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Nonlinear Models

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The Basic Approach to Age-Structured Population Dynamics

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Abstract

In the previous chapters we presented a linear model that can be considered the age-structured version of the so-called Malthus model. Thus, the criticisms of the latter also apply to the former: exponential growth is no more realistic for age-structured models than it is for unstructured ones, unless we just want to follow the growth of the population for a limited time during which the restrictive assumptions that we have made are satisfied. The golden age dreamed of by Raymond Queneau (ironically limited to the extreme cases of increasing nutrients or to absence of reproduction) has to be replaced by logistic growth and by those models that include intra-specific interactions. In Sect. 1.1.6 of Chap. 1 we have discussed some of the mechanisms through which these interactions occur, and in this chapter we shall formulate a fairly general nonlinear model that enables us to consider the main phenomenology of the single population growth . After presenting some general results on the analysis of this general model, we shall discuss some specific cases in the context of adult juvenile dynamics, corresponding to different intra-specific mechanisms such as competition, the Allee effect and cannibalism . All these effects produce stationary states that may be considered reasonable substitutes for Queneau’s utopian golden age.

XXXI

QU’IL N’Y A PAS EU DE VÉRITABLE AGE D’OR

Il n’aurait pu y avoir age d’or veritable, c’est-à-dire de durée indéfinie, que si la végétation avait pu fournir, sur place, une quantité croissante de nourriture; ou encore si l’homme ne s’était pas reproduit.

R. Queneau, Une Histoire Modèle, 1966

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Notes

  1. 1.

    XXXI: THERE HAS NOT BEEN A REAL GOLDEN AGE/ There could not have been a real golden age, that is of unlimited duration, except if the vegetation would have been able to provide, on the spot, an increasing amount of food; or else if man had not reproduced.

  2. 2.

    The assumptions on the functions Φ and Ψ are actually stronger than necessary. In fact, here we have assumed that both are strictly monotone, even though it is enough that just one of them is.

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Authors and Affiliations

  1. Department of Mathematics, University of Trento, Trento, Italy

    Mimmo Iannelli

  2. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona, USA

    Fabio Milner

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  1. Mimmo Iannelli
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Iannelli, M., Milner, F. (2017). Nonlinear Models. In: The Basic Approach to Age-Structured Population Dynamics. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1146-1_5

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