带有交叉扩散的捕食模型的非常数正稳态解的存在性

The Existence of Non-constant Positive Steady-state Solutions for a Prey-predator Model with Cross-diffusion

本文研究了下列带有交叉扩散的捕食模型的稳态问题的非常数正解的存在性,证明了当d4>1/m1v-u时存在(g1,d2,d3)使得稳态问题存在非常数正解;而当d4≤1/m1v-u或者d1≥m1v-u/u1或者a(m1b,a2(b))时稳态问题不存在非常数正解．

This paper deals with the existence of non-constant positive solutions for the following steady state problem of a prey-predator model with cross-diffusion {-d1△[u（1＋d3v）]=au-u^2-m1uv,x∈Ω, -d2△[v（1＋d4u）]=bv-v^2＋m2uv/r＋u,x∈Ω, δu/δv=δv/δv=0,x∈δΩ, It is proved that there exists （dl, d2, d3） such that the steady state problcm has non-constant positive solutions ifd4〉1/m1v-u;whereas the steady state problem has no non-constant positive solution provided d4≤1/m1v-u,or d1≥m1v-u/μ1,or a ￠（m1b,a2（b））