Modelling physiologically structured populations: renewal equations and partial differential equations

Eugenia Franco; Odo Diekmann; Mats Gyllenberg

doi:10.1007/s00028-023-00880-4

Modelling physiologically structured populations: renewal equations and partial differential equations

Eugenia Franco^*, Odo Diekmann, Mats Gyllenberg

^*Corresponding author for this work

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

Abstract

We analyse the long term behaviour of the measure-valued solutions of a class of linear renewal equations modelling physiologically structured populations. The renewal equations that we consider are characterised by a regularisation property of the kernel. This regularisation property allows to deduce the large time behaviour of the measure-valued solutions from the asymptotic behaviour of their absolutely continuous, with respect to the Lebesgue measure, component. We apply the results to a model of cell growth and fission and to a model of waning and boosting of immunity. For both models we relate the renewal equation to the partial differential equation (PDE) formulation and draw conclusions about the asymptotic behaviour of the solutions of the PDEs.

Original language	English
Article number	46
Pages (from-to)	1-62
Number of pages	62
Journal	Journal of Evolution Equations
Volume	23
Issue number	3
Early online date	6 Jun 2023
DOIs	https://doi.org/10.1007/s00028-023-00880-4
Publication status	Published - Sept 2023

Keywords

Asynchronous exponential growth
Cell growth and fission model
Laplace transform
Measure-valued solutions
Waning and boosting of the level of immunity

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10.1007/s00028-023-00880-4

s00028-023-00880-4Final published version, 824 KBLicence: Taverne

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title = "Modelling physiologically structured populations: renewal equations and partial differential equations",

abstract = "We analyse the long term behaviour of the measure-valued solutions of a class of linear renewal equations modelling physiologically structured populations. The renewal equations that we consider are characterised by a regularisation property of the kernel. This regularisation property allows to deduce the large time behaviour of the measure-valued solutions from the asymptotic behaviour of their absolutely continuous, with respect to the Lebesgue measure, component. We apply the results to a model of cell growth and fission and to a model of waning and boosting of immunity. For both models we relate the renewal equation to the partial differential equation (PDE) formulation and draw conclusions about the asymptotic behaviour of the solutions of the PDEs.",

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AU - Diekmann, Odo

AU - Gyllenberg, Mats

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AB - We analyse the long term behaviour of the measure-valued solutions of a class of linear renewal equations modelling physiologically structured populations. The renewal equations that we consider are characterised by a regularisation property of the kernel. This regularisation property allows to deduce the large time behaviour of the measure-valued solutions from the asymptotic behaviour of their absolutely continuous, with respect to the Lebesgue measure, component. We apply the results to a model of cell growth and fission and to a model of waning and boosting of immunity. For both models we relate the renewal equation to the partial differential equation (PDE) formulation and draw conclusions about the asymptotic behaviour of the solutions of the PDEs.

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Modelling physiologically structured populations: renewal equations and partial differential equations

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