Title: 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_\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 $\sigma\to0$.