Analysis of a chemotaxis system modeling ant foraging

Abstract

In this paper, we analyze a system of PDEs recently introduced in [P. Amorim, Modeling ant foraging: A chemotaxis approach with pheromones and trail formation, J. Theor. Biol. 385 (2015) 160–173], 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 focus is on the (physically relevant) two-dimensional case with boundary conditions, where we prove that the solutions remain bounded for all times. Further, we prove a hypercontractivity result (instantaneous formation of all [Formula: see text]-norms, [Formula: see text]). The proofs involve a series of fine a priori estimates in Lebesgue spaces and a De Giorgi level set technique.