Anomalous invasion speed in a system of coupled reaction-diffusion equations

Faye, Grégory; Peltier, Gwenaël

doi:10.4310/cms.2018.v16.n2.a7

Cited by 3 publications

(7 citation statements)

References 10 publications

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“…This structure is crucial for our analysis building on the Feynman-Kac formula. Equations of a similar structure have been also been studied in [12,7]. It is furthermore convenient to eliminate the parameters K and α by rescaling.…”

Section: The F-kpp Equationsmentioning

confidence: 99%

“…Systems of coupled F-KPP equations have been studied in different contexts in the literature, see e.g. [10,6,11,13,12,7,14]. There has been interest in such systems in the context of dormancy, see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…[10] use purely analytic methods involving sub-and super-solutions, while the equations appearing in [6] and [4] allow for a representation in terms of branching Brownian motion and the use of martingale methods. The equations in [20] (and [12,7]) are particularly nice, as they allow for the use of the Feynman-Kac representation. However, even the introduction of two different diffusion constants seems to spoil this feature, and it seems unclear (albeit interesting) to see how this method can be extended to more general settings.…”

Section: Introductionmentioning

confidence: 99%

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The speed of invasion in an advancing population

Bovier¹,

Hartung²

2022

Preprint

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We derive rigorous estimates on the speed of invasion of an advantageous trait in a spatially advancing population in the context of a system of one-dimensional F-KPP equations. The model was introduced and studied heuristically and numerically in a paper by . In that paper, it was noted that the speed of invasion by the mutant trait is faster faster when the resident population ist expanding in space compared to the speed when the resident population is already present everywhere. We use the Feynman-Kac representation to provide rigorous estimates that confirm these predictions.

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Section: The F-kpp Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The speed of invasion in an advancing population

Bovier¹,

Hartung²

2022

Preprint

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“…Girardin and Lam ( 2019 ), Champneys et al. ( 1995 ), Holzer ( 2012 ), Holzer and Scheel ( 2012 ), Holzer and Scheel ( 2014 ), Holzer ( 2016 ), Faye and Peltier ( 2018 ), Keenan and Cornell ( 2021 ) and Lam and Yu ( 2022 ). In particular, an analogous result to that in Venegas-Ortiz et al.…”

Section: Introductionmentioning

confidence: 99%

“…The equations in Venegas-Ortiz et al. ( 2014 ), Holzer and Scheel ( 2014 ), Holzer ( 2016 ) and Faye and Peltier ( 2018 ) are particularly nice, as they allow for the use of the Feynman–Kac representation. However, even the introduction of two different diffusion constants seems to spoil this feature, and it seems unclear (albeit interesting) to see how this method can be extended to more general settings.…”

Section: Introductionmentioning

confidence: 99%

The speed of invasion in an advancing population

Bovier,

Hartung

2023

J. Math. Biol.

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We derive rigorous estimates on the speed of invasion of an advantageous trait in a spatially advancing population in the context of a system of one-dimensional F-KPP equations. The model was introduced and studied heuristically and numerically in a paper by Venegas-Ortiz et al. (Genetics 196:497–507, 2014). In that paper, it was noted that the speed of invasion by the mutant trait is faster when the resident population is expanding in space compared to the speed when the resident population is already present everywhere. We use the Feynman–Kac representation to provide rigorous estimates that confirm these predictions.

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Travelling waves for a fast reaction limit of a discrete coagulation–fragmentation model with diffusion and proliferation

Estavoyer,

Lepoutre

2024

J. Math. Biol.

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We study traveling wave solutions for a reaction-diffusion model, introduced in the article Calvez et al (2023), describing the spread of the social bacterium Myxococcus xanthus. This model describes the spatial dynamics of two different cluster sizes: isolated bacteria and paired bacteria. Two isolated bacteria can coagulate to form a cluster of two bacteria and conversely, a pair of bacteria can fragment into two isolated bacteria. Coagulation and fragmentation are assumed to occur at a certain rate denoted by k. In this article we study theoretically the limit of fast coagulation fragmentation corresponding mathematically to the limit when the value of the parameter k tends to +∞. For this regime, we demonstrate the existence and uniqueness of a transition between pulled and pushed fronts for a certain critical ratio θ ⋆ between the diffusion coefficient of isolated bacteria and the diffusion coefficient of paired bacteria. When the ratio is below θ ⋆ , the critical front speed is constant and corresponds to the linear speed. Conversely, when the ratio is above the critical threshold, the critical spreading speed becomes strictly greater than the linear speed.

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Anomalous invasion speed in a system of coupled reaction-diffusion equations

Cited by 3 publications

References 10 publications

The speed of invasion in an advancing population

The speed of invasion in an advancing population

The speed of invasion in an advancing population

Travelling waves for a fast reaction limit of a discrete coagulation–fragmentation model with diffusion and proliferation

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