Super-linear propagation for a general, local cane toads model

Henderson, Christopher; Perthame, Benôıt; Souganidis, Panagiotis E.

doi:10.48550/arxiv.1705.04029

Cited by 2 publications

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“…The rest of the proof of Proposition 3.1 assumed that |x| > 1, which was crucial for bounding the nonlocal diffusion; see (4.18) and (4. 19).…”

Section: Proof Of Proposition 32mentioning

confidence: 99%

“…This is consistent with the fact that the expected advancing level sets (the fronts) move with algebraic (in time) velocity. That is, for every h in the range of u and all t sufficiently large, the set {x : u(x, t) = h} is comparable in a quantified way to the set {|x| ≈ ct γ } for some c; see [18], [14], and [25] for the results for linear (γ = 1) growth, and the recent works of Berestycki, Mouhot, and Raoul [7], Bouin, Henderson, and Ryzhik [10], and Henderson, Perthame, Souganidis [19] for the superlinear (γ > 1) case.…”

Section: Introductionmentioning

confidence: 99%

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Front propagation for integro-differential KPP reaction–diffusion equations in periodic media

Souganidis

Tarfulea

2019

Nonlinear Differ. Equ. Appl.

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“…The rest of the proof of Proposition 3.1 assumed that |x| > 1, which was crucial for bounding the nonlocal diffusion; see (4.18) and (4. 19).…”

Section: Proof Of Proposition 32mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Front propagation for integro-differential KPP reaction–diffusion equations in periodic media

Souganidis

Tarfulea

2019

Nonlinear Differ. Equ. Appl.

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“…argued formally that the linear problem (omitting the quadratic saturation term) should propagate super-linearly as (4/3)t 3/2 at the leading order. This prediction was rigorously confirmed for the local version of (1.3), that is, when f (1 − ρ) is replaced by f (1 − f ), by Berestycki, Mouhot and Raoul [13] using probabilistic techniques, and by Bouin, Henderson and Ryzhik [21] using PDE arguments (see also Henderson, Perthame and Souganidis [43] for a more general model). While the local model is unrealistic for the context of spatial sorting, it allows the difficulties due to the unbounded diffusion to be isolated from those caused by the non-local saturation.…”

Section: Introduction and Main Resultsmentioning

confidence: 67%

“…We point out that we have reduced to /2. The first two inequalities follow from standard arguments in the theory of viscosity solutions, see, e.g., [43,Section 3.2] for a similar setting. The third inequality follows directly from the upper bound (3.24) and the fact that ψ(x, θ) is positive for max{x, θ} > 0.…”

Section: From the Comparison Principle We Deduce Thatmentioning

confidence: 99%

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Non-local competition slows down front acceleration during dispersal evolution

Cálvez¹,

Henderson²,

Mirrahimi³

et al. 2022

Annales Henri Lebesgue

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We investigate super-linear spreading in a reaction-diffusion model analogous to the Fisher-KPP equation, but in which the population is heterogeneous with respect to the dispersal ability of individuals and the saturation factor is non-local with respect to one variable. It was previously shown that the population expands as O(t 3/2 ). We identify a constant α * , and show that, in a weak sense, the front is located at α * t 3/2 . Surprisingly, α * is smaller than the prefactor predicted by the linear problem (that is, without saturation) and analogous problem with local saturation. This hindering phenomenon is the consequence of a subtle interplay between the non-local saturation and the non-trivial dynamics of some particular curves that carry the mass to the front. A careful analysis of these trajectories allows us to characterize the value α * . The article is complemented with numerical simulations that illustrate some behavior of the model that is beyond our analysis.Résumé. -Nous examinons le phénomène de propagation surlinéaire pour un modèle de réaction-diffusion analogue à l'équation de Fisher-KPP, mais pour lequel la population est hétérogène vis-à-vis du taux de dispersion de chaque individu, et de plus, le terme de saturation est non-local par rapport à la variable de dispersion. Il avait été démontré que la population s'étend comme un O(t 3/2 ). Ici, nous identifions une constante α * telle que le front d'expansion est localisé autour de α * t 3/2 , dans un sens faible. Curieusement, la constante α * est strictement inférieure au préfacteur obtenu à partir du problème linéarisé (en omettant la saturation), ce dernier coïncidant par ailleurs avec celui obtenu à partir du problème avec saturation locale. Ce phénomène de ralentissement est la conséquence d'une interaction subtile entre la saturation non-locale et la dynamique non-triviale de certaines trajectoires qui amènent la masse au front d'invasion. Une analyse très précise de ces courbes particulières nous permet de caractériser algébriquement la valeur de α * . En complément de ce travail, des simulations numériques viennent illustrer le comportement attendu des solutions, au-delà des résultats analytiques. Strategy of proofThis section is intended to sketch the main ideas underlying the proof. As a byproduct, we explain, in rough terms, the reason for slower propagation than linearly determined.

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Super-linear propagation for a general, local cane toads model

Cited by 2 publications

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Front propagation for integro-differential KPP reaction–diffusion equations in periodic media

Front propagation for integro-differential KPP reaction–diffusion equations in periodic media

Non-local competition slows down front acceleration during dispersal evolution

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