Protection Zones For A Predator-prey Model With B-D Functional Response

In this paper we consider the following predator-prey model with Beddington-DeAngelis functional response with Ω0(?)(?)Ω(?)RN smooth, and to study the importance of protection zones Ω0 established for keeping the persistence of the prey population u, as well as the substantial effect of the size of Ω0 to the behavior of solutions.In Chapter 1, we outline the background of the studies and introduce some preliminaries of the paper.In Chapter 2, we consider effects of the sizes of protection zones to the existence of positive steady states. We prove that if the prey growth rate θ is large enough, the prey could always sur-vive. Otherwise, the survival of the prey relies on the predator growth rate μ. When the predator growth rate μ is large, protection zones must be established to prevent the prey from extinction. By using the bifurcation analysis, we obtain that there is a critical size of protection zones such that if the protection zone is over the critical size, the prey population persists regardless of the growth rate of the predator, and the dynamics of the model would be similar to the case without protection zones when the protection zone is below the critical size.Chapter 3 deals with the case where the protection zone is below the critical size. We give the bifurcation analysis to the steady states and the dynamic behavior of solutions.In chapter 4, we study a model with a modified Lotka-Volterra functional response, i.e. m=0 in the predator-prey model with B-D functional response. We find the substantial effect of the size of protection zones to the model.