Age-structured Trait Substitution Sequence Process and Canonical Equation

TL;DR

This paper develops a stochastic model for age-structured populations undergoing mutation and selection, deriving new evolutionary approximations like a generalized Trait Substitution Sequence and an age-dependent Canonical Equation, enriched by ecological PDEs.

Contribution

It introduces novel age-structured evolutionary models based on PDEs, extending existing adaptive dynamics frameworks with new mathematical tools and ecological insights.

Findings

Derived a jump process generalizing Trait Substitution Sequence for age-structured populations.

Established an age-dependent canonical equation under small mutation assumptions.

Analyzed long-term behavior and singularities of the models with numerical simulations.

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 a fitness function, describing the probability of invasion of the resident population by the mutant one, that can not 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.