Dynamic behavior of a parasite–host model with general incidence

In this paper, we consider the global dynamics of a microparasite model with more general incidences. For the model with the bilinear incidence, Ebert et al. [D. Ebert, M. Lipsitch, K.L. Mangin, The effect of parasites on host population density and extinction: Experimental epidemiology with Daphnia and six microparasites, American Naturalist 156 (2000) 459–477] observed that parasites can reduce host density, but the extinction of both host population and parasite population occurs only under stochastic perturbations. Hwang and Kuang [T.W. Hwang, Y. Kuang, Deterministic extinction effect of parasites on host populations, J. Math. Biol. 46 (2003) 17–30] studied the model with the standard incidence and found that the host population may be extinct in the absence of random disturbance. We consider more general incidences that characterize transitions from the bilinear incidence to the standard incidence to simulate behavior changes of populations from random mobility in a fixed area to the mobility with a fixed population density. Using the techniques of Xiao and Ruan [D. Xiao, S. Ruan, Global dynamics of a ratio-dependent predator–prey system, J. Math. Biol. 43 (2001) 268–290], it is shown that parasites can drive the host to extinction only by the standard incidence. The complete classifications of dynamical behaviors of the model are obtained by a qualitative analysis.