Research article Special Issues

Study of the chemostat model with non-monotonic growth under random disturbances on the removal rate

  • Received: 21 May 2020 Accepted: 19 October 2020 Published: 29 October 2020
  • We revisit the chemostat model with Haldane growth function, here subject to bounded random disturbances on the input flow rate, as often met in biotechnological or waste-water industry. We prove existence and uniqueness of global positive solution of the random dynamics and existence of absorbing and attracting sets that are independent of the realizations of the noise. We study the long-time behavior of the random dynamics in terms of attracting sets, and provide first conditions under which biomass extinction cannot be avoided. We prove conditions for weak and strong persistence of the microbial species and provide lower bounds for the biomass concentration, as a relevant information for practitioners. The theoretical results are illustrated with numerical simulations.

    Citation: Tomás Caraballo, Renato Colucci, Javier López-de-la-Cruz, Alain Rapaport. Study of the chemostat model with non-monotonic growth under random disturbances on the removal rate[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7480-7501. doi: 10.3934/mbe.2020382

    Related Papers:

  • We revisit the chemostat model with Haldane growth function, here subject to bounded random disturbances on the input flow rate, as often met in biotechnological or waste-water industry. We prove existence and uniqueness of global positive solution of the random dynamics and existence of absorbing and attracting sets that are independent of the realizations of the noise. We study the long-time behavior of the random dynamics in terms of attracting sets, and provide first conditions under which biomass extinction cannot be avoided. We prove conditions for weak and strong persistence of the microbial species and provide lower bounds for the biomass concentration, as a relevant information for practitioners. The theoretical results are illustrated with numerical simulations.


    加载中


    [1] J. Monod, La technique de culture continue: Théorie et applications, Annales de l'Institute Pasteur, 79 (1950), 390-410.
    [2] A. Novick, L. Szilard, Experiments with the chemostat on spontaneous mutations of bacteria, Proc. Natl. Acad. Sci., 36 (1950), 708-719. doi: 10.1073/pnas.36.12.708
    [3] H. W. Jannasch, Steady state and the chemostat in ecology, Limnol. Oceanogr., 19 (1974), 716- 720. doi: 10.4319/lo.1974.19.4.0716
    [4] G. D'Ans, P. Kokotovic, D. Gottlieb, A nonlinear regulator problem for a model of biological waste treatment, IEEE Trans. Autom. Control., 16 (1971), 341-347. doi: 10.1109/TAC.1971.1099745
    [5] J. W. M. La Rivière, Microbial ecology of liquid waste treatment, in Advances in Microbial Ecology, 1 (1977), 215-259, Springer US.
    [6] J. Barlow, F. de Noyelles, B. Peterson, J. Peterson, W. Schaffner, "Continuous flow nutrient bioassays with natural phytoplankton populations". G. Glass (Editor): Bioassay Techniques and Environmental Chemistry, John Wiley & Sons Ltd., 1973.
    [7] I. F. Creed, D. M. McKnight, B. A. Pellerin, M. B. Green, B. A. Bergamaschi, G. R. Aiken, et al., The river as a chemostat: fresh perspectives on dissolved organic matter flowing down the river continuum, Can. J. Fish. Aquatic Sci., 72 (2015), 1272-1285. doi: 10.1139/cjfas-2014-0400
    [8] S. Jorgensen, B. Fath, Fundamentals of Ecological Modelling Applications in Environmental Management and Research. Elsevier, 2011.
    [9] J. Kalff, R. Knoechel, Phytoplankton and their dynamics in oligotrophic and eutrophic lakes, Annu. Rev. Ecol. Syst., 9 (1978), 475-495. doi: 10.1146/annurev.es.09.110178.002355
    [10] E. Rurangwa, M. C. J. Verdegem, Microorganisms in recirculating aquaculture systems and their management, Rev. Aquac., 7 (2015), 117-130. doi: 10.1111/raq.12057
    [11] J. Harmand, C. Lobry, A. Rapaport, T. Sari, The Chemostat: Mathematical Theory of Microorganisms Cultures. Wiley, Chemical Engineering Series, John Wiley & Sons, Inc., 2017.
    [12] H. L. Smith, P. Waltman, The theory of the chemostat: dynamics of microbial competition. Cambridge University Press, 1995.
    [13] T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz, A. Rapaport, Modeling and analysis of random and stochastic input flows in the chemostat model, Discret. Continuous Dyn. Syst. Ser. B, 24 (2018), 3591-3614.
    [14] Y. Asai, P. Kloeden, Numerical schemes for random odes via stochastic differential equations, Commun. Appl. Analysis, 17 (2013), 521-528.
    [15] T. Caraballo, X. Han, P. E. Kloeden, Chemostats with random inputs and wall growth, Math. Methods Appl. Sci., 38 (2015), 3538-3550. doi: 10.1002/mma.3437
    [16] T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz, Some Aspects Concerning the Dynamics of Stochastic Chemostats, vol. 69, ch. 11, pp. 227-246. Springer International Publishing, 2016.
    [17] T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz, Dynamics of some stochastic chemostat models with multiplicative noise, Commun. Pure Appl. Analysis, 16 (2017), 1893-1914. doi: 10.3934/cpaa.2017092
    [18] J. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707-723. doi: 10.1002/bit.260100602
    [19] G. Bastin, D. Dochain, On-line estimation and adaptive control of bioreactors. Elsevier, 1990.
    [20] A. Rapaport, J. Harmand, Robust regulation of a class of partially observed nonlinear continuous bioreactors, J. Process. Control., 12 (2002), 291-302. doi: 10.1016/S0959-1524(01)00029-4
    [21] B. Satishkumar, M. Chidambaram, Control of unstable bioreactor using fuzzy tuned PI controller, Bioprocess Eng., 20, (1999), 127-132. doi: 10.1007/s004490050570
    [22] A. Schaum, J. Alvarez, T. Lopez-Arenas, Saturated PI control of continuous bioreactors with haldane kinetics, Chem. Eng. Sci., 68 (2012), 520-529. doi: 10.1016/j.ces.2011.10.006
    [23] A. Rapaport, I. Haidar, J. Harmand, Global dynamics of the buffered chemostat for a general class of response functions, J. Math. Biol., 71 (2014), 69-98.
    [24] A. Rapaport, J. Harmand, Biological control of the chemostat with nonmonotonic response and different removal rates, Math. Biosci. Eng., 5 (2008), 539-547. doi: 10.3934/mbe.2008.5.539
    [25] B. Cloez, C. Fritsch, Gaussian approximations for chemostat models in finite and infinite dimensions, J. Math. Biol., 75 (2017), 805-843. doi: 10.1007/s00285-017-1097-6
    [26] P. Collet, S. Martínez, S. Méléard, J. S. Martín, Stochastic models for a chemostat and long-time behavior, Adv. Appl. Probab., 45 (2013), 822-836. doi: 10.1017/S0001867800006595
    [27] C. Fritsch, J. Harmand, F. Campillo, A modeling approach of the chemostat, Ecol. Model., 299 (2015), 1-13. doi: 10.1016/j.ecolmodel.2014.11.021
    [28] J. Grasman, M. D. Gee, O. A. V. Herwaarden, Breakdown of a chemostat exposed to stochastic noise, J. Eng. Math., 53 (2005), 291-300. doi: 10.1007/s10665-005-9004-3
    [29] L. Imhof, S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations, 217 (2005), 26-53. doi: 10.1016/j.jde.2005.06.017
    [30] G. Stephanopoulos, R. Aris, A. Fredrickson, A stochastic analysis of the growth of competing microbial populations in a continuous biochemical reactor, Math. Biosci., 45 (1979), 99-135. doi: 10.1016/0025-5564(79)90098-1
    [31] C. Xu, S. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Appl. Math. Lett., 48 (2015), 62-68. doi: 10.1016/j.aml.2015.03.012
    [32] D. Zhao, S. Yuan, Critical result on the break-even concentration in a single-species stochastic chemostat model, J. Math. Anal. Appl., 434 (2016), 1336-1345. doi: 10.1016/j.jmaa.2015.09.070
    [33] T. Caraballo, R. Colucci, J. López-de-la-Cruz, A. Rapaport, A way to model stochastic perturba-tions in population dynamics models with bounded realizations, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 239-257. doi: 10.1016/j.cnsns.2019.04.019
    [34] T. Caraballo, X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems. Springer International Publishing, 2016.
    [35] L. Arnold, Random Dynamical Systems. Springer Berlin Heidelberg, 1998.
    [36] T. Caraballo, P. E. Kloeden, B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207. doi: 10.1007/s00245-004-0802-1
    [37] H. I. Freedman, P. Moson, Persistence definitions and their connections, Proc. Am. Math. Soc., 109 (1990), 1025-1033. doi: 10.1090/S0002-9939-1990-1012928-6
    [38] W. Walter, Ordinary Differential Equations. Springer New York, 1998.
    [39] H. R. Thieme, Convergence results and a poincare-bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
  • Reader Comments
  • © 2020 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)
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(2747) PDF downloads(110) Cited by(10)

Article outline

Figures and Tables

Figures(10)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog