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Appears in Collections:Computing Science and Mathematics Journal Articles
Peer Review Status: Refereed
Title: Structured populations with distributed recruitment: from PDE to delay formulation
Author(s): Calsina, Angel
Diekmann, Odo
Farkas, Jozsef Zoltan
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Keywords: Physiologically structured populations
distributed recruitment
delay formulation
spectral theory of positive operators
Issue Date: Dec-2016
Date Deposited: 11-Mar-2016
Citation: Calsina A, Diekmann O & Farkas JZ (2016) Structured populations with distributed recruitment: from PDE to delay formulation. Mathematical Methods in the Applied Sciences, 39 (18), pp. 5175-5191.;
Abstract: In this work first we consider a physiologically structured population model with a distributed recruitment process. That is, our model allows newly recruited individuals to enter the population at all possible individual states, in principle. The model can be naturally formulated as a first order partial integro-differential equation, and it has been studied extensively. In particular, it is well-posed on the biologically relevant state space of Lebesgue integrable functions. We also formulate a delayed integral equation (renewal equation) for the distributed birth rate of the po\-pu\-lation. We aim to illustrate the connection between the partial integro-differential and the delayed integral equation formulation of the model utilising a recent spectral theoretic result. In particular, we consider the equivalence of the steady state problems in the two different formulations, which then leads us to characterise irreducibility of the semigroup governing the linear partial integro-differential equation. Furthermore, using the method of characteristics, we investigate the connection between the time dependent problems. In particular, we prove that any (non-negative) solution of the delayed integral equation determines a (non-negative) solution of the partial differential equation and vice versa. The results obtained for the particular distributed states at birth model then lead us to present some very general results, which establish the equivalence between a general class of partial differential and delay equation, modelling physiologically structured populations.
DOI Link: 10.1002/mma.3898
Rights: This item has been embargoed for a period. During the embargo please use the Request a Copy feature at the foot of the Repository record to request a copy directly from the author. You can only request a copy if you wish to use this work for your own research or private study. This is the peer reviewed version of the following article: Calsina, À., Diekmann, O., and Farkas, J. Z. (2016) Structured populations with distributed recruitment: from PDE to delay formulation. Math. Meth. Appl. Sci., 39: 5175–5191. doi: , which has been published in final form at This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.

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