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

The wavefronts of a nonlinear nonlocal bistable reaction-diffusion equation,∂u∂t=∂2u∂x2+u2(1−Jσ⁎u)−du,(t,x)∈(0,∞)×R, with Jσ(x)=(1/σ)J(x/σ) and ∫RJ(x)dx=1 are investigated in this article. It is proven that there exists a c⁎(σ) such that for all c≥c⁎(σ), a monotone wavefront (c,ω) can be connected by the two positive equilibrium points. On the other hand, there exists a c⁎(σ) such that the model admits a semi-wavefront (c⁎(σ),ω) with ω(−∞)=0. Furthermore, it is shown that for sufficiently small σ, 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 σ→0.