Polytopic Lyapunov functions for persistence analysis of competing species
Résumé
We show that stability of the equilibrium of a family of interconnected scalar systems can be proved by using a sum of monotonic ${\mathcal C}^0$ functions as Lyapunov function. We prove this result in the general framework of nonlinear systems and then in the special case of Kolmogorov systems. As an application, it is then used to show that intra-specific competition can explain coexistence of several species in a chemostat where they compete for a single substrate. This invalidates the Competitive Exclusion Principle, that states that in the classical case (without this intra-specific competition), it is indeed known that only one of the species will survive.