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 by far the most natural one. 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 trouble with the PDE formulation is that the solution operators are NOT differentiable, precluding rigorous linearisation. (The underlying reason is the same as for state-dependent delay equations: we need to differentiate with respect to a quantity that appears as argument of a function that may not be differentiable.) The DELAY formulation does not suffer from this technical difficulty and the Principle of Linearized Stability holds [1]. The two semigroups are intertwined by a non-invertible operator. Under certain conditions this 'equivalence' yields a rather indirect proof of the Principle of Linearized Stability for the PDE formulation [2].

[1] Diekmann, O., & Gyllenberg, M. Equations with infinite delay: blending the abstract and the concrete. Journal of Differential Equations (2012) 252 : 819-851

[2] Barril, C., Calsina, À., Diekmann, O., & Farkas, J. Z. (2021). On the formulation of size-structured consumer resource models (with special attention for the principle of linearized stability). Mathematical Models and Methods in Applied Sciences (2022) 32 : 1141-1191