On The Study Of A Predator-prey Model With B-D Functional Response

In this paper we have discussed the existence of positive steady-state solutions of the predator-prey model with B-D functional response.as follow: where Ω is a bounded domain in Rn with smooth boundary (?)Ω,,all the coefficient functions except a and b are continuous positive constants,and a, b may be a continuous positive functions of x∈Ω.For predator-prey model,a crucial element of the model is the "functional response",which is the function representing the prey consumption per unit time.This model progresses the Hoiling-Tanner model and the ratio-dependent model,and it produces richer dynamics than the previous expressions. And it more reasonably reflecting the interactions between predator and prey relationships. This paper considered the existence of positive steady-state solutions under the Neumann boundary condition when establishing a biological reserve.Mainly using the tools here include fixed-point theory of topology,comparison principle, bifurcation theory.This paper is divided to two parts:homogeneous and heterogeneous conditions are discussed.In the homogeneous case,we discussed the existence of positive solution by bifurcation theory.More precisely:we present as Theoreml If μ> λ,and λ=λN((bθ[μ])/(1+kθ[μ])),then there is a neighborhood of (λ;0,θ[μ]) such that any solution of (2) either lies on this curve or is identically equal to (0,θ[μ]).In the heterogeneous case.it is divided to two chapters.First chapter gives the sufficient conditions for the existence of positive solutions with fixed-point theory of topology,when b is continuous function.And then we assume that all the coefficient functions are positive constants,except b,which is a nonconstant function of x.And b(x)≥0.In this chapter we gives a priori estimates with maximum principle,then reached the theorem on the existence of positive steady-state solutions.We present as Theorem2If b(x)∈(Ω), and λ>b/k,μ>c/m,then there is a positive steady-state solution.The chapter2,we assume the a is a non-constant function of x,and a(x)=0, x∈D,且a(x)>0, x∈Ω\D.The same with chapter1,we give the priori estimates,then concluded the theorem on the existence of positive steady-state solution with fixed-point of topology. We prove the following theorem Theorem3Suppose that an is a sequence in C(Ω) that converges to a in this space,and an=0, in D, Let μ>λ1Dm be fixed and λn∈[m’, M’] C (0, λ1D1), Then there exists a positive constant C independent of n,such that any positive solution (un,un) of (2) satisfy‖un‖∞+‖vn‖∞<C.In the last part of this chapter we construct solution with prescribed patterns.