Persistence criteria for populations with non-local dispersion

Abstract

In this article, we analyse the non-local model :where J is a positive continuous dispersal kernel and f (x, u) 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). For compactly supported dispersal kernels J, we derive an optimal persistence criteria. We prove that a positive stationary solution exists if and only if the generalised principal eigenvalue λ p of the linear problemis negative. λ p is a spectral quantity that we defined in the spirit of the generalised first eigenvalue of an elliptic operator. In addition, for any continuous non-negative initial data that is bounded or integrable, we establish the long time behaviour of the solution u(t, x). We also analyse the impact of the size of the support of the dispersal kernel on the persistence criteria. We exhibit situations where the dispersal strategy has "no impact" on the persistence of the species and other ones where the slowest dispersal strategy is not any more an "Ecological Stable Strategy". We also discuss persistence criteria for fat-tailed kernels.

Liouville type results for a nonlocal obstacle problem

“…In other words, the nonlocal dispersion operator in or would be replaced by and the kernel would be given by . Furthermore, using for example , the associated energy of , with in place of , can be thought of as an approximation of that of (see and also where similar quantities are considered in a biological framework). Now, to see how the Neumann boundary condition in can be recovered from or with as , let us consider for simplicity the case where is of class with unit normal and the bounded function is of class and is extended as a function still denoted by .…”

A spectral condition for Liouville-type result of monostable KPP equation in periodic shear flows

“…During the time, this model has been proved to be a good model to study the complexity of many natural phenomena, various aspects have been investigated and numerous interesting results were already obtained. More recently, KPP equations with a given forced speed was used to describe the dynamics of a population facing a climate change by Berestycki et al [2,3,7,8] under the additional condition lim sup |z|→∞ ∂ u f (z, 0) ≤ −m < 0, z = (x, y). (1.2) In [2,3,7,8], assumption (1.2) means that the environment of species is completely unfavorable outside a compact set and it may be favorable inside.…”

“…More recently, KPP equations with a given forced speed was used to describe the dynamics of a population facing a climate change by Berestycki et al [2,3,7,8] under the additional condition lim sup |z|→∞ ∂ u f (z, 0) ≤ −m < 0, z = (x, y). (1.2) In [2,3,7,8], assumption (1.2) means that the environment of species is completely unfavorable outside a compact set and it may be favorable inside. This kind of model is newly investigated in one dimensional space by Li et al [12] under assumption that ∂ u f (z, 0) is positive near positive infinity and is negative near negative infinity.…”

Asymptotic behaviors for nonlocal diffusion equations about the dispersal spread

“…A typical example of such a nonlinearity is given by f (x, s) := s(a(x) − s) with a ∈ C(Ω). Nonlocal dispersal equations of the form (1.1) are widely used to model dispersal phenomena which exhibit nonlocal internal interactions and have attracted much attention, see Bates and Zhao [2], Berestycki et al [4], Cao et al [7], Chasseigne et al [8], Cortázar et al [9], Coville [10,13], Coville et al [11], Fife [16], Hutson et al [18], Kao et al [19], Li et al [22][23][24][25], Shen and Xie [28], Shen and Zhang [30], Su et al [32], Sun et al [35,36], Wang and Lv [37], Yang et al [38] and Zhang et al [39][40][41]. From both mathematical and ecological points of view, nonlocal dispersal kernel functions can have a variety of forms.…”