Oscillations for a delayed predator–prey model with Hassell–Varley-type functional response

Abstract

In this paper, a delayed predator–prey model with Hassell–Varley-type functional response is investigated. By choosing the delay as a bifurcation parameter and analyzing the locations on the complex plane of the roots of the associated characteristic equation, the existence of a bifurcation parameter point is determined. It is found that a Hopf bifurcation occurs when the parameter τ passes through a series of critical values. The direction and the stability of Hopf bifurcation periodic solutions are determined by using the normal form theory and the center manifold theorem due to Faria and Maglhalaes (1995). In addition, using a global Hopf bifurcation result of Wu (1998) for functional differential equations, we show the global existence of periodic solutions. Some numerical simulations are presented to substantiate the analytical results. Finally, some biological explanations and the main conclusions are included.