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Approximate Aggregation Methods in Discrete Time StochasticPopulation 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 thatallow one to reduce a complex system involving many coupled variables obtaining a simplerʽʽaggregated systemʼʼ governed by a few variables. Moreover, they give results that allowone to extract information about the complex original system in terms of the behavior ofthe reduced one. Often, the feature that allows one to carry out such a reduction is thepresence of different time scales in the system under consideration. In this work we dealwith aggregation techniques in stochastic discrete time models and their application tothe study of multiregional models, i.e., of models for an age structured populationdistributed amongst different spatial patches and in which migration between the patchesis usually fast with respect to the demography (reproduction-survival) in each patch.Stochasticity in population models can be of two kinds: environmental and demographic. Wereview the formulation and the main properties of the dynamics of the different models forpopulations evolving in discrete time and subjected to the effects of environmental anddemographic stochasticity. Then we present different stochastic multiregional models withtwo time scales in which migration is fast with respect to demography and we review themain relationships between the dynamics of the original complex system and the aggregatedsimpler one. Finally, and within the context of models with environmental stochasticity inwhich the environmental variation is Markovian, we make use these techniques to analyzequalitatively the behavior of two multiregional models in which the original complexsystem is intractable. In particular we study conditions under which the population goesextinct 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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