
Mathematical Biosciences and Engineering, 2020, 17(5): 62176239. 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 803090526, USA
2 Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
Received: , Accepted: , Published:
Special Issues: Recent advances of mathematical modeling and computational methods in cell and developmental biology
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.
Keywords: microbial electrolysis; competitive exclusion; asymptotic stability; Differentialalgebraic 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. Mathematical Biosciences and Engineering, 2020, 17(5): 62176239. doi: 10.3934/mbe.2020329
References:
 1. L. Lu, Z. J. Ren, Microbial electrolysis cells for waste biorefinery: A state of the art review, Bioresour. Technol., 215 (2016), 254264.
 2. B. E. Logan, D. Call, S. Cheng, H. V. M. Hamelers, T. H. J. A. Sleutels, A. W. Jeremiasse, et al., Microbial electrolysis cells for high yield hydrogen gas production from organic matter, Environ. Sci. Technol., 42 (2008), 86308640.
 3. L. Lu, D. Hou, X. Wang, D. Jassby, Z. J. Ren, Active H2 harvesting prevents methanogenesis in microbial electrolysis cells, Environ. Sci. Technol. Lett., 3 (2016), 286290.
 4. L. Lu, W. Vakki, J. A. Aguiar, C. Xiao, K. Hurst, M. Fairchild, et al., Unbiased solar H_{2} production with current density up to 23 mA cm^{2} by Swisscheese black Si coupled with wastewater bioanode, Energy Environ. Sci., 12 (2019), 10881099.
 5. T. Chookaew, P. Prasertsan, Z. J. Ren, Twostage conversion of crude glycerol to energy using dark fermentation linked with microbial fuel cell or microbial electrolysis cell, N. Biotechnol., 31 (2014), 179184.
 6. L. Lu, N. Ren, D. Xing, B. E. Logan, Hydrogen production with effluent from an ethanolh2coproducing fermentation reactor using a singlechamber microbial electrolysis cell, Biosens. Bioelectron., 24 (2009), 30553060.
 7. H. Dudley, L. Lu, Z. Ren, D. Bortz, Sensitivity and bifurcation analysis of a DifferentialAlgebraic equation model for a microbial electrolysis cell, SIAM J. Appl. Dyn. Syst., 709728.
 8. R. P. Pinto, B. Srinivasan, A. Escapa, B. Tartakovsky, Multipopulation model of a microbial electrolysis cell, Environ. Sci. Technol., 45 (2011), 50395046.
 9. E G & G Services, U.S. Department of Energy, Fuel cell handbook, 7th edition, 2004.
 10. R. Pinto, B. Srinivasan, M. F. Manuel, B. Tartakovsky, A twopopulation bioelectrochemical model of a microbial fuel cell, Bioresour. Technol., 101 (2010), 52565265.
 11. B. E. Logan, Microbial fuel cells: Methodology and technology, Environ. Sci. Technol., 40 (2006), 51815192.
 12. A. Kato Marcus, C. I. Torres, B. E. Rittmann, Conductionbased modeling of the biofilm anode of a microbial fuel cell, Biotechnol. Bioeng., 98 (2007), 11711182.
 13. D. A. Noren, M. A. Hoffman, Clarifying the butlervolmer equation and related approximations for calculating activation losses in solid oxide fuel cell models, J. Power Sources, 152 (2005), 175181.
 14. S. Hsu, S. Hubbell, P. Waltman, A mathematical theory for Singlenutrient competition in continuous cultures of Microorganisms, SIAM J. Appl. Math., 32 (1977), 366383.
 15. S. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760763.
 16. S. R. Hansen, S. P. Hubbell, Singlenutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 14911493.
 17. H. L. Smith, P. Waltman, The Theory of the Chemostat: Dynamics of microbial competition, Cambridge University Press, 1995.
 18. T. Sari, F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Math. Biosci. Eng., 8 (2011), 827840.
 19. R. A. Armstrong, R. McGehee, Competitive exclusion, Am. Nat., 115 (1980), 151170.
 20. D. J. Hill, I. M. Y. Mareels, Stability theory for differential/algebraic systems with application to power systems, IEEE Trans. Circuits Syst. I, Reg. Papers, 37 (1990), 14161423.
 21. R. Riaza, Differentialalgebraic systems: Analytical aspects and circuit applications, World Scientific, 2008.
 22. R. März, Practical Lyapunov stability criteria for differential algebraic equations, HumboldtUniv., Fachbereich Mathematik, Informationsstelle, Berlin, 1991.
 23. R. E. Beardmore, Stability and bifurcation properties of index1 DAEs, Numer. Algorithms, 19 (1998), 4353.
 24. R. Riaza, Stability issues in regular and noncritical singular DAEs, Acta Appl. Math., 73 (2002), 301336.
 25. J. LaSalle, Some extensions of Liapunov's second method, IRE Trans. Circuit Theory, 7 (1960), 520527.
 26. S. Hsu, K. Cheng, S. Hubbell, Exploitative competition of microorganisms for two complementary nutrients in continuous cultures, SIAM J. Appl. Math., 41 (1981), 422444.
 27. M. M. Ballyk, G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources, Math. Biosci, 118 (1993), 127180.
 28. B. Li, G. Wolkowicz, Y. Kuang, Global Asymptotic behavior of a Chemostat model with two pPerfectly complementary resources and distributed delay, SIAM J. Appl. Math., 60 (2000), 20582086.
 29. B. Li, H. Smith, How many species can two essential resources support?, SIAM J. Appl. Math., 62 (2001), 336366.
 30. K. Brenan, S. Campbell, L. Petzold, Numerical solution of Initialvalue problems in Differentialalgebraic equations, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, 1995.
 31. A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, et al., SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers, ACM Trans. Math. Softw., 31 (2005), 363396.
 32. G. Wolkowicz, Z. Lu, Global dynamics of a mathematical model of competition in the Chemostat: General response functions and differential death rates, SIAM J. Appl. Math, 52 (1992), 222233.
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