Non-local competition slows down front acceleration during dispersal evolution

[La compétition non-locale réduit l’accélération du front d’invasion en cas d’évolution de la dispersion]

Calvez, Vincent¹ ; Henderson, Christopher² ; Mirrahimi, Sepideh³ ; Turanova, Olga⁴ ; Dumont, Thierry⁵

¹ Institut Camille Jordan, UMR 5208 CNRS & Université Claude Bernard Lyon 1, and Equipe-projet Inria Dracula, Lyon (France)
² Department of Mathematics, University of Arizona, Tucson, AZ 85721 (USA)
³ Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse; CNRS, UPS, F-31062 Toulouse Cedex 9 (France)
⁴ Michigan State University, East Lansing, MI 48824 (USA)
⁵ Institut Camille Jordan, UMR 5208 CNRS & Université Claude Bernard Lyon 1, Lyon (France)

Annales Henri Lebesgue, Tome 5 (2022), pp. 1-71.

Abstract (VO)

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 $𝒪\left(t^{3/2}\right)$. We identify a constant ${\alpha }^{*}$, and show that, in a weak sense, the front is located at ${\alpha }^{*}{t}^{3/2}$. Surprisingly, ${\alpha }^{*}$ 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 ${\alpha }^{*}$. 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 $𝒪\left(t^{3/2}\right)$. Ici, nous identifions une constante ${\alpha }^{*}$ telle que le front d’expansion est localisé autour de ${\alpha }^{*}{t}^{3/2}$, dans un sens faible. Curieusement, la constante ${\alpha }^{*}$ 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 ${\alpha }^{*}$. En complément de ce travail, des simulations numériques viennent illustrer le comportement attendu des solutions, au-delà des résultats analytiques.

Détail

Reçu le : 2019-10-15
Révisé le : 2021-02-25
Accepté le : 2021-08-07
Publié le : 2022-02-23

DOI : 10.5802/ahl.117

Classification : 35A18, 35A24, 35A30, 35B40, 35B50, 35C06, 35C07, 35D40, 35F21, 35K15, 35K57, 35Q92, 49L25, 92D15, 92D25, 92D40
Keywords: Reaction-diffusion, Dispersal evolution, Front acceleration, Linear determinacy, Approximation of geometric optics, Lagrangian dynamics, Explicit rate of expansion

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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