Abstract. We present recent results on Piecewise Deterministic Markov Processes (PDMPs), involved in biological modeling. PDMPs, first introduced in the probabilistic literature by [30], are a very general class of Markov processes and are being increasingly popular in biological applications. They also give new interesting challenges from the theoretical point of view. We give here different examples on the long time behavior of switching Markov models applied to population dynamics, on uniform sampling in general branching models applied to structured population dynamic, on time scale separation in integrate-and-fire models used in neuroscience, and, finally, on moment calculus in stochastic models of gene expression. Résumé. Nous présentons des résultats récents sur les Processus de Markov Déterministes parMorceaux (PDMPs) utilisés en modélisation en biologie. Les PDMPs, introduits pour la première fois dans la littérature probabiliste par [30], forment une classe générale de processus de Markov et sont de plus en plus populaires dans les applications en biologie. Ils fournissent également de nouveaux défis intéressant du point de vue théorique. Nous donnons ici différents exemples sur le comportement en temps long de modèles de Markov modulés appliqués à la dynamique des populations, sur le tirage uniforme dans des modèles génériques de branchement appliqués à la dynamique de populations structurées, sur les séparations d'échelles de temps dans des modèles intègre-et-tire utilisés en neuroscience, et, finalement, sur le calcul de moments dans des modèles stochastiques d'expression des gènes.

Stochastic models that incorporate birth, death and immigration (also called birth-death and innovation models) are ubiquitous and applicable to many problems such as quantifying species sizes in ecological populations, describing gene family sizes, modeling lymphocyte evolution in the body. Many of these applications involve the immigration of new species into the system. We consider the full high-dimensional stochastic process associated with multispecies birth-deathimmigration and present a number of exact and asymptotic results at steady-state. We further include random mutations or interactions through a carrying capacity and nd the statistics of the total number of individuals, the total number of species, the species size distribution, and various diversity indices. Our results include a rigorous analysis of the behavior of these systems in the fast immigration limit which shows that of the di erent diversity indices, the species richness is best able to distinguish di erent types of birth-death-immigration models. We also nd that detailed balance is preserved in the simple noninteracting birth-death-immigration model and the birth-death-immigration model with carrying capacity implemented through death. Surprisingly, when carrying capacity is implemented through the birth rate, detailed balance is violated.