Two species nonlocal diffusion systems with free boundaries

Abstract

<p style='text-indent:20px;'>We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by "nonlocal diffusion" instead of "local diffusion". We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in [<xref ref-type="bibr" rid="b7">7</xref>], and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several new difficulties, which are overcome by the use of some new techniques.</p>

Long-time dynamics of a nonlocal epidemic model with free boundaries: spreading-vanishing dichotomy

“…Hence, f 1 (x, t) = 0 in (g( t), h( t)). In particular, for x ∈ [−h 0 , h 0 ], With the help of Lemma 2.1, we can follow the approach of [10] and [21] to prove Theorem 1.1. Since the proof is very long, and does not require considerably new ideas, we postpone it to the end of the paper.…”

“…Step 4: Global existence and uniqueness. This step can be proved by the same argument used in the proof of Theorem 2.1 in [21], and we omit the details.…”

“…The proof of this step is long and tedious, but it is only a simple modification of the corresponding step in the proof of Theorem 2.1 in [21], so we leave the details to the interested reader.…”

A free boundary problem with nonlocal diffusion and unbounded initial range

“…Since the work [19], there have been lots of related researches on the free boundary problem with nonlocal diffusion, a sample of which can be referred to [21,22,23,24,25,26,27,28,29,30] and the references therein.…”

On monostable cooperative system with nonlocal diffusion and free boundaries

“…Since these two works appeared, some related research has emerged. For example, one can refer to [7] for the first attempt to the spreading speed of [4], [8,9,10] for the Lotka-Volterra competition and prey-predator models, [11] for high dimensional and radial symmetric version for Fisher-KPP equation, and [12] for the model with a fixed boundary and a moving boundary.…”