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 behavior 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 linearization. (The underlying reason for the lack of differentiability is exactly the same as in the case of 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.) 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 Linearized Stability. Next, the equivalence of the two formulations yields a rather indirect proof of this principle for the PDE formulation.

为了描述规模结构化人口及其非结构化资源的动态，我们以两种不同的方式制定簿记方程。第一个称为 PDE 公式，是相当标准的。它采用一阶偏微分方程，具有非局部边界条件，用于消费者的规模密度，耦合到资源集中度的常微分方程。第二种称为延迟公式，它采用了消费者人口水平出生率的更新方程，以及资源集中（历史）的延迟微分方程。对于这两个公式中的每一个，我们将构造定义的非线性解算子半群联系起来。这两个半群由一个不可逆的算子交织在一起。在本文中，我们描述了这两个半群在什么意义上是等价的。特别是，我们（i）确定模型成分和状态空间选择的条件，以保证交织算子是满射的，（ii）关注大时间行为和（iii）考虑全轨道，即为时间定义的轨道从$-\infty$至$+\infty$. 从概念上讲，PDE 公式是迄今为止最自然的公式。然而，它的技术缺点是解算子不可微分，排除了严格的线性化。（缺乏可微性的根本原因与状态相关延迟方程的情况完全相同：我们需要对出现为可能不可微分的函数的参数的量进行微分。）对于延迟公式，可以（在有关模型成分的某些条件下）证明解算子的可微性并建立线性稳定性原理。接下来，两个公式的等价性为 PDE 公式的这一原理提供了一个相当间接的证明。