Spatial structures and generalized
travelling waves for an integro-differential equation

Apreutesei, Narcisa; Bessonov, Nikolai; Volpert, Vitaly; Vougalter, Vitali

doi:10.3934/dcdsb.2010.13.537

Cited by 63 publications

(86 citation statements)

References 16 publications

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“…Lorsque la transformée de Fourier de φ change de signe et μ est assez grand, alors l'état stationnaire u ≡ 1 devient instable au sens de Turing [1,4]. Ceci se caractérise par plusieurs différences avec les équations de Fisher/monostable ou bistables.…”

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Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

Nadin

Perthame

Tang

2011

Comptes Rendus. Mathématique

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Presented by the Editorial BoardThis Note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in Berestycki et al. (2009) [3] but their asymptotic behavior was still unclear. We use here a new numerical approximation of these traveling waves which shows that some traveling waves connect the two homogeneous steady states 0 and 1, which is a striking fact since 0 is dynamically unstable and 1 is unstable in the sense of Turing. © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é Nous étudions dans cette Note les propriétés des solutions de type ondes progressives pour l'équation de Fisher non-locale. L'existence de telles solutions a été prouvée récemment dans Berestycki et al. (2009) [3] mais leur comportement asymptotique était encore mal compris. Nous développons ici une nouvelle méthode d'approximation numérique montrant que certaines ondes progressives connectent les deux états d'équilibre homogènes 0 et 1, ce qui est surprenant puisque 0 est dynamiquement instable et 1 est instable au sens de Turing. Pour une équation semilinéaireune onde progressive peut relier deux états stationnaires u ≡ 0 et u ≡ 1 dans différentes situations. C'est le cas lorsque 0 est instable et 1 est stable (Fisher/monostable). C'est également le cas lorsque 0 et 1 sont tous deux stables (AllenCahn/bistable). Ces ondes sont alors attractives et la dynamique (1) permet de construire l'onde progressive [6]. Par contre si 0 et 1 sont instables, il existe un point stationnaire intermédiaire stable et celui-ci interdit la construction d'une onde progressive.

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Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

Nadin

Perthame

Tang

2011

Comptes Rendus. Mathématique

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“…The notion of generalized travelling waves, which can be characterized as propagating solutions existing for all times from −∞ to ∞ [32], becomes appropriate here and allows the proof of wave existence without the assumption that the support of the kernel is sufficiently narrow [4], [11]. Numerical simulations show that nonmonotone travelling waves can be stable, and there exist periodic travelling waves [19], [22], [36].…”

Section: Nonlocal Reaction-diffusion Equations In Population Dynamicsmentioning

confidence: 99%

“…They exist for all speeds greater than or equal to the minimal speed [4]. Their structure and the patterns formed behind the waves can depend on their speed.…”

Section: Nonlocal Reaction-diffusion Equations In Population Dynamicsmentioning

confidence: 99%

Preface to the Issue Nonlocal Reaction-Diffusion Equations

Alfaro

Apreutesei

Davidson

et al. 2015

Math. Model. Nat. Phenom.

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Nonlocal reaction-diffusion equations in population dynamicsNonlocal reaction-diffusion equations are intensively studied during the last decade in relation with problems in population dynamics and other applications. In comparison with traditional reaction-diffusion equations they possess new mathematical properties and richer nonlinear dynamics. Many studies are devoted to the nonlocal reaction-diffusion equationwherewhich describes the distribution of population density in the case of nonlocal consumption of resources.Here k is a positive integer, k = 1 corresponds to asexual and k = 2 to sexual reproduction. If the kernel φ of the integral is replaced by the δ-function, then we obtain conventional reaction-diffusion equation.In the case k = 1, it is the logistic equation with the reproduction term au(1 − u) proportional to the population density u and to available resources (1 − u). In the case of nonlocal consumption of resources, the integral J(u) describes consumption of resources at the space point x by individuals located in some area around this point. The function φ(x − y) determines the efficiency of such consumption. Introduction of nonlocal consumption of resources changes the properties of solutions of this equation. Consider for certainty the case where k = 1 and σ = 0. Suppose that ∞ −∞ φ(y)dy = 1. Then u = 1 is a stationary solution of this equation. It is stable in the case of the local equation but it can lose its stability for the nonlocal equation. If it becomes unstable, then a periodic in space stationary solution bifurcates from it [13], [19], [21]. This simple result of the linear stability analysis has important consequences from the mathematical point of view and for the applications.

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“…Such modifications enables the explanation of emergence and evolution of biological species as well as speciation in a more appropriate manner [27][28][29][30][31]. The models with nonlocal consumption of resources present complex dynamics for the single species models [28,29,[32][33][34][35] as well as for competition models including two or more species [32,[36][37][38]. Furthermore, such complex dynamics cannot be found in the corresponding local models.…”

Section: Introductionmentioning

confidence: 99%

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

2018

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Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra-and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with prey-dependent functional response, specialist predator and linear intrinsic death rate for predator species. The primary goal of our present work is to consider nonlocal consumption of resources in a spatiotemporal prey-predator model with bistable reaction kinetics for prey growth in the absence of predators. We derive the conditions of the Turing and of the spatial Hopf bifurcation around the coexisting homogeneous steady-state and verify the analytical results through extensive numerical simulations. Bifurcations of spatial patterns are also explored numerically.

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Spatial structures and generalized travelling waves for an integro-differential equation

Cited by 63 publications

References 16 publications

Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

Preface to the Issue Nonlocal Reaction-Diffusion Equations

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

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