast-slow oscillation, even unbounded oscillation. Finally, a brief conclusion is given. MSC:34K13, 34K18, 34K60, 37D45, 37N25, 92D25.

【 授权许可】

CC BY