On the Formulation and Analysis of General Deterministic Structured Population Models. II. Nonlinear Theory

Diekmann, O., Gyllenberg, M., Huang, H., Kirkilionis, M., Metz, J.A.J., & Thieme, H.R. (2002). On the Formulation and Analysis of General Deterministic Structured Population Models. II. Nonlinear Theory. IIASA Interim Report. IIASA, Laxenburg, Austria: IR-02-048

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Abstract

This paper is as much about a certain modeling methodology, as it is about the constructive definition of future population states from a description of individual behavior and an initial population state. The key idea is to build a nonlinear model in two steps, by explicitly introducing the environmental condition via the requirement that individuals are independent from one another (and hence equations are linear) when this condition is given (prescribed) as a function of time.

A linear physiologically structured population model is defined by two rules, one for reproduction and one for development and survival, both depending on the initial individual state and the prevailing environmental condition. In Part I we showed how one can constructively define future population state operators from these two ingredients.

A nonlinear model is a linear model together with a feedback law that describes how the environmental condition at any particular time depends on the population size and composition at that time. When applied to the solution of the linear problem, the feedback law yields a fixed point problem. This we solve constructively by means of the contraction mapping principle, for any given initial population state. Using subsequently this fixed point as input in the linear population model, we obtain a population semiflow. We then say that we solved the nonlinear problem.

The paper is organized in a top-down spirit: We describe a general abstract setting first and then specialize, while becoming more technical.

The results are not restricted to a single population but also cover the interaction (including predation) of several structured (and unstructured) populations.

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