Population growth in discrete time: a renewal equation oriented survey

B. Boldin; O. Diekmann; J. A.J. Metz

doi:10.1080/10236198.2023.2265499

Population growth in discrete time: a renewal equation oriented survey

B. Boldin, O. Diekmann^*, J. A.J. Metz

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.

Original language	English
Pages (from-to)	1062-1090
Number of pages	29
Journal	Journal of Difference Equations and Applications
Volume	30
Issue number	8
Early online date	13 Oct 2023
DOIs	https://doi.org/10.1080/10236198.2023.2265499
Publication status	Published - 2024

Bibliographical note

Publisher Copyright:
© 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.

Funding

Barbara Boldin acknowledges the support of the Slovenian Research Agency [I0-0035, research program P1-0285 and research projects J1-9186, J1-1715]. Hans Metz benefited from the support from the \u2018Chaire Mod\u00E9lisation Math\u00E9matique et Biodiversit\u00E9 of Veolia Environnement-\u00C9cole Polytechnique-Museum National d'Histoire Naturelle-Fondation X\u2019.

Funders	Funder number
The Slovenian Research and Innovation Agency	J1-9186, I0-0035, J1-1715

Keywords

basic reproduction number
growth rate
next-generation matrix
renewal equation
reproductive value
stable distribution
Structured-population model

Access to Document

10.1080/10236198.2023.2265499Licence: CC BY-NC-ND

Population growth in discrete time a renewal equation oriented surveyFinal published version, 2.66 MBLicence: CC BY-NC-ND

Cite this

@article{96a3aa576495442f84f05d1f8f0f61b3,

title = "Population growth in discrete time: a renewal equation oriented survey",

abstract = "Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.",

keywords = "basic reproduction number, growth rate, next-generation matrix, renewal equation, reproductive value, stable distribution, Structured-population model",

author = "B. Boldin and O. Diekmann and Metz, {J. A.J.}",

note = "Publisher Copyright: {\textcopyright} 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.",

year = "2024",

doi = "10.1080/10236198.2023.2265499",

language = "English",

volume = "30",

pages = "1062--1090",

journal = "Journal of Difference Equations and Applications",

issn = "1023-6198",

publisher = "Taylor and Francis Ltd.",

number = "8",

}

TY - JOUR

T1 - Population growth in discrete time

T2 - a renewal equation oriented survey

AU - Boldin, B.

AU - Diekmann, O.

AU - Metz, J. A.J.

PY - 2024

Y1 - 2024

N2 - Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.

AB - Traditionally, population models distinguish individuals on the basis of their current state. Given a distribution, a discrete time model then specifies (precisely in deterministic models, probabilistically in stochastic models) the population distribution at the next time point. The renewal equation alternative concentrates on newborn individuals and the model specifies the production of offspring as a function of age. This has two advantages: (i) as a rule, there are far fewer birth states than individual states in general, so the dimension is often low; (ii) it relates seamlessly to the next-generation matrix and the basic reproduction number. Here we start from the renewal equation for the births and use results of Feller and Thieme to characterize the asymptotic large time behaviour. Next we explicitly elaborate the relationship between the two bookkeeping schemes. This allows us to transfer the characterization of the large time behaviour to traditional structured-population models.

KW - basic reproduction number

KW - growth rate

KW - next-generation matrix

KW - renewal equation

KW - reproductive value

KW - stable distribution

KW - Structured-population model

UR - http://www.scopus.com/inward/record.url?scp=85170855652&partnerID=8YFLogxK

U2 - 10.1080/10236198.2023.2265499

DO - 10.1080/10236198.2023.2265499

M3 - Article

AN - SCOPUS:85170855652

SN - 1023-6198

VL - 30

SP - 1062

EP - 1090

JO - Journal of Difference Equations and Applications

JF - Journal of Difference Equations and Applications

IS - 8

ER -