Analysis of a chemotaxis system modeling ant foraging

TL;DR

This paper analyzes a PDE system modeling ant foraging behavior through chemotaxis, establishing well-posedness and boundedness of solutions in two dimensions using advanced mathematical techniques.

Contribution

It provides the first rigorous analysis of a chemotaxis-based ant foraging model, proving well-posedness and solution boundedness in 2D.

Findings

Solutions remain bounded for all times in 2D

Established well-posedness for the PDE system

Developed a priori estimates in Lebesgue spaces

Abstract

In this paper we analyze a system of PDEs recently introduced in [P. Amorim, {\it Modeling ant foraging: a {chemotaxis} approach with pheromones and trail formation}], in order to describe the dynamics of ant foraging. The system is made of convection-diffusion-reaction equations, and the coupling is driven by chemotaxis mechanisms. We establish the well-posedness for the model, and investigate the regularity issue for a large class of integrable data. Our main focus is on the (physically relevant) two-dimensional case with boundary conditions, where we prove that the solutions remain bounded for all times. The proof involves a series of fine \emph{a priori} estimates in Lebesgue spaces.