One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics

Eugenia Franco; Mats Gyllenberg; Odo Diekmann

doi:10.1007/s10440-021-00440-3

One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics

Eugenia Franco^*, Mats Gyllenberg, Odo Diekmann

^*Corresponding author for this work

University of Helsinki

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

Abstract

Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller’s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.

Original language	English
Article number	12
Pages (from-to)	1-67
Number of pages	67
Journal	Acta Applicandae Mathematicae
Volume	175
Issue number	1
DOIs	https://doi.org/10.1007/s10440-021-00440-3
Publication status	Published - Oct 2021

Bibliographical note

Funding Information:
Open access funding provided by University of Helsinki including Helsinki University Central Hospital. The research was supported by the Atmospheric Mathematics (AtMath) collaboration of the Faculty of Science of the University of Helsinki, the Horizon 2020 ERC project as well as the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (project No. 307333)

Publisher Copyright:
© 2021, The Author(s).

Funding

Open access funding provided by University of Helsinki including Helsinki University Central Hospital. The research was supported by the Atmospheric Mathematics (AtMath) collaboration of the Faculty of Science of the University of Helsinki, the Horizon 2020 ERC project as well as the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (project No. 307333)

Keywords

Balanced exponential growth
Convolution
Laplace transform
Malthusian parameter
Volterra integral equations

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10.1007/s10440-021-00440-3Licence: CC BY

Franco2021_Article_OneDimensionalReductionOfAReneFinal published version, 1.61 MBLicence: CC BY

Cite this

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abstract = "Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller{\textquoteright}s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.",

keywords = "Balanced exponential growth, Convolution, Laplace transform, Malthusian parameter, Volterra integral equations",

author = "Eugenia Franco and Mats Gyllenberg and Odo Diekmann",

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N2 - Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller’s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.

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KW - Balanced exponential growth

KW - Convolution

KW - Laplace transform

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KW - Volterra integral equations

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One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics

Abstract

Bibliographical note

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Keywords

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