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 Ju^\beta),\quad\hbox{in} ;\mathbb R^N\times(0,\infty),; N\geq 1 \end{align} with $\alpha \ge 1$, $\beta ,\mu ,\kappa >0$ 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 $\alpha \in [1,{\alpha }^{\ast })$ with ${\alpha }^{\ast }=1+\beta$ for $N=1,2$ and ${\alpha }^{\ast }=1+\frac{2\beta }{N}$ for $N>2$, then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying $0\le u(x,t)\le {\kappa }^{-\frac{1}{\beta }}$ for any $(x,t)\in {\mathbb{R}}^N\times [0,+\mathrm{\infty })$ are shown to converge to ${\kappa }^{-\frac{1}{\beta }}$ 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.