Global boundedness, hair trigger effect, 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 ∂ u ∂ t = Δ u + μ u α (1 − κ J ⁎ u β) , in R N × (0 , ∞) , N ≥ 1 with α ≥ 1 , 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 ∈ [ 1 , α ⁎) with α ⁎ = 1 + β for N = 1 , 2 and α ⁎ = 1 + 2 β N for N > 2 , then the problem has a global bounded classical solution. Under further assumptions on the initial datum, the solutions satisfying 0 ≤ u (x , t) ≤ κ − 1 β for any (x , t) ∈ R N × [ 0 , + ∞) are shown to converge to κ − 1 β uniformly on any compact subset of 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. [ABSTRACT FROM AUTHOR]