Invasion dynamics for a quasi-critical birth-death process
Résumé
We study the dynamics of invasion of populations which enjoy positive density dependent effect. We start with a single individual and consider a single type birth and death process. The initial individual growth rate is zero and it increases. We prove that the probability that the population reaches macroscopic levels decreases as $\sqrt{K}$, where $K$ is the scaling parameter. We also describe the associated trajectories and show that invasion can be split into three time periods. First, the process needs to escape the neighborhood of $0$, and conditioning on survival, it grows linearly until the order $\sqrt{K}$. The renormalized process is approximated by a diffusion, as for critical branching process, with an additional drift term coming from cooperation. Second, in intermediate scale $\sqrt{K}$, we observe another diffusion, without conditioning. Finally, this diffusion can be approximated by classical fluid limit corresponding to macroscopic approximation by an ODE. The proof of the first phase involves change of probability and characterization of uniform integrability of martingales, while the two other phases need uniform approximations on polynomial time scales.