Slow and fast scales for superprocess limits of age-structured populations

TL;DR

This paper establishes a superprocess limit for an age-structured population model with trait inheritance and mutations, using a martingale approach to handle interactions and multiple time scales.

Contribution

It introduces a novel martingale problem method to analyze the joint trait-age dynamics in interacting populations with different time scales.

Findings

Trait marginals converge to a superprocess

Age distributions average to trait-dependent equilibria

Whole process convergence holds for finite-dimensional marginals

Abstract

A superprocess limit for an interacting birth-death particle system modelling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to equilibria that depend on traits. The convergence of the whole process depending on trait and age, only holds for finite-dimensional time-marginals. We apply our results to the study of examples illustrating different cases of trade-off between competition and senescence.