Global stability of perturbed chemostat systems

TL;DR

This work analyzes the global stability of a chemostat system perturbed by an exchange term between $n$ species, quantified by a small parameter $\varepsilon$ and modeled as $h(x,s,\varepsilon)$. The authors construct an invariant family of compact subsets $\Delta_{i,\alpha}$ within the invariant manifold and prove that, for each subset, a threshold $\varepsilon_{u,\alpha}$ exists so that the perturbed dynamics converge to a non-washout coexistence state in that subset, by combining the Malkin-Gorshin Theorem with Smith–Waltman perturbation results. When the exchange term is linear, $h(x,s,\varepsilon)=\varepsilon T(s)x$, a Perron–Frobenius framework yields a critical dilution rate $u_c(\varepsilon)$ separating washout from coexistence, with a locally asymptotically stable coexistence state $E_{\varepsilon,u}$ for $u<u_c(\varepsilon)$ and small $\varepsilon$; numerical cases with constant and non-constant $T(s)$ corroborate the theory and illustrate the operating diagram. Overall, the paper extends CEP-based stability analyses to perturbed, potentially nonlinear exchanges, provides rigorous thresholds for stability under small perturbations, and demonstrates persistence of all species under linear exchange through numerical experiments.

Abstract

This paper is devoted to the analysis of global stability of the chemostat system with a perturbation term representing any type of exchange between species. This conversion term depends on species and substrate concentrations but also on a positive perturbation parameter. After having written the invariant manifold as a union of a family of compact subsets, our main result states that for each subset in this family, there is a positive threshold for the perturbation parameter below which, the system is globally asymptotically stable in the corresponding subset. Our approach relies on the Malkin-Gorshin Theorem and on a Theorem by Smith and Waltman about perturbations of a globally stable steady state. Properties of steady-states and numerical simulations of the system's asymptotic behavior complete this study for two types of perturbation term between species.

Figures (9)

• Figure 1: Illustration of Theorem \ref{['CEP-thm']} for $(x_0,s_0)=(0.15,0.15,0.15,0.15,0.15,0.25)$. Fig. left: convergence to $E_1=(0.75,0,0,0,0,0.25)$ with $u=0.4$ ; Fig. right: convergence to $E_{wo}=(0,0,0,0,0,1)$ with $u=0.7$.
• Figure 2: Plot of the kinetics $\mu_i(s)= \frac{a_i s}{b_i+s}$ of Monod type for $i=1,...,5$ (coefficients are given in Table \ref{['tab:parameters']}).
• Figure 3: Plot of $s\mapsto \lambda(B(s,u,\varepsilon))$ over $[0,s_{in}]$ for $\varepsilon$ discretized between $0$ and $100$ and for $u=0.4$ (left) and $u=1.5$ (right) in the case where $T$ is given by \ref{['NC-MATRIX']}.
• Figure 4: Solutions to \ref{['eq:lineal-CS']} for 100 randomly generated initial conditions in the set $(0,10]^n \times [0,s_{in}]$
• Figure 5: Solutions to \ref{['eq:cultivated-CS']} for 100 randomly generated initial conditions in the set $(0,10]^n \times [0,s_{in}]$

Theorems & Definitions (23)

• Remark 1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.1
• proof
• Remark 2
• Theorem 2.1
• Theorem 2.2