Existence of Positive Solutions of Lotka-Volterra Competition Model with Nonlinear Boundary Conditions

Abstract

A Lotka-Volterra competition model with nonlinear boundary conditions is considered. First, by using upper and lower solutions method for nonlinear boundary problems, we investigate the existence of positive solutions in weak competition case. Next, we prove that $-d_1\mathrm{\Delta }u=u(a_1-b_{1}u-c_{1}v)$, $x\in \mathrm{\Omega }$; $-d_2\mathrm{\Delta }v=v(a_2-b_{2}u-c_{2}v)$, $x\in \mathrm{\Omega }$; $\partial u/\partial \nu +f(u)=0$, $x\in \partial \mathrm{\Omega }$; $\partial v/\partial \nu +g(v)=0$, $x\in \partial \mathrm{\Omega }$, has no positive solution when one of the diffusion coefficients is sufficiently large.