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

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_\sigma*u)-du,\quad(t,x)\in(0,\infty)\times\mathbb R, \end{align*} with $J_\sigma(x)=(1/\sigma)= J(x/\sigma)$ and $\int_{\mathbb R} J(x)dx=1$ are investigated in this article. It is proven that there exists a $c_*(\sigma)$ such that for all $c\geq c_*(\sigma)$, a monotone wavefront $(c,\omega)$ can be connected by the two positive equilibrium points. On the other hand, there exists a $c^*(\sigma)$ such that the model admits a semi-wavefront $(c^*(\sigma),\omega)$ with $\omega(-\infty)=0$. 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 $\sigma\to0$.