Attractors and confiners in demography
type: Item
instance of: scholarly article
title: Attractors and confiners in demography
zbMATH Open document ID: 0751.92007
DOI: 10.1007/BF02071046
published in: Annals of Operations Research
publication date: 1992-10-13T00:00:00Z
review text: Various dynamical models have been suggested by demographers to understand or to better forecast fertility fluctuations. Usual linear techniques have proved poor, but nonlinear techniques may help us to find out some temporal qualitative structure, and to take into account breakdown phenomena. For instance, the 1931-1985 fluctuations may be better described by drawing first return functions, which show a precise qualitative behaviour: two attractors exchanging the trajectory in the phase space under the perturbing environment of the historical context. A second example is drawn from a 17th century population, at a time when people were submitted to high and very irregular mortality. Because of the stochastic nature of the process, attractors are difficult to determine, and we resort to tools which are different from usual asymptotic attractors. These tools, called confiners, are the stochastic analogue of the notion of attractor for stochastic dynamical processes. They have been defined by \textit{M. Cosnard} and \textit{J. Demongeot} [C. R. Acad. Sci., Paris, Sér. I 300, 551-556 (1985; Zbl 0595.58032)] on the basis of the set of accumulation points of the trajectory, and rely on the essential intuitive property of invariance through the double operation consisting first in searching the confining basin of a set, then in taking the limit of this basin. The notion of confiner enables a rigorous description of the equilibrium states of the process: For fertility fluctuations during the 17th century, it reveals two confining zones exchanging the trajectory. Moreover, a simulation system shows how the existence of these two zones are directly linked to the frequency \(v\) of mortality crisis occurring, that is to say, according to the value \(v\), the population dynamics will more or less balance towards a resistance to the crisis or towards a usual renewal of the population.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 60K99
zbMATH DE Number: 68909
zbMATH Keywords: historical fluctuations
zbMATH Keywords: stochastic dynamics
zbMATH Keywords: demography
zbMATH Keywords: dynamical models
zbMATH Keywords: fertility fluctuations
zbMATH Keywords: attractors
zbMATH Keywords: confiners
zbMATH Keywords: equilibrium states
zbMATH Keywords: simulation
zbMATH Keywords: mortality
zbMATH Keywords: renewal
author: N. Bonneuil
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02071046
OpenAlex ID: W1966312491
description: scientific article; zbMATH DE number 68909