Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait

Abstract

We study an equation structured by age and a phenotypic trait describing the growth process of a population subject to aging, competition between individuals, and mutations. This leads to a renewal equation which occurs in many evolutionary biology problems. We aim to describe precisely the asymptotic behavior of the solution, to infer properties that illustrate the concentration and adaptive dynamics of such a population. This work is a continuation of [38] where the case without mutations is considered. When mutations are taken into account, it is necessary to control the corrector which is the main novelty of the present paper.Our approach consists in defining, by the Hopf transform, a Hamilton-Jacobi equation with an effective Hamiltonian as in homogenization problems. Its solution carries the singular part of the limiting density (typically Dirac masses) and the corrector defines the weights. The main new result of this paper is to prove that the corrector is uniformly bounded, using only the global Lipschitz and semi-convexity estimates for the viscosity solution of the Hamilton-Jacobi equation. We also establish the limiting equation satisfied by the corrector. To the best of our knowledge, this is the first example where such bounds can be proved in such a context.

Evolutionary dynamics of competing phenotype-structured populations in periodically fluctuating environments

“…Mathematical models formulated in terms of integrodifferential equations and non-local parabolic PDEs like those considered here have been increasingly used to achieve a more in-depth theoretical understanding of the mechanisms underlying phenotypic adaptation in a variety of biological contexts [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58]. In particular, our work follows earlier papers on non-local parabolic PDEs modelling evolutionary dynamics of populations structured by continuous traits in periodically-fluctuating environments [44,49].…”

A non-expanding transport distance for some structured equations

“…Using ( 29) and ( 30) with a function ϕ depending only on (x 1 , x 2 ) (or ( x 1 , x 2 )), we see that the marginals of v satisfy ( 27) and (28).…”

“…The formalism makes the link with the heat equation through a standard physical process used in particular to describe diffusion or anomalous diffusion, see recent analyses in [28,8,3]. We depart from the equation…”

Selected Topics on Reaction-Diffusion-Advection Models from Spatial Ecology

“…Another challenging question is to determine the limit of the timedependent solutions u ε (x, ξ, t) of (5.3) and (5.5) as → 0. See [94,132] and references therein for recent progress on the rigorous derivations of the canonical equations for the trait evolution in spatially structured mutation-selection models.…”