A Predator - Prey Model The Nature Of The Solution

Today, the so called Mathematical Biology becomes an active branch of modern science. A lot of mathematical models are established successfully, and significant achievements are obtained. 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. Studies on the coexistence, stability and persistence for predator-prey systems have very important practical significance to equilibrium of ecology, ecological environments preservation and even saving the rare and precious creature on the brink of extinction.One classical model of predator-prey systems in ecology is Lotka-Volterra model, it is a kind of the most significant models in mathematical biology and has been widely studied. A crucial element of all models is the so-called "functional response " , which is the function representing the prey consumption per unit time. Holling functional response produces richer dynamics.Modified the Holling functional response, we get a class of predator-prey systems as belowMainly 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 above predator-prey model, such as several sufficient conditions for coexistence solutions of the steady-state, the global structure of the coexistence solutions, stability of positive steady states and asymptotic behavior and stability of positive solutions. The tools used here include super-sub solutions method, comparison principle, global bifurcation theory, linear stability theory, and fixed-point theory of topology. The main contents and results in this paper are as follows:section 1 is introduction, mainly introduces the background of the question, interrelated study and the main contents and results in this paper.In section 2, 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 maximum principle, methods of the super-sub solution and 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 when the bifurcation parameters are different, stability of positive steady states.Section 3 contains asymptotic behavior and stability of positive solutions. Asymptotic behavior and stability of positive solutions are given by methods of the supersub solution and linear stability theory.