Can a population survive in a shifting environment using non-local dispersion

Jérôme Coville (1)

1 BioSP - Biostatistique et Processus Spatiaux

In this article, we analyse the non-local model : ∂ t U (t, x) = J ⋆ U (t, x) − U (t, x) + f (x − ct, 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 exists a 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 this 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 generalized principal eigenvalue λ p of the linear problem cD 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 generalized first eigenvalue of an elliptic operator. The speeds c * ,± and c * * ,pm 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).

Mots clés: Nonlocal equation; shifting environment; persitence criteria

Domaines: Equations aux dérivées partielles [math.AP]

Jérôme Coville. Can a population survive in a shifting environment using non-local dispersion. 2020. ⟨hal-03077905⟩