On an irreducibility type condition for the ergodicity of nonconservative semigroups

TL;DR

This paper introduces a new irreducibility condition for positive semi-groups that ensures exponential convergence, applicable to nonlocal population dynamics models in variable environments.

Contribution

It presents a novel irreducibility criterion inspired by Markov chains, enabling analysis of ergodicity and existence of principal eigenelements in complex dynamical systems.

Findings

Established exponential convergence under the new criterion

Proved existence and attractiveness of principal eigenelements

Applied results to nonlocal population dynamics models

Abstract

We propose a simple criterion, inspired from the irreducible aperiodic Markov chains, to derive the exponential convergence of general positive semi-groups. When not checkable on the whole state space, it can be combined to the use of Lyapunov functions. It differs from the usual generalization of irreducibility and is based on the accessibility of the trajectories of the underlying dynamics. It allows to obtain new existence results of principal eigenelements, and their exponential attractiveness, for a nonlocal selection-mutation population dynamics model defined in a space-time varying environment.