# American Institute of Mathematical Sciences

## Trait selection and rare mutations: The case of large diffusivities

 Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Paris Sorbonne Université, 4 place Jussieu, 75005 Paris, France

* Corresponding author: Idriss Mazari

Received  July 2018 Revised  March 2019 Published  July 2019

Fund Project: The author was partially supported by the Project "Analysis and simulation of optimal shapes - application to lifesciences" of the Paris City Hall

We consider a system of $N$ competing species, each of which can access a different resources distribution and who can disperse at different speeds. We fully characterize the existence and stability of steady-states for large diffusivities. Indeed, we prove that the resources distribution yielding the largest total population size at equilibrium is, broadly speaking, always the winner when species disperse quickly. The criterion also uses the different dispersal rates. The methods used rely on an expansion of the solutions of the Lotka-Volterra sytem for large diffusivities, and is an extension of the "slowest diffuser always wins" principle.

Using this method, we also study the case of an equation modelling a trait structured population, with small mutations. We assume that each trait is characterized by its diffusivity and the resources it can access. We similarly derive a criterion mixing these diffusivities and the total population size functional for the single species model to show that for rare mutations and large diffusivities, the population concentrates in a neighbourhood of a trait maximizing this criterion.

Citation: Idriss Mazari. Trait selection and rare mutations: The case of large diffusivities. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019163
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