Competitive exclusion in a DAE model for microbial electrolysis cells

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

doi:10.3934/mbe.2020329

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 6217-6239. doi: 10.3934/mbe.2020329

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Competitive exclusion in a DAE model for microbial electrolysis cells

1.
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA
2.
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA

Received: 07 July 2020 Accepted: 01 September 2020 Published: 16 September 2020

Microbial electrolysis cells (MECs) are devices that employ electroactive bacteria to perform extracellular electron transfer, enabling hydrogen generation from biodegradable substrates. In our previous work, we developed and analyzed a differential-algebraic equation (DAE) model for MECs. The model resembles a chemostat or continuous stirred tank reactor (CSTR). It consists of ordinary differential equations 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 the resulting hydrogen production. We investigate asymptotic stability in two industrially relevant versions of the model. An important aspect of many chemostat models is the principle of competitive exclusion. This states that only microbes which grow at the lowest substrate concentration will survive as t → ∞. We show that if methanogens can 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. In fact we show that local asymptotic stability of competitive 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 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 producing methane or electricity and hydrogen is favored.
- microbial electrolysis,
- competitive exclusion,
- asymptotic stability,
- Differential-algebraic equation,
- LaSalle’s invariance principle
Citation: Harry J. Dudley, Zhiyong Jason Ren, David M. Bortz. Competitive exclusion in a DAE model for microbial electrolysis cells[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 6217-6239. doi: 10.3934/mbe.2020329

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Abstract

Microbial electrolysis cells (MECs) are devices that employ electroactive bacteria to perform extracellular electron transfer, enabling hydrogen generation from biodegradable substrates. In our previous work, we developed and analyzed a differential-algebraic equation (DAE) model for MECs. The model resembles a chemostat or continuous stirred tank reactor (CSTR). It consists of ordinary differential equations 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 the resulting hydrogen production. We investigate asymptotic stability in two industrially relevant versions of the model. An important aspect of many chemostat models is the principle of competitive exclusion. This states that only microbes which grow at the lowest substrate concentration will survive as t → ∞. We show that if methanogens can 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. In fact we show that local asymptotic stability of competitive 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 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 producing methane or electricity and hydrogen is favored.

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