LIMITING CHARACTERIZATION OF STATIONARY SOLUTIONS FOR A PREY-PREDATOR MODEL WITH NONLINEAR DIFFUSION OF FRACTIONAL TYPE

We consider the following quasilinear elliptic system: {Delta u + u(a - u - cv) = 0 in Omega, Delta[(1 + gamma/1+beta u)v] + v(b + du - v) = 0 in Omega, u = v = 0 on partial derivative Omega, where Omega is a bounded domain in R-N. This system is a stationary problem of a prey-predator model with non-linear diffusion Delta(v/1 + beta u) and u (respectively v) denotes the population density of the prey (respectively the predator). Kuto [15] has studied this system for large beta under the restriction b > (1 + gamma)lambda(1), where lambda(1) is the least eigenvalue of -Delta with homogeneous Dirichlet boundary condition. The present paper studies two shadow systems and gives the complete limiting characterization of positive solutions as beta -> infinity without any restriction on b.