Approximate Aggregation Methods in Discrete Time Stochastic Population Models

Abstract

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 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.

Aggregation of Leslie matrix models with application to ten diverse animal species

“…It is useful to define the matrix Using this definition, may be written as In this form, resembles the standard aggregation of an input–output matrix from economics (Howe & Johnson, 1989), and the aggregation of discrete linear ecological models (Auger et al, 2008; Sanz & Alonso, 2010). Hooley (2000) used a similar aggregator to collapse (instead of ), leading to aggregated matrices that are not Leslie matrices, and where the age class widths do not match the projection interval.…”

Approximate reduction of linear population models governed by stochastic differential equations: application to multiregional models

“…We will use Equation (19) and for small ε will consider matrix C(ε) as a perturbation of matrix F 2 = I N ⊗ F + F ⊗ I N . First, let us study the spectral properties of F 2 .…”

“…In this way, the behaviour of the original system can be approximated, but not known with exactitude, in terms of the knowledge of the behaviour of the reduced system. Approximate aggregation techniques in population dynamics have been widely studied in the context of deterministic systems with different time scales both in continuous and discrete time (see the review in [2] and the references therein), as well as in discrete stochastic models incorporating either environmental [19] or demographic stochasticity [21].…”

Conditions for growth and extinction in matrix models with environmental stochasticity

“…As a result, in order to obtain the SGR, the main tools at the ecologist's disposal are computer simulations [18,10], certain perturbation techniques like Tuljapurkar's approximation, λ T , to the SGR (see [17] for the derivation and [8] for an application) and, in certain systems with time scales, approximate aggregation techniques that allow one to transform the original system to a scalar one [14,1,13].…”