# Global stability with selection in integro-differential Lotka-Volterra systems modelling trait-structured populations

## TL;DR

This paper studies the long-term behavior of trait-structured populations modeled by integro-differential equations, providing conditions for species persistence and trait selection, with applications to cancer drug resistance.

## Contribution

It offers new criteria for species persistence and trait selection in integro-differential Lotka-Volterra systems, extending classical models with non-local interactions.

## Key findings

- Provided a Lyapunov-based condition for species persistence.
- Established convergence rates for population dynamics.
- Characterized trait selection under mutualistic interactions.

## Abstract

We analyse the asymptotic behaviour of integro-differential equations modelling $N$ populations in interaction, all structured by different traits. Interactions are modelled by non-local terms involving linear combinations of the total number of individuals in each population. These models have already been shown to be suitable for the modelling of drug resistance in cancer, and they generalise the usual Lotka-Volterra ordinary differential equations. Our aim is to give conditions under which there is persistence of all species. Through the analysis of a Lyapunov function, our first main result gives a simple and general condition on the matrix of interactions, together with a convergence rate. The second main result establishes another type of condition in the specific case of mutualistic interactions. When either of these conditions is met, we describe which traits are asymptotically selected.