Existence and properties of Hopf bifurcation in an age-dependent predation system with prey harvesting

Abstract

In the present article, we introduce an age-dependent predation system with prey harvesting, where we assume that the predators fertility function p(x) is a piecewise function concerning their maturation period τ. System is rewritten as a semilinear Cauchy problem with an operator which is not densely defined in its domain. With the help of the integrated semigroup theory, the Hopf bifurcation theorem, the center manifold argument and normal form theory for semilinear equations with non-dense domain, we demonstrate much richer novel dynamical phenomena than the existing ones. Qualitative analysis manifests that system undergoes a Hopf bifurcation around the positive equilibrium age distribution. Furthermore, the explicit formulae are provided to determine the direction the Hopf bifurcation and the stability of the bifurcating periodic solutions, which is the main result of this paper. Last but not least, numerical simulations are given to illustrate our analytical results.