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Trait Substitution Sequence process and Canonical Equation for age-structured populations

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Abstract

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin–McCamy equation in Demography. Another particularity is that they involve an establishment probability, describing the probability of invasion of the resident population by the mutant one, that cannot always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.

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Authors and Affiliations

  1. CMAP, Ecole Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France

    Sylvie Méléard

  2. Laboratoire Paul Painlevé, Université Lille 1, 59655, Villeneuve d’Ascq Cedex, France

    Viet Chi Tran

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  1. Sylvie Méléard
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  2. Viet Chi Tran
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Correspondence to Viet Chi Tran.

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Méléard, S., Tran, V.C. Trait Substitution Sequence process and Canonical Equation for age-structured populations. J. Math. Biol. 58, 881–921 (2009). https://doi.org/10.1007/s00285-008-0202-2

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  • DOI: https://doi.org/10.1007/s00285-008-0202-2

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