On the formulation and analysis of general deterministic structured population models II. Nonlinear theory

Abstract.

This paper is as much about a certain modelling methodology, as it is about the constructive definition of future population states from a description of individual behaviour 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 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 map** 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.

For models formulated as first order partial differential equations with nonlocal boundary conditions our approach amounts to deriving an integral equation by integrating along characteristics. This integral equation is then solved by a successive approximation procedure. By formalizing the biological ingredients directly as integral kernels and using these to construct the semigroup of population state transition operators we (a) can do with rather minimal regularity conditions on the coefficients (i.e., on individual behaviour), (b) do not have to worry about regularity properties of solutions to guarantee that we really have constructed a solution of a pde.

The paper is organized in a top-down spirit: We describe a general abstract setting first and then specialise, 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.