Trait selection and rare mutations: The case of large diffusivities

Idriss Mazari

doi:10.3934/dcdsb.2019163

Article Contents

2019, Volume 24, Issue 12: 6693-6724. Doi: 10.3934/dcdsb.2019163

This issue Previous Article Boundary perturbations and steady states of structured populations Next Article Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations

Trait selection and rare mutations: The case of large diffusivities

Idriss Mazari^,

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

^* Corresponding author: Idriss Mazari
^* Corresponding author: Idriss Mazari

Received: June 30, 2018

Revised: February 28, 2019

Early access: July 2019

Published: September 2019

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

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Abstract

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.

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Mathematics Subject Classification: 35K57, 9K20, 92D15, 92D25.

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