Size-Structured Population Dynamics

TL;DR

This paper develops a mathematical framework for size-structured population dynamics with variable maturation delays, combining ODEs, renewal equations, and delay equations, and demonstrates numerical bifurcation analysis using pseudospectral methods.

Contribution

It introduces a delay equation approach to model size-structured populations with state-dependent delays, overcoming limitations of PDE methods for stability analysis.

Findings

Constructive solution approach yields weak solutions of PDEs.

Delay equation formulation simplifies stability analysis.

Numerical bifurcation analysis demonstrates model applicability.

Abstract

This chapter focuses on variable maturation delay or, more precisely, on the mathematical description of a size-structured population consuming an unstructured resource. When the resource concentration is a known function of time, we can describe the growth and survival of individuals quasi-explicitly, i.e., in terms of solutions of ordinary differential equations (ODE). Reproduction is captured by a (non-autonomous) renewal equation, which can be solved by generation expansion. After these preparatory steps, a contraction mapping argument is needed to construct the solution of the coupled consumer-resource system with prescribed initial conditions. As we shall show, this interpretation-guided constructive approach does in fact yield weak solutions of a familiar partial differential equation (PDE). A striking difficulty with the PDE approach is that the solution operators are, in general, not differentiable, precluding a linearized stability analysis of steady states. This is a manifestation of the state-dependent delay difficulty. As a (not entirely satisfactory and rather technical) way out, we present a delay equation description in terms of the history of both the $p$-level birth rate of the consumer population and the resource concentration. We end by using pseudospectral approximation to derive a system of ODE and demonstrating its use in a numerical bifurcation analysis. Importantly, the state-dependent delay difficulty dissolves in this approximation.