Stability and bifurcation in a delayed predator–prey system with Beddington–DeAngelis functional response

We consider a delayed predator–prey system with Beddington–DeAngelis functional response. The stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhaes. An example is given and numerical simulations are performed to illustrate the obtained results.

[1]  K. Cooke,et al.  Discrete delay, distributed delay and stability switches , 1982 .

[2]  Y. Kuang,et al.  Global analyses in some delayed ratio-dependent predator-prey systems , 1998 .

[3]  H. I. Freedman.  Deterministic mathematical models in population ecology , 1982 .

[4]  L. Ginzburg,et al.  The nature of predation: prey dependent, ratio dependent or neither? , 2000, Trends in ecology & evolution.

[5]  J. Beddington,et al.  Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency , 1975 .

[6]  Yang Kuang,et al.  Global qualitative analysis of a ratio-dependent predator–prey system , 1998 .

[7]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[8]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations and Applications to Bogdanov-Takens Singularity , 1995 .

[9]  Dongmei Xiao,et al.  STABILITY AND BIFURCATION IN A DELAYED RATIO-DEPENDENT PREDATOR–PREY SYSTEM , 2002, Proceedings of the Edinburgh Mathematical Society.

[10]  R. Arditi,et al.  Coupling in predator-prey dynamics: Ratio-Dependence , 1989 .

[11]  Chris Cosner,et al.  On the Dynamics of Predator–Prey Models with the Beddington–DeAngelis Functional Response☆ , 2001 .

[12]  Tzy-Wei Hwang,et al.  Global analysis of the predator–prey system with Beddington–DeAngelis functional response , 2003 .

[13]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[14]  Donald L. DeAngelis,et al.  A Model for Tropic Interaction , 1975 .

[15]  D. DeAngelis,et al.  Effects of spatial grouping on the functional response of predators. , 1999, Theoretical population biology.

[16]  Donald L. DeAngelis,et al.  A MODEL FOR TROPHIC INTERACTION , 1975 .

[17]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[18]  L. Magalhães,et al.  Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation , 1995 .