Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data

Henderson, Christopher

doi:10.1088/0951-7715/29/11/3215

Cited by 16 publications

(10 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we give a rigorous proof of this spreading rate both in the local and non-local models. This is an addition to the growing list of "accelerating fronts" that have attracted some interest in recent years [9,13,14,16,19,24,26,29].…”

Section: Introductionmentioning

confidence: 99%

Super-linear spreading in local and non-local cane toads equations

Bouin

Henderson

Ryzhik

2017

Journal de Mathématiques Pures et Appliquées

Self Cite

View full text Add to dashboard Cite

In this paper, we show super-linear propagation in a nonlocal reaction-diffusion-mutation equation modeling the invasion of cane toads in Australia that has attracted attention recently from the mathematical point of view. The population of toads is structured by a phenotypical trait that governs the spatial diffusion. In this paper, we are concerned with the case when the diffusivity can take unbounded values, and we prove that the population spreads as t 3/2 . We also get the sharp rate of spreading in a related local model.

show abstract

Section: Introductionmentioning

confidence: 99%

Super-linear spreading in local and non-local cane toads equations

Bouin

Henderson

Ryzhik

2017

Journal de Mathématiques Pures et Appliquées

Self Cite

View full text Add to dashboard Cite

show abstract

“…The mechanism is in fact closer to that of nonlocal Fisher-KPP propagation [8], [9]. It is also not so far from what happens with the classical Fisher-KPP with slowly decreasing initial data, [12], [13].…”

Section: The Underlying Mechanism Of Theorem 12 Discussionmentioning

confidence: 60%

Improved bounds for reaction-diffusion propagation with a line of nonlocal diffusion

Chalmin,

Roquejoffre

2019

Preprint

View full text Add to dashboard Cite

We consider here a model of accelerating fronts, introduced in [2], consisting of one equation with nonlocal diffusion on a line, coupled via the boundary condition with a reaction-diffusion equation of the Fisher-KPP type in the upper half-plane. It was proved in [2] that the propagation is accelerated in the direction of the line exponentially fast in time. We make this estimate more precise by computing an explicit correction that is algebraic in time. Unexpectedly, the solution mimicks the behaviour of the solution of the equation linearised around the rest state 0 in a closer way than in the classical fractional Fisher-KPP model. This paper is dedicated to S. Salsa, as the expression of our friendship and respect.

show abstract

“…For example, the case of initial data which oscillate between two decreasing exponential functions and which lead to convergent or non-convergent quantities X ± γ (t)/t has been addressed in [11,18], when f does not depend on x. For initial data decaying more slowly than any exponential function, then acceleration occurs: lim t→+∞ X ± γ (t)/t = +∞ (this has been proved when f does not depend on x in [12,13], and when f is periodic in x in [13]). On the other hand, we refer to [5] for a review of situations where X ± γ (t)/t converges as t → +∞.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Diameters of the level sets for reaction-diffusion equations in nonperiodic slowly varying media *

Hamel¹,

Nadin²

2021

Preprint

View full text Add to dashboard Cite

We consider in this article reaction-diffusion equations of the Fisher-KPP type with a nonlinearity depending on the space variable x, oscillating slowly and non-periodically. We are interested in the width of the interface between the unstable steady state 0 and the stable steady state 1 of the solutions of the Cauchy problem. We prove that, if the heterogeneity has large enough oscillations, then the width of this interface, that is, the diameter of some level sets, diverges linearly as t → +∞ along some sequences of times, while it is sublinear along other sequences. As a corollary, we show that under these conditions generalized transition fronts do not exist for this equation.

show abstract

Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data

Cited by 16 publications

References 59 publications

Super-linear spreading in local and non-local cane toads equations

Super-linear spreading in local and non-local cane toads equations

Improved bounds for reaction-diffusion propagation with a line of nonlocal diffusion

Diameters of the level sets for reaction-diffusion equations in nonperiodic slowly varying media *

Contact Info

Product

Resources

About