Hopf bifurcation in a delayed predator-prey system with modified Leslie-Gower and Holling-type III schemes. (English) Zbl 1299.34275

Zhang, Zizhen; Yang, Huizhong

Acta Autom. Sin. 39, No. 5, 610-616 (2013).

Summary: We consider a predator-prey system with modified Leslie-Gower and Holling type III schemes. By regarding the time delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. We find that Hopf bifurcations can occur as the time delay crosses some critical values. In particular, special attention is paid to the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. In addition, the global existence of periodic solutions bifurcating from the Hopf bifurcation are considered by applying a global Hopf bifurcation result. Finally, numerical simulations are carried out to illustrate the main theoretical results.

MSC:

34K60	Qualitative investigation and simulation of models involving functional-differential equations
34K18	Bifurcation theory of functional-differential equations
34K20	Stability theory of functional-differential equations
34K13	Periodic solutions to functional-differential equations
92D25	Population dynamics (general)

Keywords:

predator-prey; time delay; stability; bifurcation; periodic solution