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

Abstract

This paper deals with the stochastic modeling of a class of heterogeneous population in random environment, structured by discrete subgroups, and called birth–death–swap. In addition to demographic events modifying the population size, swap events, i.e. moves between subgroups, occur in the population. Event intensities are random functionals of the population. In the first part, we show that the complexity of the problem is significantly reduced by modeling the jump measure of the population as a multivariate counting process. 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 weak assumptions, by the thinning of a strongly dominating point process generated by the same Poisson measure. This key construction relies on a general strong comparison result, of independent interest.The second part is dedicated to averaging results when swap events are significantly more frequent than demographic events. An important ingredient is the stable convergence, which is well-adapted to the general random environment. The pathwise construction by domination yields tightness results straightforwardly. At the limit, the demographic intensity functionals are averaged against random kernels resulting from swap events. Finally, under a natural assumption, we show the convergence of the aggregated population to a birth–death process in random environment, with non-linear intensity functionals.