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Approximating the distribution of population size in stochastic multiregional matrix models with fast migration

In this work we deal with a multiregional model in discrete time for an age-structured population which lives in an environment that changes randomly with time and is distributed in different spatial patches. In addition, and as is often the case in applications, we assume that migration is fast with respect to demography. Using approximate aggregation techniques we make use of the existence of different time scales in the model and reduce the dimension of the system obtaining a stochastic Leslie model in which the variables are the total population in each age class. Literature shows that, under reasonable conditions, the distribution of population size in matrix models with environmental stochasticity is asymptotically lognormal, and is characterized by two parameters, stochastic growth rate (s.g.r.) and scaled logarithmic variance (s.l.v.), that, in most practical cases, cannot be computed exactly. We show that the s.g.r. and the s.l.v. of the original multiregional model can be approximated by those corresponding to the reduced stochastic Leslie model, therefore simplifying its analysis. Moreover, we illustrate the usefulness of the reduction procedure by presenting some practical cases in which, although the explicit computation of the s.g.r. and the s.l.v. of the original multiregional model is not feasible, we can calculate its analogues for the reduced model.