Generalizing the Keller-Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting Species

In this paper we extend the famous Keller-Segel model for the chemotactic movement of motile species to some multi-species chemotaxis equations. The presented multi-species chemotaxis models are more general than those introduced so far and also include some interaction effects that have not been studied before. For example, we consider multi-species chemotaxis models with attraction and repulsion between interacting motile species. For some of the presented new models we give sufficient conditions for the existence of Lyapunov functionals. These new results are related to those of Wolansky (Scent and sensitivity: equilibria and stability of chemotactic systems in the absence of conflicts, preprint, 1998; Eur. J. Appl. Math. 13:641-661, 2002). Furthermore, a linear stability analysis is performed for uniform steady states, and results for the corresponding steady state problems are established. These include existence and nonexistence results for non-constant steady state solutions in some special cases.