# Probabilistic and Piecewise Deterministic models in Biology

## TL;DR

This paper reviews recent advances in Piecewise Deterministic Markov Processes (PDMPs) and their applications in biological modeling, highlighting their theoretical challenges and diverse use cases in population dynamics, neuroscience, and gene expression.

## Contribution

It provides new examples and insights into the long-term behavior, sampling methods, time scale separation, and moment calculus of PDMPs in various biological contexts.

## Key findings

- Analysis of long time behavior in population models
- Application of PDMPs to structured population dynamics
- Use of moment calculus in gene expression models

## Abstract

We present recent results on Piecewise Deterministic Markov Processes (PDMPs), involved in biological modeling. PDMPs, first introduced in the probabilistic literature by Davis (1984), 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.