  To describe the dynamics of a size-structured population and its unstructured
resource, we formulate bookkeeping equations in two different ways. The first,
called the PDE formulation, is rather standard. It employs a first order
partial differential equation, with a non-local boundary condition, for the
size-density of the consumer, coupled to an ordinary differential equation for
the resource concentration. The second is called the DELAY formulation and
employs a renewal equation for the population level birth rate of the consumer,
coupled to a delay differential equation for the (history of the) resource
concentration. With each of the two formulations we associate a constructively
defined semigroup of nonlinear solution operators. The two semigroups are
intertwined by a non-invertible operator. In this paper we delineate in what
sense the two semigroups are equivalent. In particular, we (i) identify
conditions on both the model ingredients and the choice of state space that
guarantee that the intertwining operator is surjective, (ii) focus on large
time behaviour and (iii) consider full orbits, i.e., orbits defined for time
running from $-\infty $ to $+\infty $. Conceptually, the PDE formulation is by
far the most natural one. It has, however, the technical drawback that the
solution operators are not differentiable, precluding rigorous linearisation.
For the delay formulation, one can (under certain conditions concerning the
model ingredients) prove the differentiability of the solution operators and
establish the Principle of Linearised Stability. Next the equivalence of the
two formulations yields a rather indirect proof of this principle for the PDE
formulation.
