Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes

In this paper, we investigate the existence, multiplicity and stability of positive solutions to a prey-predator model with modified Leslie-Gower and Holling-type II schemes {-Delta u = u(a(1) - bu - c(1)v/u + k(1)) in Omega, -Delta v = u(a(2) - c(2)v/u + k(2)) in Omega, u >= 0, v >= 0 in Omega, u = v = 0. on partial derivative Omega, (P) where Omega subset of R-N (N >= 1) is a bounded domain with a smooth boundary partial derivative Omega, the parameters a(i), b, c(i), k(i) (i = 1,2) are positive numbers, u and v are the respective populations of prey and predator. Here, we say (u, v) with u vertical bar(partial derivative Omega) = v vertical bar(partial derivative Omega) = 0 is a positive solution of problem (P) if (u, v) is a solution of (P) and u,v > 0 in Omega. (C) 2012 Elsevier Inc. All rights reserved.)