Coexistence States And Asymptotic Behavior Of Predator-prey Model With B-D Functional Response

Lotka-Volterra model is a kind of the most significant models in mathematical biology. This model plays a very important role in ecology. In past decades, classical Lotka-Volterra model has been widely studied.The dynamic interaction between predators and their prey has long been one of the dominant themes in mathematical biology due to its universal existence and importance. A crucial element of all models is the so-called "functional response", which is the function representing the prey consumption per unit time. In this paper, a kind of predator-prey model with Beddington-DeAngelis functional response is studied. This model progresses the Holling-Tanner model and the ratio-dependent model, and it produces richer dynamics than the previous expressions.Mainly using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of parabolic equations and corresponding elliptic equations, we have systematically studied the dynamical behavior of the predator-prey model with Beddington-DeAngelis functional response, such as coexistence, multiplicity, uniqueness, stability of positive steady states and the longtime behavior of species.The tools used here include super-sub solutions method, comparison principle, global bifurcation theory, linear stability theory, and fixed-point theory of topology. The model in this paper takes the form as belowThe main contents and results in this paper are as follows:In section 1, we state our main analytical results on equilibria of the above system. First, several sufficient conditions for coexistence of the steady-state are given by the standard fixed-point index theory in cone. Second, the global structure of the coexistence solutions and their local stability are established by using bifurcation theory. Third, the multiplicity, uniqueness and stability of positive steady-state solutions to this system are derived by means of perturbation theory of eigenvalues, standard regularity theory, Sobolev embedding theorem and fixed-point index theory. Section 2 contains the longtime behavior of species. The k and m as parameters play very important roles in deciding the number of the coexistence solutions.