Birth Death Swap population in random environment and aggregation with two timescales.

Summary This paper deals with the stochastic modeling of a class of heterogeneous population dynamics in a random environment. These Birth-Death-Swap populations generalize Markov multi-type Birth-Death processes, by considering swap events (moves between subgroups) in addition to demographic events, and allowing event intensities to be random functional of the population. The complexity of the problem is significantly reduced by modeling the jumps measure of the population, described by a multivariate counting process. In the spirit of Massouli\'e (1998), this process is defined as the solution of a stochastic differential system with random coefficients, driven by a multivariate Poisson random measure. The solution is obtained under weaker assumptions than usual, by thinning of a dominating point process driven by the same Poisson measure.This key construction rely on a general comparison result of independent interest. The second part is dedicated to averaging results when swap events are more frequent than demographic events. An important ingredient is the stable convergence which extends naturally the convergence in distribution in the presence of a random environment. The pathwise construction by domination yields straightforward tightness results, in particular for the population process which is considered as a simple variable on $\Omega \times \mathbb R^+$. At the limit, the demographic intensity functionals are averaged against random kernels depending on swap events. Finally, we show under a natural assumption the convergence of the aggregated population to a ``true'' Birth-Death process in random environment with density-dependence.