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

Abstract

The wavefronts of a nonlinear nonlocal bistable reaction-diffusion equation, \begin{align*} \frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+u^2(1-J_σ*u)-du,\quad(t,x)\in(0,\infty)\times\mathbb R, \end{align*} with $J_σ(x)=(1/σ)= J(x/σ)$ and $ \int_{\mathbb R} J(x)dx=1 $ are investigated in this article. It is proven that there exists a $c_*(σ)$ such that for all $c\geq c_*(σ)$, 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 $ω(-\infty)=0$. 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 $σ\to0$.