The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries

Abstract

We introduce and study a class of free boundary models with "nonlocal diffusion", which are natural extensions of the free boundary models in [17] and elsewhere, where "local diffusion" is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant differences arise from the well known local diffusion model in [17].

The dynamics of a degenerate epidemic model with nonlocal diffusion and free boundaries

“…Using this we can follow the corresponding arguments of [4] to show that, for some sufficiently small T 0 = T 0 (µ, A, h 0 , ǫ 0 , u 0 , J) > 0 and any T ∈ (0, T 0 ], sup…”

“…Let us note that, due to (G1)-(G2), when θ > 0, f (w) := −aw + cG(w) b is a Fisher-KPP nonlinear function, namely it satisfies conditions (f3)-(f4) in [4]. Let us also recall from [4] that, lim l 2 −l 1 →+∞ λ p (L (l 1 ,l 2 ) + θ) = θ, lim…”

“…Proof of Theorem 1.1: Following the approach of [4], we make use of Lemma 2.2 and a fixed point argument. For any given T > 0 and (g * , h * )…”

“…Recently, Cao et al [4] proposed a nonlocal version of the logistic model of [9], and successfully extended many basic results of [9] to the nonlocal model. In this paper, following the approach of [4], we propose and examine a nonlocal version of (1.3), which has the form…”

“…mean that the expanding rate of the range [g(t), h(t)] is proportional to the outward flux of the population across the boundary of the range (see [4] for further explanations and justification).…”