Persistence versus extinction under a climate change in mixed environments

Abstract

This paper is devoted to the study of the persistence versus extinction of species in the reaction-diffusion equation:where Ω is of cylindrical type or partially periodic domain, f is of Fisher-KPP type and the scalar c > 0 is a given forced speed. This type of equation originally comes from a model in population dynamics (see [3], [17], [18]) to study the impact of climate change on the persistence versus extinction of species. From these works, we know that the dynamics is governed by the traveling fronts u(t, x 1 , y) = U (x 1 − ct, y), thus characterizing the set of traveling fronts plays a major role. In this paper, we first consider a more general model than the model of [3] in higher dimensional space, where the environment is only assumed to be globally unfavorable with favorable pockets extending to infinity. We consider in two frameworks: the reaction term is time-independent or time-periodic dependent. For the latter, we study the concentration of the species when the environment outside Ω becomes extremely unfavorable and further prove a symmetry breaking property of the fronts.Mathematical Subject Classification (2010): 35C07, 35J15, 35B09, 35P20, 92D25.

Nonlocal dispersal equations in time-periodic media: Principal spectral theory, limiting properties and long-time dynamics

“…These notions are originally introduced by Berestycki, Nirenberg and Varadhan in [3] to study the principal spectral theory of elliptic operators. Since then, they are widely used to study the principal spectral theory of various linear operators associated to reaction-diffusion equations and nonlocal dispersal equations (see [1,4,5,9,19,26,27] and references therein). The equivalence of λ 1 (−L Ω ), λ p (−L Ω ) and λ p (−L Ω ) under the condition (1.4) provides not only sup-inf characterizations of λ 1 (−L Ω ), but also alternative and powerful tools in the spirit of analysis to study deeper qualitative properties of λ 1 (−L Ω ) to be presented.…”

“…It is worthwhile to point out that in the case f (t, x, s) = f (x, s), the global dynamics of (1.6) based on the bifurcation result of (1.7) have attracted a lot of attention recently due to their significance in applications and underlying mathematical challenges (see e.g. [7,9,23,1,2,25,26]). In particular, to investigate the problem, the authors in [7,9,1,2] took a PDE approach, while the authors in [23,25] employed a dynamical system approach.…”

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

“…There may have favorable (∂ u f (z, 0) > 0) and unfavorable (∂ u f (z, 0) < 0) patches extending to infinity but only at infinity the unfavorable regions dominate. This situation usually happens when studying the large time behavior of the species under the effect of global warming and therefore it is useful to describe the dynamics of the species facing a climatic metamorphosis [3,7,8,13,14]. If μ(y) ≡ −m < 0, one readily has λ p = m > 0, φ ≡ 1 and thus (1.2) is recovered.…”

“…(1.1), which compensates the lack of compactness of R N and therefore it may be useful in other investigation of the problem with non-compact domains. It is worth to mentioning that in [14], the author has considered the case, where n(y) ≡ 1, β(y) ≡ 0, α(y) ≡ constant and f depends periodically also in t. Our current result confirms that, in the environment being globally unfavorable at infinity in the sense of Hypothesis 2, the species survive if the unfavorable zone is dominated by the favorable zone, namely λ 1 < 0, otherwise it must be extinct.…”

Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

“…Hypothesis (A1) refers to the environments being unfavorable, unfavorably neutral or nearly neutral near infinity, according to the cases α = 0, α ∈ (0, p) or α = p. This kind of assumption is recently used to describe the effect of global warming (see [2], [5], [10], [32]). Hypothesis (A2) means that the intrinsic growth rate decreases when the population density is increasing.…”