APPROXIMATE REDUCTION OF MULTI-TYPE GALTON–WATSON PROCESSES WITH TWO TIME SCALES

Abstract

Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler "aggregated system" governed by a few "macrovariables". Moreover, they give results that allow one to extract information about the complex original system in terms of the behavior of the reduced one. Often, the feature that allows one to carry out such a reduction is the presence of different time scales. In this work we deal with the approximate aggregation of a model for a population subjected to demographic stochasticity and whose dynamics is controlled by two processes with different time scales. There are no restrictions on the slow process while the fast process is supposed to be conservative of the total number of individuals. The incorporation of the effects of demographic stochasticity in the dynamics of the population makes both the fast and the slow processes being modelled by two multi-type Galton–Watson branching processes. We present a multi-type global model that incorporates the combined effect of the fast and slow processes and develop a method that takes advantage of the difference of time scales to reduce the model obtaining an "aggregated" simpler system. We show that, given the separation of time scales between the two processes is high enough, we can obtain relevant information about the behavior of the multi-type global model through the study of this simple aggregated system.