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Competitive Exclusion in a DAE Model for Microbial Electrolysis Cells

Harry J. Dudley, Zhiyong Jason Ren, David M. Bortz

arXiv:1906.02086v3q-bio.PEmath.CAmath.DS

TL;DR

The paper asks how methanogens and electroactive bacteria compete in MECs, where only electroactive bacteria support current and hydrogen production. It analyzes two DAE model versions using local matrix-pencil stability and global Lyapunov–LaSalle methods. Methanogen exclusion is globally asymptotically stable under the stated condition, whereas electroactive-bacteria exclusion requires additional stability conditions and is not guaranteed.

  • Problem

    The paper studies whether methanogens or electroactive bacteria competitively exclude one another in MECs, because only electroactive bacteria contribute to current and hydrogen production.

  • Method

    The paper analyzes local stability with matrix-pencil spectra and global stability with Lyapunov–LaSalle methods in simplified and full semi-explicit index 1 DAE models.

  • Results

    Methanogen exclusion is globally asymptotically stable when a methanogen survives at the lowest substrate concentration, while electroactive-bacteria exclusion is not necessarily locally asymptotically stable.

  • Takeaways & Limitations

    MEC operation should identify the microbe able to grow at the lowest substrate concentration and check matrix-pencil conditions at the electroactive-only equilibrium.

  • Takeaways & Limitations

    For the full MEC system, the matrix-pencil spectrum for electroactive-bacteria exclusion cannot readily be evaluated even with one species of each type.

Abstract

from arXiv · show

Microbial electrolysis cells (MECs) employ electroactive bacteria to perform extracellular electron transfer, enabling hydrogen generation from biodegradable substrates. In previous work, we developed and analyzed a differential-algebraic equation (DAE) model for MECs. The model resembles a chemostat with ordinary differential equations (ODEs) for concentrations of substrate, microorganisms, and an extracellular mediator involved in electron transfer. There is also an algebraic constraint for electric current and hydrogen production. Our goal is to determine the outcome of competition between methanogenic archaea and electroactive bacteria, because only the latter contribute to electric current and resulting hydrogen production. We investigate asymptotic stability in two industrially relevant versions of the model. An important aspect of chemostats models is the principle of competitive exclusion -- only microbes which grow at the lowest substrate concentration will survive as $t\to\infty$. We show that if methanogens grow at the lowest substrate concentration, then the equilibrium corresponding to competitive exclusion by methanogens is globally asymptotically stable. The analogous result for electroactive bacteria is not necessarily true. We show that local asymptotic stability of exclusion by electroactive bacteria is not guaranteed, even in a simplified version of the model. In this case, even if electroactive bacteria can grow at the lowest substrate concentration, a few additional conditions are required to guarantee local asymptotic stability. We also provide numerical simulations supporting these arguments. Our results suggest operating conditions that are most conducive to success of electroactive bacteria and the resulting current and hydrogen production in MECs. This will help identify when methane production or electricity and hydrogen production are favored.

1. Introduction.

MECs use electroactive bacteria to convert organic substrates into current and hydrogen, but methanogens compete for the same substrate and reduce system efficiency. This paper extends a DAE framework to analyze stability and competitive exclusion in that competition.

  • 1. Introduction.: MECs recover energy from organic waste by using electroactive bacteria to oxidize biodegradable substrate, transfer electrons, and release protons for hydrogen production.A small applied voltage of 0.2–0.8 V overcomes the thermodynamic barrier, compared with 1.8–3.5 V for traditional water electrolysis.
  • 1. Introduction.: The paper extends prior MEC modeling by treating current production through an extracellular mediator within a regular, semi-explicit, index 1 DAE framework.The algebraic constraint may only admit a local representation, complicating global analysis.
  • 1. Introduction.: The model couples ODEs for substrate, microbial populations, and an extracellular mediator with an algebraic current constraint accounting for practical voltage losses.The constraint relates current to electroactive bacteria and mediator concentrations and includes voltage losses in the external circuit.
  • 1. Introduction.: Competitive exclusion is complicated because electroactive-bacteria growth depends nonlinearly on both substrate and mediator concentrations.The paper builds on chemostat theory, where microbes growing at lower substrate concentrations can globally exclude competitors.
  • 1.1. Model.: The extended model omits fermenting microorganisms and a separate methanogen-only outer biofilm layer, focusing on competition in a single simple-substrate MEC.The model includes finitely many methanogen and electroactive-bacteria species with substrate consumption, growth, and decay dynamics.
  • 1. Introduction.: Methanogens consume substrate while producing methane, whereas electroactive bacteria transfer electrons through a mediator and support current and hydrogen production.Methanogens are also present in the anodic biofilm, and the model represents mediator replenishment as proportional to electric current.

2. Asymptotic stability in semi-explicit DAEs.

The paper analyzes semi-explicit DAEs using regularity, reduced ODE representations, matrix-pencil spectra, and LaSalle’s invariance principle. These tools support local and global asymptotic-stability results for MEC equilibria.

  • 2. Asymptotic stability in semi-explicit DAEs.: The DAE framework is necessary because the MEC algebraic current constraint does not admit a global solution.The system can nevertheless be represented locally as a semi-explicit DAE.
  • 2. Asymptotic stability in semi-explicit DAEs.: Local asymptotic stability can be determined from the matrix-pencil spectrum, requiring every relevant eigenvalue σ to satisfy Re(σ) < 0.The paper applies this criterion to simplified MEC models.
  • 2. Asymptotic stability in semi-explicit DAEs.: A regular semi-explicit index 1 DAE requires a nonsingular g_z, allowing the constraint to be written locally as z = ψ(y).The implicit function theorem then yields the reduced ODE ẏ = f(y, ψ(y)).
  • 2. Asymptotic stability in semi-explicit DAEs.: LaSalle’s invariance principle states that trajectories in a compact positively invariant set approach the largest invariant set where V̇(x) = 0.If that invariant set is an isolated equilibrium, the equilibrium is asymptotically stable.
  • 2. Asymptotic stability in semi-explicit DAEs.: A modified Lyapunov function is used later to analyze global stability for the semi-explicit DAE with multiple species, Monod kinetics, and a solvable constraint.This applies LaSalle’s principle rather than requiring V̇ to be strictly negative everywhere.

3. Local asymptotic stability in a simplified model .

The simplified MEC model analyzes local asymptotic stability of extinction and competitive-exclusion equilibria using general monotone kinetics and an algebraic current constraint. Methanogen exclusion is locally stable under its growth condition, whereas electroactive exclusion additionally requires spectral conditions.

  • Model assumptions: The simplified model assumes one microbial compartment of each type, equal decay and dilution rates, general monotone kinetics, and a general algebraic constraint.These assumptions yield concise local-stability conditions for the equilibria.
  • Model formulation: The rescaled system includes substrate dynamics, microbial growth and consumption, mediator dynamics, and the algebraic constraint g(x_e, m, I) = 0.The total variable u = s + x_m + x_e approaches 1, so asymptotically stable equilibria satisfy u = 1.
  • Equilibria: The extinction and competitive-exclusion equilibria are characterized by s = 1, s + x_m = 1, or s + x_e = 1.These correspond respectively to extinction, methanogen exclusion, and electroactive-bacteria exclusion.
  • Stability cases: If neither microbe can grow at the relevant substrate threshold, the extinction equilibrium p0 is locally asymptotically stable.The extinction equilibrium is generally unstable when either λ_m or λ_e(m0) lies in (0, 1).
  • Stability cases: If λ_m < λ_e(m0), the methanogen-only equilibrium p_m is locally asymptotically stable, with only methanogens attaining positive net growth nearby.This is the local counterpart of methanogen competitive exclusion in the simplified model.
  • Stability cases: Electroactive-only exclusion p_e is locally asymptotically stable when λ_e(m*) < λ_m and the additional discriminant or real-part condition is satisfied.Thus, electroactive bacteria growing at the lowest substrate concentration alone does not guarantee local asymptotic stability.

4. Global asymptotic stability with Monod kinetics .

Under Monod kinetics, the smallest attainable zero-net-growth substrate concentration determines limiting behavior. Methanogen exclusion is globally asymptotically stable, whereas electroactive-bacteria exclusion can require additional stability conditions and is not generally guaranteed.

  • Equilibria: The equilibrium points considered represent either extinction of all microbes or competitive exclusion by one microbial species.Their microbial concentrations are determined from the substrate balance, while mediator concentrations are obtained from the algebraic constraint.
  • Extinction conditions: A microbe that cannot achieve zero net growth over attainable substrate and mediator values has concentration tending to zero as t→∞.The result follows because equilibrium substrate concentrations must lie within the attainable interval.
  • Competitive exclusion: The DAE system’s limiting behavior is determined by the smallest zero-net-growth substrate concentration among methanogens and electroactive bacteria.For electroactive bacteria, this threshold depends on mediator concentration through λe,j(M).
  • Competitive exclusion by methanogens: If methanogen Xm,1 has the uniquely smallest λ value, all solutions approach the equilibrium corresponding to exclusion by Xm,1.This result applies to the full MEC system with multiplicative Monod kinetics, different decay rates, and a Nernst–Butler-Volmer-based constraint.
  • Methanogen coexistence: When several methanogens share the smallest λ value, they can coexist while competitively excluding electroactive bacteria and other microbes.The long-term state approaches an invariant set containing those methanogens, with the remaining microbial concentrations zero.
  • Electroactive-bacteria stability: Electroactive-bacteria exclusion is not necessarily locally asymptotically stable, even when electroactive bacteria can grow at the lowest substrate concentration.The complication arises from dual substrate–mediator limitation and the algebraic constraint determining current.

5. Numerical Simulations .

The simulations support competitive exclusion by methanogens when they grow at the lowest substrate concentration, while electroactive exclusion may occur but is not guaranteed.

  • Figure 5.1a: Methanogen 1 competitively excludes the other microbes when it can grow at the lowest substrate concentration.Figure 5.1a converges to a methanogen-only equilibrium, Pm.
  • Figure 5.1b: Multiple methanogens can coexist when they survive at the same lowest substrate concentration.
  • Figure 5.1c: Electroactive bacteria may exclude methanogens when they can grow at the lowest substrate concentration.The simulations do not establish an analogous general result for electroactive bacteria.
  • Figure 5.1: Figure 5.1 plots substrate and microorganism concentrations on a semi-logarithmic scale using distinct lines for substrate, methanogens, and electroactive bacteria.

6. Conclusion .

The conclusion identifies operating conditions that favor either methanogen exclusion or electroactive-bacteria exclusion in MECs. Methanogen exclusion is stable when methanogens grow at the lowest substrate value, whereas electroactive exclusion additionally requires a matrix-pencil condition.

  • Stability conclusions: If a methanogen grows at the lowest substrate value, methanogen competitive exclusion is locally or globally asymptotically stable.
  • Operating guidance: At the electroactive-only equilibrium, the matrix-pencil spectrum should be computed to assess local asymptotic stability.
  • Operating guidance: Electroactive bacteria are most likely to outcompete methanogens when they survive at the lowest substrate concentration and the discriminant condition is satisfied.
  • Practical implications: Ensuring methanogen growth at the lowest substrate concentration favors methane production, while satisfying the electroactive stability condition favors current and hydrogen production.

Appendix A. Proof of Lemma 4.2 .

The proof establishes positivity and boundedness of substrate, microbial concentrations, mediator, and current for admissible DAE solutions.

  • Positivity: Substrate and microorganism concentrations remain positive because their boundaries cannot be reached from positive initial conditions.
  • Boundedness: The substrate concentration is bounded above by its influent concentration, S0.
  • Current bound: The logarithmic mediator relation implies a positive bound on the current under the stated resistance conditions.
  • Mediator and current: The mediator variables M and I are positive and bounded within a positively invariant set consistent with the DAE constraint.

Appendix B. Proof of Lemma 4.3.

The proof shows that microbial populations whose growth cannot overcome decay are bounded by decaying exponentials and therefore vanish asymptotically.

  • Methanogens: Methanogen X_m,j converges to zero when its maximum growth rate does not exceed its decay rate.
  • Methanogens: Methanogen X_m,j also converges to zero when its break-even substrate concentration exceeds the influent substrate concentration.
  • Electroactive bacteria: Electroactive X_e,j converges to zero when its mediator-dependent growth cannot exceed decay across the admissible mediator range.
  • Electroactive bacteria: Under the alternative electroactive growth condition, X_e,j is likewise bounded by a decaying exponential and tends to zero.
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