Threatening excursions in large population quasi-stationary birth and death systems. On a question of Antonio Galves
We consider time continuous multispecies birth and death processes in a regime of large populations. The jump rates depend on a large scaling parameter K modeling the charge capacity. When K tends to infinity, the process is close (in finite time) to a dynamical system containing a non zero global attracting equilibrium and zero as unstable equilibrium. For each fixed K, extinction in finite time occurs almost surely and a quasi-stationary distribution occurs naturally in the study of the statistics over times scales which are large but smaller than the extinction time scale. Before this catastrophic event the process makes many unsuccessful large deviations attempts with time scales corresponding to how far it deviates from the quasi-equilibrium. The paper concerns the statistical description of these typical trajectories starting from the quasi-stationary distribution until extinction. An unusual mixing property yields large time scale behavior for the process starting from a fixed state. We give a precise statistical description of the successive exit times of the process rescaled by K from a neighborhood of the equilibrium of the dynamical system in a clumping time scale and prove their asymptotic Poisson distribution. We also give a precise description of the asymptotic distribution of the successive records until extinction.