Finite dimensional state representation of physiologically structured populations

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 ODE-reducible. 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 environment-dependent 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.

Numerical methods and hypoexponential approximations for gamma distributed delay differential equations

“…depends on if we are using the smoothed or fixed hypoexponential approximation. In both cases, the convolution integral ∞ 0 Y (t − s)f P (s)ds will satisfy a system of n ordinary differential equations in a similar manner to the linear chain technique [Cassidy, 2020;Diekmann et al, 2018Diekmann et al, , 2020a. To show that this is indeed the case, we introduce n auxiliary variables B i (t) satisfying…”

“…Inspired by the lack of existing appropriate numerical methods for problems such as (1.1), there has also been considerable interest in approximating infinite delay DDEs by forms that are more convenient for simulation Diekmann et al, 2018Diekmann et al, , 2020aHurtado and Kirosingh, 2019;Koch and Schropp, 2015;Krzyzanski, 2019]. The most well-known of these is the previously mentioned linear chain technique, wherein modellers often make the simplifying assumption that j ∈ N when implementing gamma distributed DDE models.…”

Towards a replicator dynamics model of age structured populations

“…However, how to construct a general description of the relationships between demographic structure and population dynamics is still an unsolved problem (Caswell 2011). More precise than matrix models are continuous approaches arising from Lotka's renewal equation (Lotka 1911, Diekmann et al 2020a and the McKendrick von Foerster model (McKendrick 1926). The combination of demography with a game theoretic perspective focused on frequency dependent selection, advocated by McNamara (2013), can be very useful since demographers are interested in the patterns produced by heterogeneity in the populations (Vaupel et al 1979;Vaupel and Yashin 1983;Hougaard 1984;Vaupel and Yashin 1985).…”

Distributed Delay Differential Equation Representations of Cyclic Differential Equations

“…It is well established that, when the relationship between stages is linear, these compartmental models "hide" delays [4,6,24,47,48]. Recently, there has been increased interest establishing the equivalence between models that explicitly include delays, like renewal or distributed delay differential equations (DDEs), and multi-stage ordinary differential equation (ODE) models [8,12,13,14,27].…”

“…The linear chain trick, or linear chain technique (LCT), establishes the equivalence between Erlang distributed DDEs and transit compartment ODE models with constant transition rate [33,47]. Recently, a number of authors have generalized the LCT to other distributions and model formulations [13,14,27]. Often, these transit compartment ODE models take the form…”