Approximate reduction of multiregional birth-death models with fast migration

In this work, we deal with the reduction of a time discrete model for a population distributed among N spatial patches and whose dynamics is controlled both by reproduction and by migration. These processes take place at different time scales in the sense of the latter being much faster than the former. We incorporate the effect of demographic stochasticity into the population, which results in both dynamics being modelled by multitype Galton-Watson branching processes. We present a multitype global model that incorporates the effect of the two processes and develop a method that takes advantage of the difference of time scales to reduce the model obtaining a unitype “aggregated” process that approximates the evolution of the total size of the population. We show that, given the separation of time scales between the birth-death process and the migration process is sufficiently high, we can obtain both qualitative and quantitative information about the behavior of the multitype global model through the study of this simple aggregated model.