On eigenvalue problems arising from nonlocal diffusion models

Li, Fang; Coville, Jérôme; Wang, Xuefeng

doi:10.3934/dcds.2017036

Cited by 60 publications

(33 citation statements)

References 21 publications

(28 reference statements)

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“…In this paper, we are particularly interested in the properties of λ p (L (ℓ 1 ,ℓ 2 ) + a 0 ), with a 0 a positive constant and −∞ ≤ ℓ 1 < ℓ 2 ≤ +∞. In this special case, it is well known (see, e.g., [5,11,29]) that λ p (L (ℓ 1 ,ℓ 2 ) + a 0 ) is a principal eigenvalue. Moreover, we show that the following conclusions hold.…”

Section: Comparison Principlesmentioning

confidence: 99%

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

Cao

et al. 2019

Journal of Functional Analysis

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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].

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Section: Comparison Principlesmentioning

confidence: 99%

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

Cao

et al. 2019

Journal of Functional Analysis

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110

249

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“…The characterization of µ (θ d ,0) and ν (0,η D ) plays an important role in this paper. We present the related results from [11] as follows for the convenience of readers.…”

Section: Properties Of Nonlocal Eigenvalue Problemsmentioning

confidence: 99%

Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small

Bai¹,

Li²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we study the global dynamics of a general 2 × 2 competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper continues the work in [2], where competition models with symmetric nonlocal operators are considered. 2where d, D > 0, Ω is a bounded domain in R n , n ≥ 1, K and P represent nonlocal operators, which will be defined later.When nonlocal operators are replaced by differential operators, two-component systems like (1.1) allow for a large range of possible phenomena in chemistry, biology, ecology, physics and so on and have been extensively studied. One of the most famous phenomena is the idea, which was first proposed by Alan Turing, that a stable state in the local system can become unstable in the presence of diffusion. This remarkable idea is called "diffusion-driven instability", which is one of the most classical theories in the studies of pattern formations. Another important phenomena comes from ecology, where random diffusion is introduced to model dispersal strategies [18] and there are tremendous studies in this direction, see the books [4,17]. Indeed, dispersal strategies of organisms have been a central topic in ecology. However, when a long range dispersal is considered, nonlocal reaction diffusion equations are commonly used [5,6,9,15,16,19], where the nonlocal operator takes the following form:To be more specific, this paper is motivated by the studies of Lotka-Volterra type weak competition models with spatial inhomogeneity, that is f (are the population densities of two competing species, d, D > 0 are their dispersal rates, which measure the total number of dispersal individuals per unit time, respectively, while m(x) is nonconstant and represents spatial distribution of resources. This type of models reflects the interactions among dispersal strategies, spatial heterogeneity of resources and interspecific competition abilities on the persistence of species and has received extensive studies from both mathematicians and ecologists for the last three decades. For models with random diffusion, see [3,7,10,13,14] and the references therein, while for models with nonlocal dispersals, see [1,2,8,12] and the references therein.Inspired by the nature of this type of models, in [14], an insightful conjecture was proposed and partially verified:Conjucture A. The locally stable steady state is globally asymptotically stable.It is known that this conjecture is true for ODE systems. Recently, for symmetric PDE case, this conjecture has been completely resolved in [7]. Moreover, if random diffusion is replaced by symmetric nonlocal operators, this conjecture is also verified in [2]. In the proofs of these results, the symmetry property of operators and the particular form of reaction terms are crucial. These naturally lead us to investigate the system (1.1) with nonsymmetric operators and more general reaction terms. Now let us designate the de...

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“…Proof of Theorem 2.1. Since the spectrum of the operator L+f u (x, 0) has been thoroughly studied in [34], the arguments in [15,Section 6] can be applied. In particular, we just explain how to obtain the global convergence of the positive steady state when λ * > 0.…”

Section: Single-species Modelmentioning

confidence: 99%

“…Remark that as explained in Section 2.1, in general µ (u d ,0) and ν (0,v D ) might not be principal eigenvalues of the corresponding linearized operators. See [34] and its references for more discussions.…”

Section: Competition Modelsmentioning

confidence: 99%

Classification of global dynamics of competition models with nonlocal dispersals I: symmetric kernels

Bai

2018

Calc. Var.

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In this paper, we gives a complete classification of the global dynamics of twospecies Lotka-Volterra competition models with nonlocal dispersals:where K, P represent nonlocal operators, under the assumptions that the nonlocal operators are symmetric, the models admit two semi-trivial steady states and 0 < bc ≤ 1. In particular, when both semi-trivial steady states are locally stable, it is proved that there exist infinitely many steady states and the solution with nonnegative and nontrivial initial data converges to some steady state in C(Ω) × C(Ω). Furthermore, we generalize these results to the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals. u(y, t)dy≥ δσδ a J t = σδ 2 a J t.

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On eigenvalue problems arising from nonlocal diffusion models

Cited by 60 publications

References 21 publications

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

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

Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small

Classification of global dynamics of competition models with nonlocal dispersals I: symmetric kernels

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