On the long-time behaviour of age and trait structured population dynamics

TL;DR

This paper investigates the long-term dynamics of populations structured by age and traits, demonstrating conditions for convergence to equilibrium or the emergence of singular stationary solutions through spectral analysis.

Contribution

It introduces a spectral analysis framework for understanding long-time behavior of age and trait-structured populations, including conditions for convergence and singular solutions.

Findings

Existence of eventually singular stationary solutions.

Convergence to a unique equilibrium when stationary measures are absolutely continuous.

Spectral properties determine long-term population dynamics.

Abstract

We study the long-time behaviour of a population structured by age and a phenotypic trait under a selection-mutation dynamics. By analysing spectral properties of a family of positive operators on measure spaces, we show the existence of eventually singular stationary solutions. When the stationary measures are absolutely continuous with a continuous density, we show the convergence of the dynamics to the unique equilibrium.