# Asymptotic behavior of age-structured and delayed Lotka-Volterra models

## TL;DR

This paper analyzes the long-term behavior of an age-structured Lotka-Volterra model with delay, establishing conditions for stability, coexistence, and periodic solutions, and extending results to the PDE formulation.

## Contribution

It introduces a novel analysis of asymptotic properties and stability for a delayed, age-structured Lotka-Volterra model, including the existence of periodic solutions.

## Key findings

- Existence of a unique coexistence equilibrium.
- Characterization of conditions for periodic solutions.
- Asymptotic stability of the equilibrium and basin of attraction.

## Abstract

In this work we investigate some asymptotic properties of an age-structured Lotka-Volterra model, where a specific choice of the functional parameters allows us to formulate it as a delayed problem, for which we prove the existence of a unique coexistence equilibrium and characterize the existence of a periodic solution. We also exhibit a Lyapunov functional that enables us to reduce the attractive set to either the nontrivial equilibrium or to a periodic solution. We then prove the asymptotic stability of the nontrivial equilibrium where, depending on the existence of the periodic trajectory, we make explicit the basin of attraction of the equilibrium. Finally, we prove that these results can be extended to the initial PDE problem.