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Competitive Exclusion in a DAE Model for Microbial Electrolysis Cells

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.