Wavefronts for a nonlinear nonlocal bistable reaction-diffusion equation in population dynamics

The wavefronts of a nonlinear nonlocal bistable reaction-diffusion equation, partial derivative u/partial derivative t = partial derivative(2)u/partial derivative x(2) + u(2) (1 J(sigma) *u) - du, (t, x) is an element of (0, infinity) x R, with J(sigma) (x) = (1/sigma)J(x/sigma) and integral J(x)dx =1 are investigated in this article. It is proven that there exists a c*(sigma) such that for all c >= c*(sigma), a monotone wavefront (c, omega)) can be connected by the two positive equilibrium points. On the other hand, there exists a c*(a) such that the model admits a semi-wavefront (c*(sigma), omega) with omega(-infinity) = 0. Furthermore, it is shown that for sufficiently small sigma, the semi-wavefronts are in fact wavefronts connecting 0 to the largest equilibrium. In addition, the wavefronts converge to those of the local problem as sigma -> 0. (C) 2017 Elsevier Inc. All rights reserved.