Finite dimensional state representation of physiologically structured populations
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Mathematical Biology
Finite dimensional state representation of physiologically structured populations Odo Diekmann1 · Mats Gyllenberg2
· Johan A. J. Metz3,4
Received: 21 August 2018 / Revised: 15 October 2019 © The Author(s) 2019
Abstract In a physiologically structured population model (PSPM) individuals are characterised by continuous variables, like age and size, 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 model consists of submodels for (i) the dynamics of the i-state, e.g. growth and maturation, (ii) survival, (iii) reproduction, with the relevant rates described as a function of (i-state, environmental condition), (iv) functions of (i-state, environmental condition), like biomass or feeding rate, that integrated over the i-state distribution together produce the output of the population model. When the environmental condition is treated as a given function of time (input), the population model becomes linear in the state. Density dependence and interaction with other populations is captured by feedback via a shared environment, i.e., by letting the environmental condition be influenced by the populations’ outputs. This yields a systematic methodology for formulating community models by coupling nonlinear input–output relations defined by state-linear population models. For some combinations of submodels an (infinite dimensional) PSPM can without loss of relevant information be replaced by a finite dimensional ODE. We then call the model ODEreducible. The present paper provides (a) a test for checking whether a PSPM is ODE reducible, and (b) a catalogue of all possible ODE-reducible models given certain restrictions, to wit: (i) the i-state dynamics is deterministic, (ii) the i-state space is one-dimensional, (iii) the birth rate can be written as a finite sum of environmentdependent distributions over the birth states weighted by environment independent ‘population outputs’. So under these restrictions our conditions for ODE-reducibility are not only sufficient but in fact necessary. Restriction (iii) has the desirable effect that it guarantees that the population trajectories are after a while fully determined by the solution of the ODE so that the latter gives a complete picture of the dynamics of the population and not just of its outputs. Keywords ODE-reducibility · Linear chain trick · Evolutionary system · Input–output system
Extended author information available on the last page of the article
123
O. Diekmann et al.
Mathematics Subject Classification 92D25 · 93B11
1 Introduction From the very beginning of community modelling, ordinary differential equations (ODEs) have been its main tool. This notwithstanding the fact that much earlier Euler (1760) and other mathematicians working on population dynamics had already considered age structured models, see (Bacaër 2008, 2011; Gyllenberg 2007) for more information on the history of population dynamics. This probably
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