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Optimal Separation of Two Microbial Species Competing for Two Substitutable Resources

Author

Listed:
  • Walid Djema

    (CNRS-Sorbonne Université (LOV), GreenOwl project-team)

  • Térence Bayen

    (Avignon Université)

  • Jean-Luc Gouzé

    (CNRS, Macbes project-team)

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.

Suggested Citation

  • Walid Djema & Térence Bayen & Jean-Luc Gouzé, 2025. "Optimal Separation of Two Microbial Species Competing for Two Substitutable Resources," Journal of Optimization Theory and Applications, Springer, vol. 205(3), pages 1-32, June.
    • Handle: RePEc:spr:joptap:v:205:y:2025:i:3:d:10.1007_s10957-025-02672-x
    DOI: 10.1007/s10957-025-02672-x
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    References listed on IDEAS

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    1. Hanan H. Almuashi & Nada A. Almuallem & Miled El Hajji, 2024. "The Effect of Leachate Recycling on the Dynamics of Two Competing Bacteria with an Obligate One-Way Beneficial Relationship in a Chemostat," Mathematics, MDPI, vol. 12(23), pages 1-19, December.
    2. Joaquín Gutiérrez Mena & Sant Kumar & Mustafa Khammash, 2022. "Dynamic cybergenetic control of bacterial co-culture composition via optogenetic feedback," Nature Communications, Nature, vol. 13(1), pages 1-16, December.
    Full references (including those not matched with items on IDEAS)

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