Stability and bifurcation analysis on a ratio-dependent predator-prey model with prey dispersal and time delay. (English) Zbl 1186.34123

Xu, Rui; Gan, Qintao; Ma, Zhien

Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 16, No. 5, 699-726 (2009).

Summary: A ratio-dependent predator-prey model with prey dispersal and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium and the existence of Hopf bifurcations are established. Using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction and stability of bifurcating periodic solutions. By means of an iteration technique, sufficient conditions are obtained to guarantee the positive equilibrium to be globally attractive. Numerical simulations are carried out to illustrate the main results.

MSC:

34K60	Qualitative investigation and simulation of models involving functional-differential equations
92D25	Population dynamics (general)
34K20	Stability theory of functional-differential equations
34K18	Bifurcation theory of functional-differential equations
34K13	Periodic solutions to functional-differential equations
34K19	Invariant manifolds of functional-differential equations
34K17	Transformation and reduction of functional-differential equations and systems, normal forms

Keywords:

ratio-dependence; predator-prey model; dispersal; time delay; Hopf bifurcation; stability