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Approximate Aggregation Methods in Discrete Time Stochastic Population Models

Published online by Cambridge University Press:  08 April 2010

L. Sanz  and
J. A. Alonso
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Abstract

Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler ʽʽaggregated systemʼʼ governed by a few variables. 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 the system under consideration. In this work we deal with aggregation techniques in stochastic discrete time models and their application to the study of multiregional models, i.e., of models for an age structured population distributed amongst different spatial patches and in which migration between the patches is usually fast with respect to the demography (reproduction-survival) in each patch. Stochasticity in population models can be of two kinds: environmental and demographic. We review the formulation and the main properties of the dynamics of the different models for populations evolving in discrete time and subjected to the effects of environmental and demographic stochasticity. Then we present different stochastic multiregional models with two time scales in which migration is fast with respect to demography and we review the main relationships between the dynamics of the original complex system and the aggregated simpler one. Finally, and within the context of models with environmental stochasticity in which the environmental variation is Markovian, we make use these techniques to analyze qualitatively the behavior of two multiregional models in which the original complex system is intractable. In particular we study conditions under which the population goes extinct or grows exponentially.

Keywords

approximate aggregationenvironmental stochasticitydemographic stochasticitytime scalesmultiregional modelslognormal distribution
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 6: Ecology (Part 2) , 2010 , pp. 38 - 69
Copyright
© EDP Sciences, 2010

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