Non-local competition slows down front acceleration during dispersal evolution

Calvez, Vincent; Henderson, Christopher; Mirrahimi, Sepideh; Turanova, Olga; Dumont, Thierry

doi:10.5802/ahl.117

Calvez, Vincent; Henderson, Christopher; Mirrahimi, Sepideh; Turanova, Olga; Dumont, Thierry

Non-local competition slows down front acceleration during dispersal evolution

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

Metadata

DOI10.5802/ahl.117

KeywordsReaction-diffusion, Dispersal evolution, Front acceleration, Linear determinacy, Approximation of geometric optics, Lagrangian dynamics, Explicit rate of expansion

Abstract

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.

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