Approximations and Behavior of stochastic Individual-based Models – ABIM

The ABIM project aims at understanding and quantifying two major features in the modeling of biological systems: the behavior (in short and long time) of stochastic individual-based models and their links with macroscopic models. Deriving macroscopic approximations from individual-based models is the key to define precisely the parameters of the macroscopic dynamics in terms of measurable quantities and to use powerful tools for prediction and sensitivity analysis. We focus on the convergence of stochastic models to dynamical systems with functions of interactions at the population level, typically called functional responses, networks of intraspecific and interspecific interactions and inhomogeneous environment. We want both to control the error terms in such approximations via functional limit theorems (diffusion approximations and central limit theorems) and specify the time-interval during which such approximations hold. Moreover, we study the persistence and stability of these individual-based stochastic models and compare their behavior to the macroscopic approximation. From a mathematical point of view, it raises challenging problems such as the approximations of interacting systems beyond the mean field regime, concentration inequalities for transient multidimensional Markov processes, long-time behavior for Markov chains in random environment and short-time behavior for Markov processes with non-Lipschitz drift. This mathematical framework in theoretical biology will allow us to answer basic and general questions in biology and ecology. In particular, how stochastic functional responses can be derived from the individuals, for instance based on observations or experiments ? How these functional responses may fluctuate and how they affect population dynamics and predictions ? Then we can derive applications in ecology, for instance for predicting the effects of environmental or management changes on the extinction of populations.