Research article Special Issues

Growth on two limiting essential resources in a self-cycling fermentor

  • Received: 02 April 2018 Accepted: 23 August 2018 Published: 06 December 2018
  • A system of impulsive differential equations with state-dependent impulses is used to model the growth of a single population on two limiting essential resources in a self-cycling fermentor. Potential applications include water purification and biological waste remediation. The self-cycling fermentation process is a semi-batch process and the model is an example of a hybrid system. In this case, a well-stirred tank is partially drained, and subsequently refilled using fresh medium when the concentration of both resources (assumed to be pollutants) falls below some acceptable threshold. We consider the process successful if the threshold for emptying/refilling the reactor can be reached indefinitely without the time between successive emptying/refillings becoming unbounded and without interference by the operator. We prove that whenever the process is successful, the model predicts that the concentrations of the population and the resources converge to a positive periodic solution. We derive conditions for the successful operation of the process that are shown to be initial condition dependent and prove that if these conditions are not satisfied, then the reactor fails. We show numerically that there is an optimal fraction of the medium drained from the tank at each impulse that maximizes the output of the process.

    Citation: Ting-Hao Hsu, Tyler Meadows, LinWang, Gail S. K. Wolkowicz. Growth on two limiting essential resources in a self-cycling fermentor[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 78-100. doi: 10.3934/mbe.2019004

    Related Papers:

  • A system of impulsive differential equations with state-dependent impulses is used to model the growth of a single population on two limiting essential resources in a self-cycling fermentor. Potential applications include water purification and biological waste remediation. The self-cycling fermentation process is a semi-batch process and the model is an example of a hybrid system. In this case, a well-stirred tank is partially drained, and subsequently refilled using fresh medium when the concentration of both resources (assumed to be pollutants) falls below some acceptable threshold. We consider the process successful if the threshold for emptying/refilling the reactor can be reached indefinitely without the time between successive emptying/refillings becoming unbounded and without interference by the operator. We prove that whenever the process is successful, the model predicts that the concentrations of the population and the resources converge to a positive periodic solution. We derive conditions for the successful operation of the process that are shown to be initial condition dependent and prove that if these conditions are not satisfied, then the reactor fails. We show numerically that there is an optimal fraction of the medium drained from the tank at each impulse that maximizes the output of the process.
    加载中


    [1] D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1989.
    [2] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993.
    [3] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, vol. 28 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co., Inc., River Edge, NJ, 1995, Translated from the Bulgarian manuscript by V. Covachev [V. Khr. Kovachev].
    [4] G. Butler and G. S. K. Wolkowicz, Exploitative competition in a chemostat for two complementary, and possibly inhibitory, resources, Math. Biosci., 83 (1987), 1–48.
    [5] A. Cinar, S. J. Parulekar, C. Ündey and G. Birol, Batch Fermentors: Modeling, Monitoring, and Control, Marcel Dekker, New York, 2003.
    [6] F. Córdova-Lepe, R. D. Valle and G. Robledo, Stability analysis of a self-cycling fermentation model with state-dependent impulses, Math. Methods Appl. Sci., 37 (2014), 1460–1475.
    [7] G. Fan and G. S. K. Wolkowicz, Analysis of a model of nutrient driven self-cycling fermentation allowing unimodal response functions, Discrete Contin. Dyn. Syst., 8 (2007), 801–831.
    [8] J. P. Grover, Resource Competition, Population and community biology series 19, Chapman and Hall, New York, 1997.
    [9] A. Halanay and D. Wexler, Teoria Calitativa a Sistemelor du Impulsuri, Ed. Acad. RSR, Bucharedt, 1968.
    [10] S. Hughes and D. Cooper, Biodegradation of phenol using the self-cycling fermentation process, Biotechnol. Bioeng., 51 (1996), 112–119.
    [11] J. A. León and D. B. Tumpson, Competition between two species for two complementary or substitutable resources, J. Theor. Biol., 50 (1975), 185–2012.
    [12] B. Li, G. S. K. Wolkowicz and Y. Kuang, Global asymptotic behavior of a chemostat model with two perfectly complementary resources with distributed delay, SIAM J. Appl. Math., 60 (2000), 2058–2086.
    [13] MATLAB, version 9.3.0 (R2017b), The MathWorks Inc., Natick, Massachusetts, 2017.
    [14] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 1996.
    [15] A. Samoilenko and N. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
    [16] B. E. Sarkas and D. G. Cooper, Biodegradation of aromatic compounds in a self-cycling fermenter, Can. J. Chem. Eng., 72 (1994), 874–880.
    [17] D. Sauvageau, Z. Storms and D. G. Cooper, Sychronized populations of Escherichia coli using simplified self-cycling fermentation, J. Biotechnol., 149 (2010), 67–73.
    [18] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition., vol. 13 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1995.
    [19] R. J. Smith and G. S. K. Wolkowicz, Analysis of a model of the nutrient driven self-cycling fermentation process, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 11 (2004), 239–265.
    [20] Z. J. Storms, T. Brown, D. Sauvageau and D. G. Cooper, Self-cycling operation increases productivity of recombinent protein in Escherichia Coli, Biotechnol. Bioeng., 109 (2012), 2262–2270.
    [21] D. Tilman, Resource Competition and Community Structure, Princeton University Press, New Jersey, 1982.
    [22] J. Von Liebig, Die Organische Chemie in Ihrer Anwendung auf Agrikultur und Physiologie, Friedrich Vieweg, Braunschweig, 1840.
    [23] J. Wang, M. Chae, D. Sauvageau and D. C. Bressler, Improving ethanol productivity through selfcycling fermentation of yeast: a proof of concept, Biotechnol. Biofuels, 10 (2017), 193.
    [24] J. Wu, H. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat, SIAM J. Appl. Math., 65 (2004), 209–229.

    © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
  • Reader Comments
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(620) PDF downloads(612) Cited by(1)

Article outline

Figures and Tables

Figures(6)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog