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

On the formulation of size-structured consumer resource models (with special attention for the principle of linearised stability)