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

TL;DR

This paper models a heterogeneous population with birth, death, and swap events in a random environment, using stochastic differential systems and averaging techniques to analyze the population dynamics and their convergence to a simplified birth-death process.

Contribution

It introduces a novel stochastic modeling framework for birth-death-swap populations in random environments, including new averaging results and convergence analysis.

Findings

Reduction of complex jump measure to a multivariate counting process

Establishment of averaging results for fast swap events

Convergence of the population to a non-linear birth-death process

Abstract

This paper deals with the stochastic modeling of a class of heterogeneous population in a random environment, called birth-death-swap. In addition to demographic events, swap events, i.e. moves between subgroups, occur in the population. Event intensities are random functionals of the multi-type population. In the first part, we show that the complexity of the problem is significantly reduced by modeling the jumps measure of the population, described by a multivariate counting process. This process is defined as a 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 depending on swap events. Finally, under a natural assumption, we show the convergence of the aggregated population to a "true" birth-death process in random environment, with non-linear intensity functionals.