Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Demin, Ivan; Volpert, Vitaly

doi:10.1051/mmnp/20105506

Cited by 21 publications

(15 citation statements)

References 14 publications

(24 reference statements)

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“…We do not address the problem of existence of such a solution. This problem has been investigated for instance in [2,3,6]. The stability result is related to some spectral properties of the linearized operator around the wave solution w 0 , that reads…”

Section: Linear Stability Of Travelling Wave Solutionsmentioning

confidence: 99%

“…Similar models can describe evolution of cell populations where cells can produce mitosis factors diffusing in the extra-cellular matrix and stimulating proliferation of surrounding cells. The existence of travelling waves for such integro-differential equations is proved in [2], [3] in the case of functions φ with a small support and in [6] for an arbitrary support.…”

Section: Population Dynamics With Nonlocal Stimulation Of Reproductionmentioning

confidence: 99%

“…and where f = f (u, v) is some regular map. This kind of equations arises in many applications (see for instance [6,14,15] and the references cited therein).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectrum of some integro-differential operators and stability of travelling waves

Ducrot¹,

Marion²,

Volpert³

2011

Nonlinear Analysis: Theory, Methods & Applications

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Section: Linear Stability Of Travelling Wave Solutionsmentioning

confidence: 99%

Section: Population Dynamics With Nonlocal Stimulation Of Reproductionmentioning

confidence: 99%

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Spectrum of some integro-differential operators and stability of travelling waves

Ducrot¹,

Marion²,

Volpert³

2011

Nonlinear Analysis: Theory, Methods & Applications

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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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“…Besides, this model was also numerically investigated in [7,9,5,24,17,20,16], and those numerical results showed more behaviors than the theoretical ways. For more results on traveling wave solutions of (2) and other nonlocal reaction-diffusion equations, we refer to [1,3,4,12,13,37,21,22,26,29,31,32,33,34] and the references therein. It is clear that the study of Gourley et al [24] and Billingham [9] on traveling wave solutions of (1) was mainly based on numerical and asymptotic methods, which result in the lack of rigorously mathematical proof on the existence of traveling wave solutions of (1) with general kernel function φ and general admissible speed c (i.e.…”

mentioning

confidence: 99%

Traveling wave solutions in a nonlocal reaction-diffusion population model

Han¹,

Wang²

2016

CPAA

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This paper is concerned with a nonlocal reaction-diffusion equation with the form ∂u ∂t = ∂ 2 u ∂x 2 + u 1 + αu − βu 2 − (1 + α − β)(φ * u) , (t, x) ∈ (0, ∞) × R, where α and β are positive constants, 0 < β < 1 + α. We prove that there exists a number c * ≥ 2 such that the equation admits a positive traveling wave solution connecting the zero equilibrium to an unknown positive steady state for each speed c > c *. At the same time, we show that there is no such traveling wave solutions for speed c < 2. For sufficiently large speed c > c * , we further show that the steady state is the unique positive equilibrium. Using the lower and upper solutions method, we also establish the existence of monotone traveling wave fronts connecting the zero equilibrium and the positive equilibrium. Finally, for a specific kernel function φ(x) := 1 2σ e − |x| σ (σ > 0), by numerical simulations we show that the traveling wave solutions may connects the zero equilibrium to a periodic steady state as σ is increased. Furthermore, by the stability analysis we explain why and when a periodic steady state can appear.

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Existence of Waves for a Nonlocal Reaction-Diffusion Equation

Cited by 21 publications

References 14 publications

Spectrum of some integro-differential operators and stability of travelling waves

Spectrum of some integro-differential operators and stability of travelling waves

Preface to the Issue Nonlocal Reaction-Diffusion Equations

Traveling wave solutions in a nonlocal reaction-diffusion population model

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