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

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 t o build a nonlinear model in two steps, by explicitly introducing the envir onmental condition via the requirement that individuals are independent fro m one another (and hence equations are linear) when this condition is presc ribed as a function of time. A linear physiologically structured population model is defined by two rule s, one for reproduction and one for development and survival, both dependin g on the initial individual state and the prevailing environmental conditio n. 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 descr ibes 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 t hen say that we solved the nonlinear problem. O. Diekmann: Department of Mathematics, University of Utrecht, P.O. Box 800 10, 3580 TA Utrecht, The Netherlands M. Gyllenberg (corresponding author): Department of Mathematics, University of Turku, 20014 Turku. Finland. e-mail: matsgy1@utu. fi H. Huang: Department of Mathematics, Beijing Normal University, Beijing 100 875, P.R. of China M. Kirkilionis: Universitv of Heidelberg, Interdisziplinaeres Inst. f, wiss . Rechnen, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany J.A.J. Metz: Institute for Evolutionary and Ecological Sciences, Leiden Uni versity, Kaiserstruat 63. NL-2311 GP Leiden, The Netherlands and Adaptive D ynamics Network, IIASA, A-2361 Laxenburg. Austria H.R. Thieme: Department of' Mathematics, Arizona State University, Tempe, A Z 852871804, USA