Two Kinds Of Reaction Diffusion Equations

In this paper,we focus on the existence and quality of steady-state solutions.The results are as following:(1)A predator-prey model with non-monotonic response functional under ho-mogeneous Robin boundary condition ((?)u)/((?)t)-Δu=u(a-u-bve-mu),x∈Ω, t＞0, ((?)v)/((?)t)-Δv=v(c-v+due-mu),x∈Ω, t＞0, 1((?)u)/((?)n)+u=0, x∈(?)Ωt＞0,κ2((?)v)/((?)n)+v=0, x∈(?)Ω, t＞0, u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, are discussed.Firstly,by the maximum principle,Leray-Schauder degree theory,we discuss the conditions for the existence of steady-state solutions and the sufficient conditions for the inexistence of steady-state solutions.Secondly,some results of the multiplicity, stability and some uniqueness of coexistence states depending on some parameters are obtained by the spectrum analysis of operators,bifurcation theory and rayleigh's formula;Finally,we make some numerical simulation for positive solutions by Matlab.(2)A predator-prey model with HollingⅡresponse functional and cross-diffusion term under homogeneous Neumann boundary condition ((?)u)/((?)t)-Δu=u(1-u/k)-(uv)/(1+au), x∈Ω, t＞0, ((?)v)/((?)t)-Δ[(1+α/(μ+u))v]=-rv+(cuv)/(1+au), t＞0. ((?)u)/((?)n)=((?)v)/((?)n)=0, x∈(?)Ω, t＞0, u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, are discussed. Firstly, by the maximum principle, the lower-upper solution method and Harnack inequality, we discuss the solutions of priori upper and lower bounds. Secondly, some results of the sufficient conditions for the existence of steady-state solutions are obtained by the priori upper and lower bounds and Leray-Schauder degree theory.