Hans Metz: Conditions for the ODE reducibility of physiologically structured population models

In a physiologically structured population model, individuals are characterised by certain variables, collectively called their i-state. The world in which these individuals live is characterised by another set of variables, collectively called the environmental condition. The population model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival and (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) population outputs, i.e., observables and variables that may act as input for a model for the dynamics of the environment (generating the environmental condition). When the environmental condition is a given function of time, the population model is linear. Density dependence and interaction with other populations is captured by feedback via the environment. This yields a systematic methodology for formulating a community model by combining building blocks that use a state-linear population models to define a nonlinear input-output maps. For some combinations of model ingredients a (infinite dimensional) physiologically structured population model may be replaced by a finite dimensional ODE without loss of relevant information. We then call that model ODE-reducible.The talk discusses a generally applicable constructive test for ODE-reducibility, and a complete catalogue of ODE-reducible models for the case of one-dimensional i-state,deterministic i-state dynamics and birth rates that are generated by a finite number of functionals that are subsumed under the population output.