Speed-up of reaction-diffusion fronts by a line of fast diffusion

Berestycki, Henri; Coulon, Anne-Charline; Roquejoffre, Jean‐Michel; Rossi, Luca

doi:10.5802/slsedp.62

Cited by 16 publications

(21 citation statements)

References 29 publications

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“…A more general formula for the diffusion threshold, including the case when transport or reproduction take place on the road, was given in [10]: in particular, when reproduction rates are identical on the road and in the field, then acceleration occurs as soon as D > d. Related results were also obtained in various frameworks such as fractional diffusion [4], ignition type nonlinearities [14,15] and nonlocal exchange [23]. The issue of the spreading speed in oblique directions away from the road was investigated in [4,12] and revealed interesting properties where the shape of the propagation is no longer dictated by a Huygens principle, and may not even be convex. Lastly, we refer to [26] where the restriction of the KPP problem (1.3) to a truncated field, with a Dirichlet boundary condition and constant exchange coefficients, was studied.…”

Section: Introductionmentioning

confidence: 98%

A KPP road–field system with spatially periodic exchange terms

2015

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We take interest in a reaction-diffusion system which has been recently proposed [11] as a model for the effect of a road on propagation phenomena arising in epidemiology and ecology. This system consists in coupling a classical Fisher-KPP equation in a half-plane with a line with fast diffusion accounting for a straight road. The effect of the line on spreading properties of solutions (with compactly supported initial data) was investigated in a series of works starting from [11]. We recover these earlier results in a more general spatially periodic framework by exhibiting a threshold for road diffusion above which the propagation is driven by the road and the global speed is accelerated. We also discuss further applications of our approach, which will rely on the construction of a suitable generalized principal eigenvalue, and investigate in particular the spreading of solutions with exponentially decaying initial data.

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Section: Introductionmentioning

confidence: 98%

A KPP road–field system with spatially periodic exchange terms

2015

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“…He proved that their speed grows like √ D as D grows to infinity. We refer the reader to the survey [4].…”

Section: Introduction Presentation Of the Modelsmentioning

confidence: 99%

Travelling waves, spreading and extinction for Fisher–KPP propagation driven by a line with fast diffusion

2016

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In an earlier work [9], we introduced a parabolic system to describe biological invasions in the presence of a line (the "road") with a specific diffusion, included in a domain (the "field") subject to Fisher-KPP propagation. We determined the asymptotic spreading speed in the direction of the road, w * , in terms of the various parameters. The new result we establish here is the existence of travelling fronts for this system for any speed c w *. In addition, we show that no front can travel with a speed c < w *. We further extend the results to a new model where there is Fisher-KPP growth on the road, embedded in an elsewhere unfavourable plane. This is relevant to discuss certain invasions that have been observed along roads. We establish sharp conditions for extinction/invasion and describe the invasion dynamics.

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“…For example, the case where the exchanges coefficients

μ, ν

are not constant but periodic functions of

x

is treated in , non‐local exchanges are studied in , a combustion non‐linearity instead of a KPP one is considered in . The effect of non‐local diffusion is studied in . The case where the field is a cylinder with its boundary playing the role of the road is treated in .…”

Section: Introduction Known Results and Presentation Of The Modelmentioning

confidence: 99%

Influence of the geometry on a field-road model: the case of a conical field

Ducasse

2018

J. London Math. Soc.

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Field-road models are reaction-diffusion systems, recently introduced, designed to account for the effect of a road on propagation phenomena arising in epidemiology and ecology. Such systems consist in coupling a classical Fisher-Kolmogorov-Petrovski-Piskunov equation with a line with fast diffusion accounting for a road. A series of works investigate the spreading properties of such systems when the road is a straight line and the field a half-plane. Here, we take interest in the case where the field is a cone. Our main result is that the spreading speed is not influenced by the angle of the cone. Introduction, known results and presentation of the model Field-road modelsThe study of reaction-diffusion equations and systems is motivated by a wide range of applications, in particular in ecology and epidemiology. Such equations can indeed model propagation phenomena arising from the combined effects of diffusion, which accounts in a biological context for random motion of individuals, and reaction (resulting from reproduction and mortality). The most iconic example is the Fisher-KPP equation (see [10] for the biological motivations and [13] for a mathematical study):The function f is supposed to satisfy the KPP hypothesis, that is, f is a locally Lipschitz continuous function on [0, +∞), f > 0 on ]0, 1[, f < 0 on ]1, +∞[, f (0) = f (1) = 0 and v → f (v) v is non-increasing (hence, f is differentiable at 0 and f (0) > 0). This is called the KPP property, and this setting will be assumed through all the paper. A typical example is the logistic non-linearity f (v) = v(1 − v). In the KPP setting, it is proven in [13] (see also [1] for further properties) that, if u 0 ≡ 0 is a non-negative compactly supported initial datum, then the solution u(t, x) of the Fisher-KPP equation arising from this datum goes to 1 as t goes to infinity, locally uniformly in x. This property is called invasion. One can then define the speed of invasion as a quantity c KP P 0 such that ∀ c > c KP P , sup |x| ctMoreover, the speed of invasion can be explicitly computed in the case of the Fisher-KPP equation: c KP P = 2 df (0), see [1]. In [6], Berestycki, Roquejoffre and Rossi proposed a new system specifically devised for modeling the role of roads in biological invasions (informally, a road will be a region where

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Speed-up of reaction-diffusion fronts by a line of fast diffusion

Cited by 16 publications

References 29 publications

A KPP road–field system with spatially periodic exchange terms

A KPP road–field system with spatially periodic exchange terms

Travelling waves, spreading and extinction for Fisher–KPP propagation driven by a line with fast diffusion

Influence of the geometry on a field-road model: the case of a conical field

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