A NONLOCAL DISPERSAL LOGISTIC EQUATION WITH SPATIAL DEGENERACY

In this paper, we study the nonlocal dispersal Logistic equation {u(t) = Du + lambda m(x)u - c(x)u(p) in Omega x (0, +infinity), u(x, 0) - u(0)(x) >= 0 in Omega, where Omega subset of R-N is a bounded domain, lambda > 0 and p > 1 are constants. Du(x, t) = integral(Omega)J(x y)(u(y, t) u(x, t))dy represents the nonlocal dispersal operator with continuous and nonnegative dispersal kernel J, m is an element of C((Omega) over bar) and may change sign in Omega. The function c is nonnegative and has a degeneracy in some subdomain of Omega. We establish the existence and uniqueness of positive stationary solution and also consider the effect of degeneracy of c on the long-time behavior of positive solutions. Our results reveal that the necessary condition to guarantee a positive stationary solution and the asymptotic behaviour of solutions are quite different from those of the corresponding reaction-diffusion equation.