Moments of density-dependent branching processes and their genealogy

Abstract

A density-dependent branching process is a particle system in which individuals reproduce independently, but in a way that depends on the current population size. This feature can model a wide range of ecological interactions at the cost of breaking the branching property. We propose a general approach for studying the genealogy of these models based on moments. Building on a recent work of Bansaye, we show how to compute recursively these moments in a similar spirit to the many-to-few formula in the theory of branching processes. These formulas enable one to deduce the convergence of the genealogy by studying the population density, for which stochastic calculus techniques are available. As a first application of these ideas, we consider a density-dependent branching process started close to a stable equilibrium of the ecological dynamics. We show that, under a finite second moment assumption, its genealogy converges to Kingman's coalescent when the carrying capacity of the population goes to infinity.