Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait
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Dynamics of Concentration in a Population Model Structured by Age and a Phenotypical Trait Samuel Nordmann1 · Benoît Perthame2 · Cécile Taing2
Received: 22 March 2017 / Accepted: 11 December 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017
Abstract We study a mathematical model describing the growth process of a population structured by age and a phenotypical trait, subject to aging, competition between individuals and rare mutations. Our goals are to describe the asymptotic behavior of the solution to a renewal type equation, and then to derive properties that illustrate the adaptive dynamics of such a population. We begin with a simplified model by discarding the effect of mutations, which allows us to introduce the main ideas and state the full result. Then we discuss the general model and its limitations. Our approach uses the eigenelements of a formal limiting operator, that depend on the structuring variables of the model and define an effective fitness. Then we introduce a new method which reduces the convergence proof to entropy estimates rather than estimates on the constrained Hamilton-Jacobi equation. Numerical tests illustrate the theory and show the selection of a fittest trait according to the effective fitness. For the problem with mutations, an unusual Hamiltonian arises with an exponential growth, for which we prove existence of a global viscosity solution, using an uncommon a priori estimate and a new uniqueness result. Keywords Adaptive evolution · Asymptotic behavior · Dirac concentrations · Hamilton-Jacobi equations · Mathematical biology · Renewal equation · Viscosity solutions Mathematics Subject Classification 35B40 · 35F21 · 35Q92 · 49L25
B B. Perthame
[email protected] S. Nordmann [email protected] C. Taing [email protected]
1
Ecole des Hautes Etudes en Sciences Sociales, CAMS, 190-198, avenue de France, 75244 Paris Cedex 13, France
2
Sorbonne Universités, UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions UMR CNRS 7598, UPD, Inria de Paris, F75005 Paris, France
S. Nordmann et al.
1 Introduction The mathematical description of competition between populations and selection phenomena leads to the use of nonlocal equations that are structured by a quantitative trait. A mathematical way to express the selection of the fittest trait is to prove that the population density concentrates as a Dirac mass (or a sum of Dirac masses) located on this trait. This result has been obtained for various models with parabolic [6, 9, 32] and integro-differential equations [8, 20, 31]. More generally, convergence to positive measures in selection-mutation models has been studied by many authors, see [1, 11, 13] for example. The question that we pose in the present paper is the long time behavior of the population density when the growth rate depends both on phenotypical fitness and age. This question brings up to consider the aging parameter and to use renewal type equations. Accordingly, the aim of this paper is to study the asymptotic behavior of the solut
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