Stochastic and Deterministic Population Dynamics

Investigating the dynamical evolution of an ensemble made of microscopic entities in mutual interaction constitutes a rich and fascinating problem, of paramount importance and cross-disciplinary interest. The intimate discreteness of any individual based stochastic models results in finite size corrections to the ideal mean-field dynamics. Under specific conditions, such microscopic disturbances can amplify as follows a complex resonance mechanism and yield to organized spatio-temporal patterns. More specifically, the measured concentration which reflects the distribution of the interacting entities (e.g. chemical species, biomolecules) can oscillate regularly in time and/or display spatially patched profile, collective phenomena which testify on a surprising degree of macroscopic order, as mediated by the stochastic component of the dynamics. Our research is aimed at exploring these effects into details and so contribute to elaborate on a comprehensive theoretical picture. This task is pursued with reference to specific models, of broad theoretical and applied relevance.  More concretely,  we have revisited the reference mechanism for patterns formation in biology, the so called Turing instability, classically assumed to rely on a mean-field description. By explicitly accounting for the stochastic nature of the microscopic medium, and consequently resolving the inherent finite size corrections, we extended the concept of Turing instability to a generalized setting that holds promise to bridge the gap between theory and observation.