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Journal Articles Nonlinear Analysis: Theory, Methods and Applications Year : 2021

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

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Jérôme Coville
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#### Abstract

In this article, we analyse the non-local model: partial derivative U-t(t, x) = J * U(t, x) - U(t, x) + f (x - ct,U (t, x)) fort > 0, and x is an element of 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 lambda(p) of the linear problem cD(x)[rho] + J * rho - rho + partial derivative(s)f (x, 0)rho lambda(p)rho = 0 in R, is negative. lambda(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 lambda(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, partial derivative(s)f(x,o) > 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.

#### Domains

Mathematics [math]

### Dates and versions

hal-03766079 , version 1 (31-08-2022)

### Identifiers

• HAL Id : hal-03766079 , version 1
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### Cite

Jérôme Coville. Can a population survive in a shifting environment using non-local dispersion?. Nonlinear Analysis: Theory, Methods and Applications, 2021, 212, pp.112416. ⟨10.1016/j.na.2021.112416⟩. ⟨hal-03766079⟩

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