Probabilistic and Piecewise Deterministic models in Biology

Abstract

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.

On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors

“…We stress that the distribution of (Y (q) s ) 0≤s≤t under the conditional law P (q) x (· | ζ > t) is absolutely continuous with respect to that (X s ) 0≤s≤t under P x , and as a consequence, Y (q) inherits irreducibility from (7). (ii) Suppose further that the assumption (ii) of Theorem 1.1 also holds, and set Y := Y (ρ) .…”

“…The comparison of those conditions displays certain resemblance and but also differences. For instance, (12) and (13) in [11] can be loosely related to (25) and (26) here; we do not need here to make assumptions such as (5,7,10) in [11]; on the other hand [11] also covers the situation where the growth rate c does not fulfill (4).…”

“…Assumption (5) guaranties that the total jump rate (2) remains bounded, so the jump times of X never accumulate. One says X is piecewise deterministic (see [7] and references therein), in the sense that the trajectory t → X t is driven by the steady flow velocity c between two consecutive jumps, and jump times and locations are the sole source of randomness. We finally assume that the Markov process X is irreducible,…”

“…Throughout the rest of this work, we always assume that (4), (5) and (7) hold. We have two main purposes.…”

The competitive exclusion principle in stochastic environments

“…The dynamics of the switched system can be constructed as follows: We follow the dynamics of E u and switch between environments E u and E v at the rate q uv . It is interesting to note that in the limit case where the switching between the different states is fast, the dynamics can be approximated (Cloez et al (2017), Benaïm & Strickler (2019)) by the 'mixed' deterministic dynamics…”