Stability and Hopf Bifurcation of a Delayed Density-Dependent Predator-Prey System with Beddington-DeAngelis Functional Response

In this paper, we investigate the stability and Hopf bifurcation of a delayed density-dependent predator-prey system with Beddington-DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator-prey system conforms to the realistically biological environment. We start with the geometric criterion introduced by Beretta and Kuang [2002] and then investigate the stability of the positive equilibrium and the stability switches of the system with respect to the delay parameter τ. Especially, we generalize the geometric criterion in [Beretta & Kuang, 2002] by introducing the condition (i′) which can be assured by the condition (H2′), and adopting the technique of lifting to define the function Son(τ) for alternatively determining stability switches at the zeroes of Sn(τ)s. Afterwards, by the Poincaré normal form for Hopf bifurcation in [Kuznetsov, 1998] and the bifurcation formulae in [Hassard et al., 1981], we qualitatively analyze the properties for the occurring Hopf bifurcations of the system (3). Finally, an example with numerical simulations is given to illustrate the obtained results.