Travelling waves for integro-differential equations in population
dynamics

Apreutesei, Narcisa; Ducrot, Arnaud; Volpert, Vitaly

doi:10.3934/dcdsb.2009.11.541

Cited by 48 publications

(55 citation statements)

References 13 publications

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“…If τ = 0 then we obtain operator A 0 with function φ(x), which has a small support, and, thus, the existence of a solution of equation A 0 (u) = 0 is known [3]. When τ = 1 we obtain equation (1.3).…”

Section: Homotopymentioning

confidence: 99%

“…In this case, the properties of the equation become quite different. It possesses an interesting nonlinear dynamics [4,8] but the wave existence can be proved only in the case of functions φ with a small support where the perturbation methods are applicable [1,2,3,5]. In this work we do not assume that the support is small.…”

Section: Theorem 1 There Exists a Monotone Travelling Wave That Is mentioning

confidence: 99%

“…The proof of this assertion follows the same lines as for elliptic operators (cf. [3]). We should verify that the Fredholm property remains valid in the weighted spaces,…”

Section: Fredholm Operator With Zero Indexmentioning

confidence: 99%

See 2 more Smart Citations

Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Demin

Volpert

2010

Math. Model. Nat. Phenom.

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Section: Homotopymentioning

confidence: 99%

Section: Theorem 1 There Exists a Monotone Travelling Wave That Is mentioning

confidence: 99%

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Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Demin

Volpert

2010

Math. Model. Nat. Phenom.

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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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“…The proof of wave existence in the case of nonlocal equation becomes much more involved, and there are only partial results [2], [5], [6], [11], [15]. 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].…”

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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Travelling waves for integro-differential equations in population dynamics

Cited by 48 publications

References 13 publications

Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

Preface to the Issue Nonlocal Reaction-Diffusion Equations

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