Global Stability Analysis of the Age-Structured Chemostat With Substrate Dynamics

TL;DR

This paper analyzes the global stability of an age-structured chemostat model with substrate dynamics, deriving explicit conditions for stability using Lyapunov functionals and comparing them with existing local results.

Contribution

It introduces a novel Lyapunov functional approach to establish global stability conditions for a complex infinite-dimensional chemostat model.

Findings

Global exponential decay estimates established

Explicit sufficient conditions for stability derived

Comparison with existing local stability results

Abstract

In this paper we study the stability properties of the equilibrium point for an age-structured chemostat model with renewal boundary condition and coupled substrate dynamics under constant dilution rate. This is a complex infinite-dimensional feedback system. It has two feedback loops, both nonlinear. A positive static loop due to reproduction at the age-zero boundary of the PDE, counteracted and dominated by a negative dynamic loop with the substrate dynamics. The derivation of explicit sufficient conditions that guarantee global stability estimates is carried out by using an appropriate Lyapunov functional. The constructed Lyapunov functional guarantees global exponential decay estimates and uniform global asymptotic stability with respect to a measure related to the Lyapunov functional. From a biological perspective, stability arises because reproduction is constrained by substrate availability, while dilution, mortality, and substrate depletion suppress transient increases in biomass before age-structure effects can amplify them. The obtained results are applied to a chemostat model from the literature, where the derived stability condition is compared with existing results that are based on (necessarily local) linearization methods.