Deux modèles de population dans un environnement périodique lent ou rapide

Two problems in population dynamics are addressed in a slow or rapid periodic environment. We first obtain a Taylor expansion for the probability of non-extinction of a supercriticial linear birth-and-death process with periodic coefficients when the period is large or small. If the birth rate is lower than the mortality for part of the period and the period tends to infinity, then the probability of non-extinction tends to a discontinuous limit related to a "canard" in a slow-fast system. Secondly, a nonlinear S-I-R epidemic model is studied when the contact rate fluctuates rapidly. The final size of the epidemic is close to that obtained by replacing the contact rate with its average. An approximation of the correction can be calculated analytically when the basic reproduction number of the epidemic is close to 1. The correction term, which can be either positive or negative, is proportional to both the period of oscillations and the initial fraction of infected people.