Global existence, asymptotic behavior, and pattern formation driven by the parametrization of a nonlocal Fisher-KPP problem
Jing Li, Li Chen, Christina Surulescu
TL;DR
The paper analyzes the global existence and long-time behavior of the nonlocal Fisher–KPP equation $u_t=\Delta u+\mu u^{\alpha}(1-\kappa J*u^{\beta})$ on $\mathbb{R}^N$, with $\alpha\ge1$, $\beta,\mu,\kappa>0$ and a competition kernel $J$. It identifies a subcritical exponent threshold $\alpha^*$, namely $\alpha^*=1+\beta$ for $N=1,2$ and $\alpha^*=1+\frac{2\beta}{N}$ for $N>2$, which guarantees global bounded solutions for all nonnegative bounded initial data. The authors prove hair-trigger convergence to the carrying capacity $\kappa^{-1/\beta}$ under additional initial-data assumptions, and provide a detailed numerical study in 1D to reveal how kernel shape and parameter choices affect boundedness, convergence, and pattern formation, complemented by a mesoscopic derivation in the Appendix. These results illustrate how nonlocal competition reshapes classical blow-up thresholds, enabling global existence on unbounded domains and offering insights into ecological and cellular pattern formation.
Abstract
The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation \begin{align*} u_t=Δu+μu^α(1-κJ*u^β),\quad\hbox{in} \;\mathbb R^N\times(0,\infty),\; N\geq 1 \end{align*} with $α\geq1$, $β,μ,κ>0$ 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 $α\in[1,α^*)$ with $α^*=1+β$ for $N=1,2$ and $α^*=1+\frac{2β}{N}$ for $N>2$, then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying $0\leq u(x,t)\leqκ^{-\frac1β}$ for any $(x,t)\in\mathbb R^N\times[0,+\infty)$ are shown to converge to $κ^{-\frac1β}$ 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.