Stability and bifurcation in a delayed predator-prey model with Holling-type IV response function and age structure

Abstract

In this article, we study a predator-prey model with age structure, Holling-type IV response, and two time delays. By an algebraic method, we determine all the critical values for these two delays, such that the characteristic equation has purely imaginary roots. This provides a sharp stability region on the parameter plane of the positive equilibrium. Applying integrated semigroup theory and Hopf bifurcation theorem for abstract Cauchy problems with non-dense domain, we can show the occurrence of Hopf bifurcation as the time delays pass through these critical values. In particular, the phenomenon of stability switches can also be observed as the time delays vary. Numerical simulations are carried out to illustrate the theoretical results.