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

TL;DR

This paper compares two mathematical models for size-structured populations with resources, establishing conditions for their equivalence and analyzing stability properties, with implications for understanding population dynamics.

Contribution

It introduces and compares PDE and delay formulations for size-structured models, proving their equivalence and analyzing stability and differentiability of solutions.

Findings

Conditions for semigroup equivalence are identified.

Differentiability of solution operators is established for the delay formulation.

The Principle of Linearised Stability is proved via the delay model.

Abstract

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.