RETRACTED: Stability and Hopf bifurcations in a delayed Leslie–Gower predator–prey system

By Sanling Yuan, Yongli Song. 2009

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In this paper, we consider the following delayed Leslie-Gower predator-prey system x'(t)=r 1 x(t)(1-x(t-τ K))-mx(t)y(t), y'(t)=r 2 y(t)(1-y(t) γx(t)). By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838] for functional differential equations, we may show the global existence of periodic solutions. Computer simulations illustrate the results.

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