On the Stochastic Modelling of Interacting Populations : a Multiscale Approach Leading to Hybrid Models

In this paper a review by the research work of the authors on the stochastic modelling of interacting individuals is presented. Both cases of direct and indirect interaction (via underlying fields) are considered. Due to the strong coupling among individuals, the evolution of each individual is governed by a stochastic equation whose parameters are themselves stochastic; as a consequence we are dealing with a doubly stochastic system, and this is a source of complexity which may tremendously increase as the number of individuals becomes extremely large. A possible way to reduce complexity is to apply suitable laws of large numbers, at a mesoscale, in order to obtain a mean field governed now by deterministic PDEs. In this way we may obtain an approximation of the driving fields which are deterministic at the macroscale, thus driving, at the microscale, a simply stochastic evolution for the individuals. Such models are called hybrid models.