Dynamical behavior of a ratio dependent predator-prey system with distributed delay

Pages: 719 - 738, Volume 16, Issue 3, October 2011

doi:10.3934/dcdsb.2011.16.719       Abstract        References        Full Text (405.9K)       Related Articles

Canan Çelik - Department of Mathematics and Computer Sciences, Bahçeşehir University, Istanbul, 34353, Turkey (email)

Abstract: In this paper, 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 [1] and [2], 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 central manifold argument, we establish the direction and the stability of Hopf bifurcation. Some numerical simulations for justifying the theoretical analysis are also presented.

Keywords:  Predator-prey system, distributed delay, hopf bifurcation, stability.
Mathematics Subject Classification:  Primary: 34K18, 34K20, 37D25; Secondary: 92D25.

Received: April 2010;      Revised: July 2010;      Published: June 2011.

 References