Can a population survive in a shifting environment using non-local dispersion?. (November 2021) - Coville, Jérôme
Abstract: In this article, we analyse the non-local model: ∂ t U ( t, x ) = J ⋆ U ( t, x ) − U ( t, x ) + f ( x − c t, U ( t, x ) ) for t > 0,  and  x ∈ R, where J is a positive continuous dispersal kernel and f ( x, s ) is a heterogeneous KPP type non-linearity describing the growth rate of the population. The ecological niche of the population is assumed to be bounded (i.e. outside a compact set, the environment is assumed to be lethal for the population) and shifted through time at a constant speed c . For compactly supported dispersal kernels J, assuming that for c = 0 the population survive, we prove that there exist critical speeds c ∗, ± and c ∗ ∗, ± such that for all − c ∗, − < c < c ∗, + then the population will survive and will perish when c ≥ c ∗ ∗, + or c ≤ − c ∗ ∗, − . To derive these results we first obtain an optimal persistence criteria depending of the speed c for non local problem with a drift term. Namely, we prove that for a positive speed c the population persists if and only if the generalised principal eigenvalue λ p of the linear problem c D x [ φ ] + J ⋆ φ − φ + ∂ s f ( x, 0 ) φ + λ p φ = 0 in R, is negative. λ p is a spectral quantity that we defined in the spirit of the generalised first eigenvalue of an elliptic operator. The speeds c ∗, ± and c ∗ ∗, ± are then obtained through a fine analysis of the properties of λ p with respect to c . In particular, we establish its continuity with respect to the speed c . In addition, for any continuous bounded non-negative initial data, we establish the long time behaviour of the solution U ( t, x ) . In the specific situation, ∂ s f ( x, 0 ) > 1 and J symmetric we also investigate the behaviour of the critical speeds c ∗ and c ∗ ∗ with respect to the tail of the kernel J . We show in particular that even for very fat tailed kernel these two critical speeds exist.