On models of physiologically structured populations and their reduction to ordinary differential equations

Diekmann, Odo; Gyllenberg, Mats; Metz, J.A.J.

doi:10.1007/s00285-019-01431-7

Cited by 12 publications

(3 citation statements)

References 27 publications

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“…More recently we have taken the abstract variant of (8.22) as the starting point for a discussion of ODE-reducibility. In the paper (Diekmann et al 2019) we provide new examples, recap the present paper and explain the usefulness of asymptotic ODE-reducibility.…”

Section: Discussionmentioning

confidence: 98%

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Finite dimensional state representation of physiologically structured populations

2019

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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.

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Section: Discussionmentioning

confidence: 98%

“…In this section we give a preview of the main content of the paper, first for theoretical biologists and probabilists and then for all kinds of mathematicians. The much shorter paper (Diekmann et al 2019) provides additional examples and may serve as a more friendly user guide to ODE-reducibility of structured population models.…”

Section: Preview Of Sects 3 Tomentioning

confidence: 99%

Finite dimensional state representation of physiologically structured populations

2019

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“…A PSPM typically involves solving a transport equation, which is a partial differential equation (PDE) that describes the flow of individuals within a state space. Another approach for describing PSPMs based on renewal equations has also been developed (Diekmann et al, 2020), but it is better suited to model inter-generational change, whereas PDEs can describe the dynamics of a population in continuous time.…”

Section: Introductionmentioning

confidence: 99%

libpspm: A feature-rich numerical package for solving physiologically structured population models

Joshi,

Zhang,

Stefaniak

et al. 2023

Preprint

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For a vast majority of organisms, life-history processes depend on their physiological state, such as body size, as well as on their environment. Size-structured population models, or more generally, physiologically structured population models (PSPMs), have emerged as powerful tools for modelling the population dynamics of organisms, as they account for the dependences of growth, mortality, and fecundity rates on an organism's physiological state and capture feedbacks between a population's structure and its environment, including all types of density regulation. However, despite their widespread appeal across biological disciplines, few numerical packages exist for solving PSPMs in an accessible and computationally efficient way. The main reason for this is that PSPMs typically involve solving partial differential equations (PDEs), and no single numerical method works universally best, or even at all, for all PDEs. Here, we present libpspm, a general-purpose numerical library for solving user-defined PSPMs. libpspm provides eight different methods for solving the PDEs underlying PSPMs, including four semi-implicit solvers that can be used for solving stiff problems. Users can choose the desired method without changing the code specifying the PSPM. libpspm allows for predicting the dynamics of multiple physiologically structured or unstructured species, each of which can have its own distinct set of physiological states and demographic functions. By separating model definition from model solution, libpspm can make PSPM-based modelling accessible to non-specialists and thus promote the widespread adoption of PSPMs.

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Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods

et al. 2019

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We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.

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On models of physiologically structured populations and their reduction to ordinary differential equations

Cited by 12 publications

References 27 publications

Finite dimensional state representation of physiologically structured populations

Finite dimensional state representation of physiologically structured populations

libpspm: A feature-rich numerical package for solving physiologically structured population models

Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods

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