In this work, the global stability of a continuous bioreactor model is studied, with the concentrations of biomass and substrate as state variables, a general non-monotonic function of substrate concentration for the specific growth rate, and constant inlet substrate concentration. Also, the dilution rate is time varying but bounded, thus leading to state convergence to a compact set instead of an equilibrium point. Based on the Lyapunov function theory with dead-zone modification, the convergence of the substrate and biomass concentrations is studied. The main contributions with respect to closely related studies are: i) The convergence regions of the substrate and biomass concentrations are determined as function of the variation region of the dilution rate (D) and the global convergence to these compact sets is proved, considering monotonic and non-monotonic growth functions separately; ii) several improvements are proposed in the stability analysis, including the definition of a new dead zone Lyapunov function and the properties of its gradient. These improvements allow proving convergence of substrate and biomass concentrations to their compact sets, while tackling the interwoven and nonlinear nature of the dynamics of biomass and substrate concentrations, the non-monotonic nature of the specific growth rate, and the time-varying nature of the dilution rate. The proposed modifications are a basis for further global stability analysis of bioreactor models exhibiting convergence to a compact set instead of an equilibrium point. Finally, the theoretical results are illustrated through numerical simulation, showing the convergence of the states under varying dilution rate.
Citation: Alejandro Rincón, Fredy E. Hoyos, Gloria Restrepo. Global stability of a continuous bioreactor model under persistent variation of the dilution rate[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 3396-3424. doi: 10.3934/mbe.2023160
In this work, the global stability of a continuous bioreactor model is studied, with the concentrations of biomass and substrate as state variables, a general non-monotonic function of substrate concentration for the specific growth rate, and constant inlet substrate concentration. Also, the dilution rate is time varying but bounded, thus leading to state convergence to a compact set instead of an equilibrium point. Based on the Lyapunov function theory with dead-zone modification, the convergence of the substrate and biomass concentrations is studied. The main contributions with respect to closely related studies are: i) The convergence regions of the substrate and biomass concentrations are determined as function of the variation region of the dilution rate (
| [1] |
T. Salmi, P. Tolvanen, K. Eränen, J. Wärnå, S. Leveneur, H. Haario, Determination of kinetic constants by using transient temperature data from continuous stirred tank reactors, Chem. Eng. Sci., 248 (2022), 117164. https://doi.org/10.1016/j.ces.2021.117164 doi: 10.1016/j.ces.2021.117164
|
| [2] |
M. A. Siddiqui, Md. N. Anwar, S. H. Laskar, Control of nonlinear jacketed continuous stirred tank reactor using different control structures, J. Process Control, 108 (2021), 112–124. https://doi.org/10.1016/j.jprocont.2021.11.005 doi: 10.1016/j.jprocont.2021.11.005
|
| [3] |
T. Caraballo, R. Colucci, J. López-de-la-Cruz, A. Rapaport, Study of the chemostat model with non-monotonic growth under random disturbances on the removal rate, Math. Biosci. Eng., 17 (2020), 7480–7501. https://doi.org/10.3934/mbe.2020382 doi: 10.3934/mbe.2020382
|
| [4] |
K. Bizon, B. Tabiś, Problems in volumetric flow rate and liquid level control of a continuous stirred tank bioreactor with structured and unstructured kinetics, Chem. Eng. Res. Design, 175 (2021), 309–319. https://doi.org/10.1016/j.cherd.2021.09.015 doi: 10.1016/j.cherd.2021.09.015
|
| [5] |
A. I. Bayu, R. A. Lestary, N. Dewayanto, M. Mellyanawaty, A. Wicaksono, R. W. Alvania-Kartika, et al., Kinetic study of thermophilic anaerobic digestion of sugarcane vinasse in a single-stage continuous stirred tank reactor, Results Eng., 14 (2022), 100432. https://doi.org/10.1016/j.rineng.2022.100432 doi: 10.1016/j.rineng.2022.100432
|
| [6] |
P. C. Ri, J. S. Kim, T. R. Kim, C. H. Pang, H. G. Mun, G. C. Pak, et al., Effect of hydraulic retention time on the hydrogen production in a horizontal and vertical continuous stirred-tank reactor, Int. J. Hydrogen Energy, 44 (2019), 17742–17749/ https://doi.org/10.1016/j.ijhydene.2019.05.136 doi: 10.1016/j.ijhydene.2019.05.136
|
| [7] |
J. Harmand, A. Rapaport, D. Dochain, Increasing the dilution rate can globally stabilize two-step biological systems, J. Process Control, 95 (2020), 67–74. https://doi.org/10.1016/j.jprocont.2020.08.009 doi: 10.1016/j.jprocont.2020.08.009
|
| [8] |
H. de Battista, M. Jamilis, F. Garelli, J. Picó, Global stabilisation of continuous bioreactors: Tools for analysis and design of feeding laws, Automatica, 89 (2018), 340–348. https://doi.org/10.1016/j.automatica.2017.12.041 doi: 10.1016/j.automatica.2017.12.041
|
| [9] |
N. Dimitrova, P. Zlateva, Global stability analysis of a bioreactor model for phenol and cresol mixture degradation, Processes, 9 (2021), 124. https://doi.org/10.3390/pr9010124 doi: 10.3390/pr9010124
|
| [10] |
L. Sridhar, Analysis of microalgae dynamics models, Biofuels, 11 (2020), 637–641. https://doi.org/10.1080/17597269.2017.1387747 doi: 10.1080/17597269.2017.1387747
|
| [11] |
F. Mairet, A. Rojas-Palma, Modeling and stability analysis of a microalgal pond with nitrification, Appl. Math. Model, 51 (2017), 448–468. https://doi.org/10.1016/j.apm.2017.07.008 doi: 10.1016/j.apm.2017.07.008
|
| [12] |
M. Sbarciog, A. V. Wouwer, A constructive approach to assess the stability of anaerobic digestion systems, Chem. Eng. Sci., 227 (2020), 115931. https://doi.org/10.1016/j.ces.2020.115931 doi: 10.1016/j.ces.2020.115931
|
| [13] |
P. Skupin, M. Metzger, Oscillatory behaviour control in a continuous culture under double-substrate limitation, J. Biol. Dyn., 12 (2018), 663–682. https://doi.org/10.1080/17513758.2018.1502368 doi: 10.1080/17513758.2018.1502368
|
| [14] |
M. I. Nelson, H. S. Sidhu, Analysis of a chemostat model with variable yield coefficient, J. Math. Chem., 38 (2005), 605–615. https://doi.org/10.1007/s10910-005-6914-2 doi: 10.1007/s10910-005-6914-2
|
| [15] |
J. Alvarez-Ramirez, J. Alvarez, A. Velasco, On the existence of sustained oscillations in a class of bioreactors, Comput. Chem. Eng., 33 (2009), 4–9. https://doi.org/10.1016/j.compchemeng.2008.05.017 doi: 10.1016/j.compchemeng.2008.05.017
|
| [16] |
T. Sari, B. Benyahia, The operating diagram for a two-step anaerobic digestion model, Nonlinear Dyn., 105 (2021), 2711–2737. https://doi.org/10.1007/s11071-021-06722-7 doi: 10.1007/s11071-021-06722-7
|
| [17] |
T. Sari, Best operating conditions for biogas production in some simple anaerobic digestion models, Processes, 10 (2022), 258. https://doi.org/10.3390/pr10020258 doi: 10.3390/pr10020258
|
| [18] |
T. Meadows, M. Weedermann, G. S. K. Wolkowicz, Global analysis of a simplified model of anaerobic digestion and a new result for the chemostat, SIAM J. Appl. Math., 79 (2019), 668–689. https://doi.org/10.1137/18M1198788 doi: 10.1137/18M1198788
|
| [19] |
C. Martínez, J. L. Gouzé, Global dynamics of the chemostat with overflow metabolism, J. Math. Biol., 82 (2021), 13. https://doi.org/10.1007/s00285-021-01566-6 doi: 10.1007/s00285-021-01566-6
|
| [20] |
R. Huang, Y. Wang, Global stability of a chemostat system with dispersal between multiple patches, Nonlinear Dyn., 108 (2022), 4511–4529. https://doi.org/10.1007/s11071-022-07386-7 doi: 10.1007/s11071-022-07386-7
|
| [21] |
T. Sari, F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields, Math. Biosci. Eng., 8 (2011), 827–840. https://doi.org/10.3934/mbe.2011.8.827 doi: 10.3934/mbe.2011.8.827
|
| [22] |
A. Rincón, G. Y. Florez, G. Olivar, Convergence assessment of the trajectories of a bioreaction system by using asymmetric truncated vertex functions, Symmetry, 12 (2020), 513. https://doi.org/10.3390/sym12040513 doi: 10.3390/sym12040513
|
| [23] |
M. Dellal, B. Bar, Global analysis of a model of competition in the chemostat with internal inhibitor, Discrete Contin. Dyn. Syst. B, 26 (2021), 1129–1148. https://doi.org/10.3934/dcdsb.2020156 doi: 10.3934/dcdsb.2020156
|
| [24] |
E. Ranjbar, M. Yaghubi, A. A. Suratgar, Robust adaptive sliding mode control of a MEMS tunable capacitor based on dead-zone method, Automatika, 61 (2020), 587–601. https://doi.org/10.1080/00051144.2020.1806011 doi: 10.1080/00051144.2020.1806011
|
| [25] |
Q. Hong, Y. Shi, Z. Chen, Dynamics modeling and tension control of composites winding system based on ASMC, IEEE Access, 8 (2020), 102795–102810. https://doi.org/10.1109/ACCESS.2020.2997340 doi: 10.1109/ACCESS.2020.2997340
|
| [26] |
A. Rincón, G. M. Restrepo, Ó. J. Sánchez, An improved robust adaptive controller for a fed-batch bioreactor with input saturation and unknown varying control gain via dead-zone quadratic forms, Computation, 9 (2021), 100. https://doi.org/10.3390/computation9090100 doi: 10.3390/computation9090100
|
| [27] |
A. Rincón, G. M. Restrepo, F. E. Hoyos, A robust observer—based adaptive control of second—order systems with input saturation via dead-zone Lyapunov functions, Computation, 9 (2021), 82. https://doi.org/10.3390/computation9080082 doi: 10.3390/computation9080082
|
| [28] |
S. Roy, S. Baldi, L. M. Fridman, On adaptive sliding mode control without a priori bounded uncertainty, Automatica, 111 (2020), 108650. https://doi.org/10.1016/j.automatica.2019.108650 doi: 10.1016/j.automatica.2019.108650
|
| [29] |
Z. Ming, H. Zhang, Y. Liang, H. Su, Nonzero-sum differential games of continuous-time nonlinear systems with uniformly ultimately ε-bounded by adaptive dynamic programming, Appl. Math. Comput., 430 (2022), 127248. https://doi.org/10.1016/j.amc.2022.127248 doi: 10.1016/j.amc.2022.127248
|
| [30] |
G. O. Tysse, A. Cibicik, L. Tingelstad, O. Egeland, Lyapunov-based damping controller with nonlinear MPC control of payload position for a knuckle boom crane, Automatica, 140 (2022), 110219. https://doi.org/10.1016/j.automatica.2022.110219 doi: 10.1016/j.automatica.2022.110219
|
| [31] |
M. C. Valentino, F. A. Faria, V. A. Oliveira, L. F. C. Alberto, Sufficient conditions in terms of linear matrix inequalities for guaranteed ultimately boundedness of solutions of switched Takagi-Sugeno fuzzy systems using the s-procedure, Inf. Sci., 572 (2021), 501–521. https://doi.org/10.1016/j.ins.2021.04.103 doi: 10.1016/j.ins.2021.04.103
|
| [32] |
A. Boulkroune, M. M'saad, M. Farza, Adaptive fuzzy system-based variable-structure controller for multivariable nonaffine nonlinear uncertain systems subject to actuator nonlinearities, Neural Comput. Appl., 28 (2017), 3371–3384. https://doi.org/10.1007/s00521-016-2241-8 doi: 10.1007/s00521-016-2241-8
|
| [33] |
M. Cai, Z. Xiang, Adaptive practical finite-time stabilization for uncertain nonstrict feedback nonlinear systems with input nonlinearity, IEEE Trans. Syst. Man Cybern. Syst., 47 (2017), 1668–1678. https://doi.org/10.1109/TSMC.2017.2660761 doi: 10.1109/TSMC.2017.2660761
|
| [34] |
E. Asiain, R. Garrido, Anti-chaos control of a servo system using nonlinear model reference adaptive control, Chaos Solitons Fractals, 143 (2021), 110581. https://doi.org/10.1016/j.chaos.2020.110581 doi: 10.1016/j.chaos.2020.110581
|
| [35] |
W. Xue, M. Zhang, S. Liu, Y. Li, S. Cang, Mechanical analysis and ultimate boundary estimation of the chaotic permanent magnet synchronous motor, J. Franklin Inst., 356 (2019), 5378–5394. https://doi.org/10.1016/j.jfranklin.2019.05.007 doi: 10.1016/j.jfranklin.2019.05.007
|
| [36] |
M. Fiaz, M. Aqeel, S. Ahmad, J. Ayub, The analysis of NSG system for existence of Si'lnikov chaos, Chin. J. Phys., 62 (2019), 43–53. https://doi.org/10.1016/j.cjph.2019.09.013 doi: 10.1016/j.cjph.2019.09.013
|
| [37] |
F. Zhang, X. Liao, G. Zhang, C. Mu, Dynamical analysis of the generalized Lorenz systems, J. Dyn. Control Syst., 23 (2017), 349–362. https://doi.org/10.1007/s10883-016-9325-8 doi: 10.1007/s10883-016-9325-8
|
| [38] |
X. Liao, G. Zhou, Q. Yang, Y. Fu, G. Chen, Constructive proof of Lagrange stability and sufficient—necessary conditions of Lyapunov stability for Yang—Chen chaotic system, Appl. Math. Comput., 309 (2017), 205–221. https://doi.org/10.1016/j.amc.2017.03.033 doi: 10.1016/j.amc.2017.03.033
|
| [39] |
S. Zeinali, M. Shahrokhi, Observer-based singularity free nonlinear controller for uncertain systems subject to input saturation, Eur. J. Control, 52 (2020), 49–58. https://doi.org/10.1016/j.ejcon.2019.08.001 doi: 10.1016/j.ejcon.2019.08.001
|
| [40] |
J. Picó, H. de Battista, F. Garelli, Smooth sliding-mode observers for specific growth rate and substrate from biomass measurement, J. Process Control, 19 (2009), 1314–1323. https://doi.org/10.1016/j.jprocont.2009.04.001 doi: 10.1016/j.jprocont.2009.04.001
|
| [41] | G. Bastin, D. Dochain, On-Line Estimation and Adaptive Control of Bioreactors, Elsevier, 1990. |
| [42] |
A. Rapaport, T. Nidelet, S. El-Aida, J. Harmand, About biomass overyielding of mixed cultures in batch processes, Math. Biosci., 322 (2020), 108322. https://doi.org/10.1016/j.mbs.2020.108322 doi: 10.1016/j.mbs.2020.108322
|
| [43] |
J. L. Gouzé, G. Robledo, Feedback control for nonmonotone competition models in the chemostat, Nonlinear Anal. Real World Appl., 6 (2005), 671–690. https://doi.org/10.1016/j.nonrwa.2004.12.003 doi: 10.1016/j.nonrwa.2004.12.003
|
| [44] |
G. J. Butler, G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math., 45 (1985), 138–151. https://doi.org/10.1137/0145006 doi: 10.1137/0145006
|
| [45] |
F. Mazenc, M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurements, Automatica, 46 (2010), 1428–1436. https://doi.org/10.1016/j.automatica.2010.06.012 doi: 10.1016/j.automatica.2010.06.012
|
| [46] |
J. H. T. Luong, Generalization of Monod kinetics for analysis of growth data with substrate inhibition, Biotechnol. Bioeng., 29 (1987), 242–248. https://doi.org/10.1002/bit.260290215 doi: 10.1002/bit.260290215
|
| [47] |
S. K. C. Lin, C. Du, A. Koutinas, R. Wang, C. Webb, Substrate and product inhibition kinetics in succinic acid production by Actinobacillus succinogenes, Biochem. Eng. J., 41 (2008), 128–135. https://doi.org/10.1016/j.bej.2008.03.013 doi: 10.1016/j.bej.2008.03.013
|
| [48] |
J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707–723. https://doi.org/10.1002/bit.260100602 doi: 10.1002/bit.260100602
|
| [49] |
O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi, J. P. Steyer, Dynamical model development and parameter identification for an anaerobic wastewater treatment process, Biotechnol. Bioeng., 75 (2001), 424–438. https://doi.org/10.1002/bit.10036 doi: 10.1002/bit.10036
|
| [50] |
J. Ribes, K. Keesman, H. Spanjers, Modelling anaerobic biomass growth kinetics with a substrate threshold concentration, Water Res., 38 (2004), 4502–4510. https://doi.org/10.1016/j.watres.2004.08.017 doi: 10.1016/j.watres.2004.08.017
|
| [51] | J. Slotine; W. Li, Applied Nonlinear Control, Englewood Cliffs, 1991. |
| [52] |
K. M. Koo, Stable adaptive fuzzy controller with time-varying dead-zone, Fuzzy Sets Syst., 121 (2001), 161–168. https://doi.org/10.1016/S0165-0114(99)00157-8 doi: 10.1016/S0165-0114(99)00157-8
|
| [53] |
X. S. Wang, C. Y. Su, H. Hong, Robust adaptive control of a class of nonlinear systems with unknown dead-zone, Automatica, 40 (2004), 407–413. https://doi.org/10.1016/j.automatica.2003.10.021 doi: 10.1016/j.automatica.2003.10.021
|
| [54] | P. Ioannou, J. Sun, Robust Adaptive Control, Prentice-Hall PTR, 1996. |
| [55] |
E. D. Sontag, Y. Wang, On characterizations of the input-to-state stability property, Syst. Control Lett., 24 (1995), 351–359. https://doi.org/10.1016/0167-6911(94)00050-6 doi: 10.1016/0167-6911(94)00050-6
|
| [56] |
A. Rincón, J. Villa, F. Angulo, G. Olivar, A dynamic analysis for an anaerobic digester: Stability and bifurcation branches, Math. Probl. Eng., 2014 (2014), 1–14. https://doi.org/10.1155/2014/514797 doi: 10.1155/2014/514797
|
| [57] | Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, 112 (2004). |
| [58] | M. Sbarciog, M. Loccufier, E. Noldus, Convergence and stability of biochemical reaction systems of rank one, in Proceedings of the Proceedings of the 44th IEEE Conference on Decision and Control, (2005), 5540–5545. https://doi.org/10.1109/CDC.2005.1583044 |
| [59] |
J. Hess, O. Bernard, Design and study of a risk management criterion for an unstable anaerobic wastewater treatment process, J. Process Control, 18 (2008), 71–79. https://doi.org/10.1016/j.jprocont.2007.05.005 doi: 10.1016/j.jprocont.2007.05.005
|
| [60] |
B. Haegeman, A. Rapaport, How flocculation can explain coexistence in the chemostat, J. Biol. Dyn., 2 (2008), 1–13. https://doi.org/10.1080/17513750801942537 doi: 10.1080/17513750801942537
|
| [61] |
I. R. Moyles, J. G. Donohue, A. C. Fowler, Quasi-steady uptake and bacterial community assembly in a mathematical model of soil-phosphorus mobility, J. Theor. Biol., 509 (2021), 110530. https://doi.org/10.1016/j.jtbi.2020.110530 doi: 10.1016/j.jtbi.2020.110530
|
| [62] |
I. R. Moyles, A. C. Fowler, Production of nitrate spikes in a model of ammonium biodegradation, Theor. Ecol., 11 (2018), 333–350. https://doi.org/10.1007/s12080-018-0370-7 doi: 10.1007/s12080-018-0370-7
|
| [63] |
M. J. McGuinness, L. B. Cribbin, H. F. Winstanley, A. C. Fowler, Modelling spatial oscillations in soil borehole bacteria, J. Theor. Biol., 363 (2014), 74–79. https://doi.org/10.1016/j.jtbi.2014.08.017 doi: 10.1016/j.jtbi.2014.08.017
|
| [64] |
A. C. Fowler, H. F. Winstanley, M. J. McGuinness, L. B. Cribbin, Oscillations in soil bacterial redox reactions, J. Theor. Biol., 2014, 342 (2014), 33–38. https://doi.org/10.1016/j.jtbi.2013.10.010 doi: 10.1016/j.jtbi.2013.10.010
|
| [65] | H. Zhang, Y. Cai, S. Fu, W. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Appl. Math. Comput., 2019, 356 (2019), 328–337. https://doi.org/10.1016/j.amc.2019.03.034 |
| [66] |
Z. Shang, Y. Qiao, L. Duan, J. Miao, Bifurcation analysis and global dynamics in a predator–prey system of Leslie type with an increasing functional response, Ecol. Modell., 455 (2021), 109660. https://doi.org/10.1016/j.ecolmodel.2021.109660 doi: 10.1016/j.ecolmodel.2021.109660
|
| [67] |
S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwan. J. Math., 9 (2005), https://doi.org/10.11650/twjm/1500407791 doi: 10.11650/twjm/1500407791
|
| [68] |
X. Gan, H. Wang, P. Ao, Existence of a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle, Nonlinear Dyn., 105 (2021), 3117–3130. https://doi.org/10.1007/s11071-021-06775-8 doi: 10.1007/s11071-021-06775-8
|
| [69] |
A. Rincón-Santamaría, J. A. Cuellar-Gil, L. F. Valencia-Gil, O. J. Sánchez-Toro, Cinética de Crecimiento de Gluconacetobacter Diazotrophicus Usando Melaza de Caña y Sacarosa: Evaluación de Modelos, Acta Biol. Colomb., 24 (2019), 38–57. https://doi.org/10.15446/abc.v24n1.70857 doi: 10.15446/abc.v24n1.70857
|
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Alejandro Rincón, Fredy E. Hoyos, Gloria Restrepo. Global stability of a continuous bioreactor model under persistent variation of the dilution rate[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 3396-3424. doi: 10.3934/mbe.2023160
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