Two Kinds Of Predator - Prey Coexistence State

Lotka-Volterra model is the nuclear contents of population dynamics. This model is used universally in ecology and in portection of plants and creatures and in the control and exploiture of environment. Therefore Lotka-Volterra model has been widely studied.The dynamic interaction between predator and their prey has long been one of the dominant themes in mathematical biology due to its universal existence and importance. More attention on the predator-prey model is whether two species coexists or one species is persistent existence but another species is extinct. Co-existents and asympotic behavior of solutions, which are about the positive steady states of the model, have a close relation with the properties of the steady states. Therefore existence and stability of positive solutions of the positive steady states of the predator-prey model are mostly analysed in this paper.According to the classic Lotka-Volterra model two kinds of biological models, i.e. a predator-prey model with non-monotonic function response and a predator-prey model incorporating a constant prey refuge have been studied in this paper by using super-sub solutions method, comparison principle, Harnack inequality, topological degree theory and bifurcation theory.In section 1, we study a predator-prey model with non-monotonic functional response whereΩis a bounded domain in Rn with smooth boundary (?)Ω, u, v represent the densities of two species, a, b, c, d are positive constants,β, m are non-negative con-stants. This chapter can be divided into three parts. First, the existence of the local bifurcation solutions is discussed by using bifurcation theory. Second, the global bi-furcation structure of the steady state solutions is investigated in detail. Third, the stability of the local bifurcation solutions is discussed by using perturbation technique.In section 2, we study a predator-prey model incorporating a constant prey refuge where△stands for the Laplace,(?) is the unit outnormal direction derivatives, a, k, b, c, d, e, m are positive constants. This chapter can be divided into four parts. First, the stability of positive constant solution of steady state system is discussed by the methods of eigenvalue theory. Second, a priori estimate of the postive so-lutions is given through the maximum principle and Harnack inequality. Third, the non-existence of non-constant positive steady states is given by using energy method. Forth, the existence of non-constant positive steady states is obtained through topological degree theory.