Optimal Separation of Two Microbial Species Competing for Two Substitutable Resources

Walid Djema (), Térence Bayen () and Jean-Luc Gouzé ()
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Walid Djema: CNRS-Sorbonne Université (LOV), GreenOwl project-team
Térence Bayen: Avignon Université
Jean-Luc Gouzé: CNRS, Macbes project-team

Journal of Optimization Theory and Applications, 2025, vol. 205, issue 3, No 20, 32 pages

Abstract: Abstract We address a selection problem between two microbial species by controlling the dilution rate in a continuous stirred tank reactor. Unlike earlier studies, we focus on the competition between species for two substitutable substrates instead of a single limiting resource, with each species capable of dominating the culture. The selection of the most suitable strain is formulated as a minimum-time optimal control problem. After examining the reachability of the target set, we conduct a thorough analysis of optimal solutions and singular arcs, using Pontryagin’s Maximum Principle and geometric control theory. Notably, unlike studies involving a single substrate, the target is not always reachable, and we construct a counterexample to prove it. In the case where the target can be reached, we highlight that optimal solutions exhibit turnpike-like phenomena by combining the approach with a direct optimization method, and these properties differ from those in the single-substrate case. Although there is no natural static optimization problem associated with a minimum-time control problem, our analysis enables us to characterize the turnpike point as the solution to a static optimization problem associated with an auxiliary Lagrange-type optimal control problem. These results are extensively illustrated throughout this work using direct optimization.

Keywords: Pontryagin’s maximum principle; Singular arcs; Turnpike phenomenon; Chemostat system; Strain selection; Microbial competition; 49K15; 37N25; 92C75 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02672-x

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