Global existence, asymptotic behavior, and pattern formation driven by the parametrization of a nonlocal Fisher-KPP problem

The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation \begin{align*} u_t=\Delta u+\mu u^\alpha(1-\kappa J*u^\beta),\quad\hbox{in} \;\mathbb R^N\times(0,\infty),\; N\geq 1 \end{align*} with $\alpha\geq1$, $\beta,\mu,\kappa>0$ and $u(x,0)=u_0(x)$ are investigated. Under appropriate assumptions on $J$, it is proved that for any nonnegative and bounded initial condition, if $\alpha\in[1,\alpha^*)$ with $\alpha^*=1+\beta$ for $N=1,2$ and $\alpha^*=1+\frac{2\beta}{N}$ for $N>2$, then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying $0\leq u(x,t)\leq\kappa^{-\frac1\beta}$ for any $(x,t)\in\mathbb R^N\times[0,+\infty)$ are shown to converge to $\kappa^{-\frac1\beta}$ uniformly on any compact subset of $\mathbb R^N$, which is known as the hair trigger effect. 1D numerical simulations of the above nonlocal reaction-diffusion equation are performed and the effect of several combinations of parameters and convolution kernels on the solution behavior is investigated. The results motivate a discussion about some conjectures arising from this model and further issues to be studied in this context. A formal deduction of the model from a mesoscopic formulation is provided as well.