Structured Populations with Distributed Recruitment: from Pde to Delay Formulation

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 population. 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. 1. Prologue Structured population models are of great interest both from the mathematical and the application point of view. Traditionally they have been formulated as partial differential equations (PDEs for short) often with non-local (and nonlinear) boundary conditions. The early monograph [41] provided a comprehensive mathematical treatment of nonlinear age-structured population dynamics; while [12] provides valuable insight into the modelling and analysis of size-structured populations, which is accessible for theoretical biologists alike. We also refer the interested reader to the “green book” [37], which provides a systematic introduction to structured population models. Date: January 29, 2016. 1991 Mathematics Subject Classification. 92D25, 35L04, 34K30.