2017/02/17 by Jean-René Chazottes, Chazottes, J. -R., Pierre Collet +3
Mathematics · Medicine · #FOS: Biological sciences #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Populations and Evolution (q-bio.PE) #Probability (math.PR) #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1702.05369
openalex publication_date 2017/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a class of birth-and-death processes describing a population made of d sub-populations of different types which interact with one another. The state space is ℤ+d (unbounded). We assume that the population goes almost surely to extinction, so that the unique stationary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameter K which can be thought as the order of magnitude of the total size of the population at time 0. For any fixed finite time span, it is well-known that such processes, when renormalized by K, are close, in the limit K→+∞, to the solutions of a certain differential equation in ℝ+d whose vector field is determined by the birth and death rates. We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, for K large, the process will stay in the vicinity of the fixed point for a very long time before being absorbed at the origin. To precisely describe this behavior, we prove the existence of a quasi-stationary distribution (qsd). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before time t and the qsd. This bound is exponentially small in t, for t≫ log K. As a by-product, we obtain an estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than log K and much smaller than the mean time to extinction, which is exponentially large as a function of K. Let us stress that we are interested in what happens for finite K. We obtain results much beyond what large deviation techniques could provide.