Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

Xinhui Wang, Haihong Liu

Department of Mathematics, Yunnan Normal University, Kunming, China.

DOI: 10.4236/am.2012.36099                

Abstract

A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, some numerical simulations are carried out for illustrating the theoretical results.

Keywords

Lotka-Volterra Predator-Prey System; Discrete Delay; Allelopathy; Stability; Hopf Bifurcation; Periodic Solution

Conflicts of Interest

The authors declare no conflicts of interest.

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