pith. sign in

arxiv: 2604.18903 · v1 · submitted 2026-04-20 · 🧮 math.DS

AM2 model with a series configuration of interconnected chemostats and distinct removal rates

Thamer Hmidhi , Radhouane Fekih-Salem This is my paper

Pith reviewed 2026-05-10 03:01 UTC · model grok-4.3

classification 🧮 math.DS
keywords AM2 modelchemostatsinterconnected bioreactorssteady stateslocal stabilitydilution ratescoexistence equilibriaJacobian trace
0
0 comments X

The pith

The eight-dimensional AM2 system in serial chemostats with unequal dilution rates admits nine equilibrium types whose existence, multiplicity, and local stability are characterized by operating parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper performs a complete qualitative analysis of the AM2 model in a series of two interconnected chemostats that operate with different dilution rates. Because the dilution rates are distinct, the standard reduction of the eight-dimensional ODE system to four dimensions is impossible, so the authors work directly with the full system. They solve explicitly for all steady states and supply necessary and sufficient conditions, in terms of dilution rates and removal rates, for when each of nine equilibrium types exists, when several equilibria coexist, and when each type is locally stable. They additionally show that coexistence equilibria in the second bioreactor are not automatically stable even when they exist and require a further inequality involving the trace of the Jacobian matrix. A reader would care because the results clarify how flow-rate choices affect microbial survival and stability in bioreactor networks used for anaerobic digestion and wastewater treatment.

Core claim

In the AM2 model placed in a serial configuration of two chemostats with distinct dilution rates, the dynamics are governed by an eight-dimensional system of nonlinear ordinary differential equations that cannot be reduced. All steady states are found explicitly, and necessary and sufficient conditions on the key operating parameters are derived for the existence, multiplicity, and local asymptotic stability of nine distinct equilibrium types, including washout states, single-species states, and various forms of coexistence. Equilibria in which both microbial species coexist inside the second bioreactor are locally stable only when they exist and the trace of the Jacobian at that point is of

What carries the argument

The irreducible eight-dimensional nonlinear ODE system for the AM2 model with distinct dilution rates; explicit solution of the steady-state algebraic equations combined with Jacobian eigenvalue analysis to obtain local stability conditions.

If this is right

  • Parameter regions exist in which multiple equilibria of different types are simultaneously possible.
  • Coexistence of both species in the second bioreactor is possible only for certain ranges of the dilution and removal rates and is stable only when an additional trace condition holds.
  • Washout of all species occurs when both dilution rates exceed the maximum specific growth rates of the organisms.
  • The stability conclusions are local; global convergence or the presence of limit cycles cannot be decided from the given analysis.
  • Identifying parameter values that produce Hopf bifurcations remains open.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Bioreactor design should deliberately choose unequal dilution rates to reach desired stable coexistence states rather than assuming symmetry.
  • Numerical continuation software could trace the curves in parameter space that separate stable and unstable coexistence regions.
  • Laboratory experiments with controlled but unequal flow rates between two chemostats could test whether oscillations appear when the trace condition is violated.
  • Global stability tools such as Lyapunov functions or Dulac criteria might be applied to rule out periodic orbits in some of the identified regions.

Load-bearing premise

The eight-dimensional system of ODEs accurately captures the biological interactions and that local linearization at equilibria determines the relevant long-term behavior.

What would settle it

Numerical simulation of the ODE system at a parameter value where a coexistence equilibrium exists but the Jacobian trace is positive, showing that trajectories fail to converge to that equilibrium.

read the original abstract

We investigate the dynamics of the AM2 model in a serial configuration of two interconnected chemostats with distinct dilution rates. The system is described by nonlinear differential equations, for which the usual reduction from an eight-dimensional system to a four-dimensional one is no longer possible due to the distinct dilution rates. We provide a complete qualitative study by characterizing all steady states and establishing necessary and sufficient conditions for the existence, multiplicity, and local stability of nine types of equilibria, expressed in terms of key operating parameters. We show that coexistence equilibria involving two species in the second bioreactor are not always stable, even when they exist, and require an additional condition related to the trace of the Jacobian matrix. Identifying parameter sets leading to Hopf bifurcations and limit cycles remains an open problem. These results provide new insights into the dynamics of interconnected bioreactors and motivate further investigations on oscillatory behaviors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript examines the AM2 model in a serial configuration of two interconnected chemostats with distinct dilution rates D1 and D2. This prevents the standard reduction from eight to four dimensions, so the authors retain the full 8D nonlinear ODE system. They classify all steady states into nine types according to which of the two species are present or absent in each bioreactor, and derive necessary-and-sufficient conditions (in terms of the operating parameters) for existence, multiplicity, and local stability of each type. For coexistence equilibria in the second bioreactor an extra trace-of-the-Jacobian condition is required for stability. Global questions such as Hopf bifurcations and limit cycles are explicitly left open.

Significance. If the algebraic characterizations hold, the work supplies a concrete, parameter-explicit local analysis of an 8D chemostat model that cannot be reduced by the usual conservation laws. The case-by-case treatment of the nine equilibrium types and the clear separation of local stability from unresolved global dynamics constitute a useful extension of classical chemostat theory to interconnected systems with unequal removal rates. The explicit listing of open problems on oscillatory behavior is a strength that delineates scope and guides future research.

minor comments (3)
  1. A compact table listing the nine equilibrium types together with their existence and stability conditions (in terms of D1, D2 and input concentrations) would improve readability and allow quick reference.
  2. The notation for the eight state variables (substrate and biomass concentrations in each reactor) should be introduced once in a single displayed equation block early in the model section and then used consistently.
  3. Several parameter inequalities are stated without an accompanying numerical example; adding one or two concrete parameter sets that realize each of the nine types would help readers verify the conditions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment, accurate summary of the manuscript, and recommendation for minor revision. The description of the 8D system, the nine equilibrium types, and the open global questions matches our work precisely.

read point-by-point responses

  1. Referee: The manuscript examines the AM2 model in a serial configuration of two interconnected chemostats with distinct dilution rates D1 and D2. This prevents the standard reduction from eight to four dimensions, so the authors retain the full 8D nonlinear ODE system. They classify all steady states into nine types according to which of the two species are present or absent in each bioreactor, and derive necessary-and-sufficient conditions (in terms of the operating parameters) for existence, multiplicity, and local stability of each type. For coexistence equilibria in the second bioreactor an extra trace-of-the-Jacobian condition is required for stability. Global questions such as Hopf bifurcations and limit cycles are explicitly left open.

    Authors: We appreciate the referee's precise recapitulation of the model setup and our main results. The necessity of retaining the full 8D system due to unequal dilution rates is correctly noted, as is the case-by-case analysis of the nine equilibrium types with explicit necessary-and-sufficient conditions on the operating parameters. The additional trace-of-the-Jacobian requirement for local stability of the relevant coexistence equilibria in the second bioreactor is indeed part of our analysis, and we explicitly identify the global questions (Hopf bifurcations and limit cycles) as open. revision: no

Circularity Check

0 steps flagged

No significant circularity in algebraic characterization of equilibria

full rationale

The paper derives all steady states by directly setting the eight ODE right-hand sides to zero and solving the resulting algebraic system case-by-case for the nine equilibrium types (zero/nonzero population combinations). Necessary-and-sufficient conditions for existence, multiplicity, and local stability are obtained from the explicit parameter expressions and from the Jacobian trace/determinant criteria evaluated at those equilibria; all quantities remain functions of the externally chosen operating parameters (dilution rates, feed concentrations, etc.). No fitted quantities are renamed as predictions, no self-citation supplies a load-bearing uniqueness or ansatz result, and the analysis stays within the model equations without reduction to prior author work. The open-problem statement on global dynamics further confirms the scope is limited to the local algebraic study.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The report rests on the standard AM2 chemostat ODEs and treats dilution rates and input concentrations as externally set operating parameters; no new entities are postulated.

free parameters (2)
  • dilution rates D1 and D2
    Distinct removal rates that prevent dimensional reduction and enter all existence and stability conditions.
  • input substrate concentrations
    Key parameters used to express thresholds for existence and multiplicity of equilibria.
axioms (1)
  • domain assumption The biological process is exactly described by the eight-dimensional AM2 system of autonomous ODEs with constant parameters.
    Invoked as the starting model whose equilibria are classified.

pith-pipeline@v0.9.0 · 5450 in / 1336 out tokens · 50187 ms · 2026-05-10T03:01:46.067774+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Benyahia, T

    B. Benyahia, T. Sari, B. Cherki and J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes, J. Proc. Control, 221008–1019 (2012)

    work page 2012

  2. [2]

    Bernard, Z

    O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi, and J.P. Steyer, Dynam- ical model development and parameter identification for an anaerobic wastewater treatment process,Biotechnol. Bioeng.,75(2001), 424–438

    work page 2001

  3. [3]

    Dali-Youcef, A

    M. Dali-Youcef, A. Rapaport and T. Sari, Study of performance criteria of serial configuration of two chemostats,Math. Biosci. Eng.,176278–6309 (2020)

    work page 2020

  4. [4]

    Dali-Youcef, A

    M. Dali-Youcef, A. Rapaport and T. Sari, Performance study of two serial inter- connected chemostats with mortality, Bull. Math. Biol.84, 110 (2022)

    work page 2022

  5. [5]

    Fekih-Salem, Y

    R. Fekih-Salem, Y. Daoud, N. Abdellatif, T. Sari, A mathematical model of anaero- bic digestion with syntrophic relationship, substrate inhibition and distinct removal rates, SIAM J. Appl. Dyn. Syst20, 1625–1654 (2021)

    work page 2021

  6. [6]

    Haidar, A

    I. Haidar, A. Rapaport, and F. Gérard, Effects of spatial structure and diffusion on the performances of the chemostat,Math. Biosci. Eng.,8(2011), 953–971

    work page 2011

  7. [7]

    Hmidhi, R

    T. Hmidhi, R. Fekih-Salem and J. Harmand, Analysis of anaerobic digestion model withtwoserialinterconnectedchemostats,Bull. Math. Biol.,87,95(2025),901–926

    work page 2025

  8. [8]

    Sari, Best operating conditions for biogas production in some simple anaerobic digestion models, Processes,10, 258 (2022)

    T. Sari, Best operating conditions for biogas production in some simple anaerobic digestion models, Processes,10, 258 (2022)

    work page 2022

  9. [9]

    Sari and B

    T. Sari and B. Benyahia, The operating diagram for a two-step anaerobic digestion model,Nonlinear Dyn.,105(2021), 2711–2737

    work page 2021