A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

TL;DR

This paper studies a multi-species chemotaxis system with complex interactions, establishing Lyapunov functionals, duality, and a critical mass threshold using variational methods and inequalities.

Contribution

It introduces a novel multi-species chemotaxis model with a probabilistic production/destruction mechanism and analyzes its variational and duality structures.

Findings

Existence of Lyapunov functionals for the system

Duality properties of the model established

Identification of a critical mass via a Hardy-Littlewood-Sobolev inequality

Abstract

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted vs. repelled by a single chemical substance. The production vs. destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model we investigate the variational structures, in particular we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy-Littlewood-Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.