Global stability and Hopf bifurcation of a host–parasite system

Abstract

Understanding the dynamical mechanism of the host–parasite interactions is one of important issues on host–parasite association. In this paper, we formulate a three-dimensional host–macroparasite system to describe the host–parasite interactions, which includes the logistic growth rate of host population, the important free-living stage and the host fecundity reduction due to parasite infection. The purpose of the paper is to investigate the asymptotical behavior of the system. By using the properties of the solution to non-autonomous linear system, the basic production number [Formula: see text] is proved to be a threshold which determines the outcome of the parasites. If [Formula: see text], the parasite will eventually die out, and if [Formula: see text] the parasite will be uniformly persistent. Hopf bifurcation of the system is further studied, and sufficient conditions for the Hopf bifurcation are obtained. By using the singular perturbation techniques, the system is separated into two time scales with a faster time scale for the free-living infective particles and a slower time scale for the population dynamics of host and parasite, and then a complete analysis of the dynamics on the slow manifold is conducted. The theoretical results show that the level of aggregation of parasites within host may influence the persistence and stability of the system.