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

Nadin, Grégoire; Perthame, Benoı̂t; Tang, Min

doi:10.1016/j.crma.2011.03.008

Cited by 55 publications

(65 citation statements)

References 7 publications

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“…Suppose that supp N c ∩ (0, +∞) = ∅ and let φ(t) be a positive monotone wavefront. Set y(t) = 1 − φ(t) and let σ * be defined as in (20). Then χ + (σ * , c, τ ) is finite and non-negative.…”

Section: Necessity Of the Condition Imposed On χ + (Z C τ )mentioning

confidence: 99%

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

Trofimchuk¹,

Pinto²,

Trofimchuk³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of 'sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar et al, we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front. Keywords: food-limited model, monotone wave, nonlocal interaction, existence, uniqueness 2010 MSC: 34K60, 35K57, 92D25 R K(s, y, τ )u(t − s, x − y)dy, and K : R + × R × R + → R + is a function satisfyingI c,τ (λ) := R+×R K(s, y, τ )e −λ(cs+y) dsdy ∈ R, I c,τ (0) = 1, lim λ→−λ0(c) + I c,τ (λ) = +∞,

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Section: Necessity Of the Condition Imposed On χ + (Z C τ )mentioning

confidence: 99%

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

Trofimchuk¹,

Pinto²,

Trofimchuk³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Berestycki et al [8] proved that equation (1.4) admits traveling wave solutions connecting 0 to an unknown positive state for all speeds c ≥ c * = 2 √ μ and there exists no such traveling wave solution with the wave speed c < 2 √ μ. Then, Nadin et al [21] showed that the unknown state is just the equilibrium 1 for some traveling wave solutions. Fang and Zhao [10] further gave a sufficient and necessary condition for the existence of monotone traveling waves of equation (1.4) connecting two equilibria 0 and 1.…”

Section: Introductionmentioning

confidence: 99%

Traveling waves for the nonlocal diffusive single species model with Allee effect

Han

Wang

Feng

2016

Journal of Mathematical Analysis and Applications

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In this paper, we study a nonlocal diffusive single species model with Allee effect. We prove that the model admits positive traveling wave solutions connecting the equilibrium 0 to some unknown positive steady state if and only if the wave speed c ≥ 2 √ r, where r > 0 is the intrinsic rate of increase of the species. For the sufficient large wave speed c, we show that the unknown steady state is the unique positive equilibrium. For two types of convolution kernel functions, we investigate the change of the wave profile as the non-locality increases, and illustrate that the unknown steady state may be a positive periodic solution.

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“…In more sophisticated contexts (reaction–diffusion systems, nonlocal equations, etc. ), propagating terraces with unstable intermediate states are observed numerically [, among others]. Rigorous analytical studies are, however, very difficult and have only been carried out in simple cases and with Heavyside‐like or compactly supported initial data .…”

Section: Introductionmentioning

confidence: 99%

Invasion of open space by two competitors: spreading properties of monostable two‐species competition‐diffusion systems

Girardin

Lam

2019

Proc. London Math. Soc.

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This paper is concerned with some spreading properties of monostable Lotka–Volterra two‐species competition‐diffusion systems when the initial values are null or exponentially decaying in a right half line. Thanks to a careful construction of super‐solutions and sub‐solutions, we improve previously known results and settle open questions. In particular, we show that if the weaker competitor is also the faster one, then it is able to evade the stronger and slower competitor by invading first into unoccupied territories. The pair of speeds depends on the initial values. If these are null in a right half‐line, then the first speed is the KPP speed of the fastest competitor and the second speed is given by an exact formula describing the possibility of nonlocal pulling. Furthermore, the unbounded set of pairs of speeds achievable with exponentially decaying initial values is characterized, up to a negligible set.

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

Cited by 55 publications

References 7 publications

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

Traveling waves for the nonlocal diffusive single species model with Allee effect

Invasion of open space by two competitors: spreading properties of monostable two‐species competition‐diffusion systems

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