Ergodicity and stability of a dynamical system perturbed by impulsive random interventions

We determine all ergodic measures and their stability properties of a Markov operator that is associated to a Markov chain which ensues from impulsive random interventions in a one-dimensional deterministic dynamical system at equally spaced time points. This setting is inspired by a biological application in population dynamics, where samples (‘catches’) are drawn regularly from a growing population or part of a bacterial population is eradicated, e.g. through antibiotics. On the way, we formulate and prove a version of Orey’s convergence theorem and exponential ergodicity using essentially Banach lattice arguments and Banach’s Fixed Point Theorem, valid in the generality of a Polish state space. We use the Krylov–Bogoliubov–Beboutov–Yosida decomposition to show that we found all ergodic measures. Finally, we prove that the extinction probability is a continuous function of the initial population size that is strictly positive on part of the state space.