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

Shun-yi Li,Zuoliang Xiong

Published 2013 in Advances in Differential Equations

ABSTRACT

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.

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