Dynamics of a Nonlocal Dispersal Model with a Nonlocal Reaction Term

In this paper, we study a class of nonlocal dispersal problem with a nonlocal term arising in population dynamics: {u(t) = Du + u[lambda - f(u) - integral(Omega) K(x, y)g(u(y))dy], in Omega x (0, +infinity), u(x, 0) = u(0)(x) >= 0, in Omega, u = 0, in R-N\Omega x (0, +infinity), where Omega subset of R-N (N >= 1) is a bounded domain, lambda is an element of R, Du(x, t) = integral(Omega) J(x - y)[u(y, t)-u(x, t)]dy represents the nonlocal dispersal operator with continuous and non-negative dispersal kernel. The kernel K is an element of C((Omega) over bar x (Omega) over bar) is assumed to be non-negative and is allowed to have a degeneracy in a smooth subdomain Omega(0) of Omega. When K is either positive or vanishes in a subdomain, we respectively investigate the existence, multiplicity and asymptotical stability of positive steady states under the local/global variation of parameter by means of sub-supersolution method, Lyapunov-Schmidt reduction, and bifurcation theory.