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Dynamical behavior of a ratio dependent predator-prey system with distributed delay. (English) Zbl 1228.34127
Summary: We consider a predator-prey system with distributed time delay where the predator dynamics is logistic with the carrying capacity proportional to prey population. In [Chaos Solitons Fractals 37, No. 1, 87–99 (2008; Zbl 1152.34059); ibid. 42, No. 3, 1474–1484 (2009; Zbl 1198.34149)], we studied the impact of the discrete time delay on the stability of the model; however, in this paper, we investigate the effect of the distributed delay for the same model. By choosing the delay time $\tau$ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time $\tau$ passes some critical values. Using normal form theory and the center manifold theorem, we establish the direction and the stability of Hopf bifurcation. Some numerical simulations justifying the theoretical analysis are also presented.
##### MSC:
 34K60 Qualitative investigation and simulation of models 34K13 Periodic solutions of functional differential equations 34K18 Bifurcation theory of functional differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general)