Non-local competition slows down front acceleration during dispersal evolution
Annales Henri Lebesgue, Volume 5 (2022), pp. 1-71.

Metadata

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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