Sep 16, 2022 - Quasi-stationary distributions, extinction and resilience in ecological models - Speaker: Jean-René Chazottes (Ecole Polytechnique, France)

We consider birth-and-death processes going eventually to extinction with probability one, accounting for the finiteness of resources. A classical result says that if we rescale such a process by a parameter K and fix a finite horizon time, then the resulting sample paths are very close to the solutions of a differentiable equation (deterministic dynamical system) with high probability, when K goes to infinity. The corresponding vector field is given by the difference between the birth and death rates. In particular, the limit when times goes to infinity does not commute with the limit when K goes to infinity.

In this talk, I will try to explain what can be said for finite times and for finite K. A central object is the so-called quasi-stationary distribution which plays the role of a stationary distribution when conditioning upon non-extinction. Another one is the mean-time to extinction. Finally, I will touch upon the following question. Observing a realization of the process, can we determine the so-called engineering resilience? This quantity can be defined as the reciprocal of the characteristic return time to, say, a fixed point after a (small) perturbation.To answer this question, we establish two relations which intermingle the resilience, which is a macroscopic quantity defined for the dynamical system, and the fluctuations of the process, which are microscopic quantities.