Dynamics Of A Predator-Prey System With Watt Type Functional Response

The relationship between predator and prey is one of the basic relationships among species. Understanding the dynamical relationship between predator and prey is a central goal in ecology and mathematical biology. Most of the predator-prey models we have investigated are the systems with functional responses. In general, the functional response can be classified into two types: prey-dependent and predator-dependent. There is much significant evidence to suggest that predator dependence in the functional response occurs quite frequently in laboratory and natrual systems. Recently, a lot of the predator-dependent functional responses have been well studied such as: Beddington-DeAngelis, Crowley-Martin, Hassell-Varley and so on. We find that the results of Watt type functinal response are rare in the literatures. In this paper, we systematically study the dynamics of a predator-prey system with Watt type functional response. The model is described byFirst, with the help of qualitative theory of ODEs, we discuss the existence and stability of its positive equilibrium, and also the exsitence of the limit cycle. Then, we discuss the positive invariance and permanence for the genaral non-autonomous case. Finally, by using the continuation theorem of coincidence degree theory, we establish sufficient criteria for the existence of a positive periodic solution. We also carry numerical simulations for some concrete models, which strongly support our theoretical studies.