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.Comment: 28 pages, to appear in Mathematical Methods in the Applied Science
