A stability analysis of a time-varying chemostat with pointwise delay

Frédéric Mazenc; Gonzalo Robledo; Daniel Sepúlveda; Frédéric Mazenc; Gonzalo Robledo; Daniel Sepúlveda

doi:10.3934/mbe.2024119

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2691-2728. doi: 10.3934/mbe.2024119

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Research article

A stability analysis of a time-varying chemostat with pointwise delay

1.
EPI DISCO Inria-Saclay, Laboratoire des Signaux et Systèmes (UMR CNRS 8506), CNRS, CentraleSupélec, Université Paris-Sud, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France
2.
Departamento de Matemáticas, Universidad de Chile, Casilla 653, Santiago, Chile
3.
Departamento de Matemática, Facultad de Ciencias Naturales, Matemática y del Medio Ambiente, Universidad Tecnológica Metropolitana, Santiago, Chile

Academic Editor: Zhi-An Wang

Received: 31 October 2023 Revised: 19 December 2023 Accepted: 04 January 2024 Published: 22 January 2024

This paper revisits a recently introduced chemostat model of one–species with a periodic input of a single nutrient which is described by a system of delay differential equations. Previous results provided sufficient conditions ensuring the existence and uniqueness of a periodic solution for arbitrarily small delays. This paper partially extends these results by proving—with the construction of Lyapunov–like functions—that the evoked periodic solution is globally asymptotically stable when considering Monod uptake functions and a particular family of nutrient inputs.
- chemostat,
- nonautonomous differential delay equations,
- Lyapunov like functions,
- Asymptotic stabilty,
- periodic solutions
Citation: Frédéric Mazenc, Gonzalo Robledo, Daniel Sepúlveda. A stability analysis of a time-varying chemostat with pointwise delay[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2691-2728. doi: 10.3934/mbe.2024119

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Abstract

This paper revisits a recently introduced chemostat model of one–species with a periodic input of a single nutrient which is described by a system of delay differential equations. Previous results provided sufficient conditions ensuring the existence and uniqueness of a periodic solution for arbitrarily small delays. This paper partially extends these results by proving—with the construction of Lyapunov–like functions—that the evoked periodic solution is globally asymptotically stable when considering Monod uptake functions and a particular family of nutrient inputs.

References

[1]	P. Amster, G. Robledo, D. Sepúlveda, Dynamics of a chemostat with periodic nutrient supply and delay in the growth, Nonlinearity, 33 (2020), 5839–5860. https://doi.org/10.1088/1361-6544/ab9bab doi: 10.1088/1361-6544/ab9bab
[2]	N. Ye, Z. Hu, Z. Teng, Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth, Commun. Pure Appl. Anal., 21 (2022), 1361–1384. https://doi.org/10.3934/cpaa.2022022 doi: 10.3934/cpaa.2022022
[3]	N. Ye, L. Zhang, Z. Teng, The dynamical behavior and periodic solution in delayed nonautonomous chemostat models, J. Appl. Anal. Comput., 13 (2023), 156–183. https://doi.org/10.11948/20210452 doi: 10.11948/20210452
[4]	J. Monod, The growth of bacterial cultures, Annu. Rev. Microbiol., 3 (1949), 371–394. https://doi.org/10.1146/annurev.mi.03.100149.002103 doi: 10.1146/annurev.mi.03.100149.002103
[5]	J. Monod, La technique de culture continue, théorie et applications, Ann. l'Inst. Pasteur, 79 (1950), 390–410. https://doi.org/10.1016/B978-0-12-460482-7.50023-3 doi: 10.1016/B978-0-12-460482-7.50023-3
[6]	A. Novick, L. Slizard, Description of the chemostat, Science, 112 (1950), 715–716. https://doi.org/10.1126/science.112.2920.715 doi: 10.1126/science.112.2920.715
[7]	A. Ajbar, K. Alhumaizi, Dynamics of the Chemostat. A Bifurcation Theory Approach, Chapman and Hall/CRC, New York, 2011. https://doi.org/10.1201/b11073
[8]	J. Harmand, C. Lobry, A. Rapaport, T. Sari, The Chemostat: Mathematical Theory of Microorganism Cultures, ISTE, London; John Wiley & Sons, Inc., Hoboken, 2017. https://doi.org/10.1002/9781119437215
[9]	H. L. Smith, P. Waltman, The Theory of the Chemostat, Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511530043
[10]	P. J. Wangersky, J. W. Cunningham, On time lags in equations of growth, Proc. Nat. Acad. Sci., 42 (1956), 699–702. https://doi.org/10.1073/pnas.42.9.699 doi: 10.1073/pnas.42.9.699
[11]	J. Caperon, Time lag in population growth response of isochrysis Galbana to a variable nitrate environment, Ecology, 50 (1969), 188–192. https://doi.org/10.2307/1934845 doi: 10.2307/1934845
[12]	T. F. Thingstad, T. I. Langeland, Dynamics of chemostat culture: The effect of a delay in cell response, J. Theor. Biol., 48 (1974), 149–159. https://doi.org/10.1016/0022-5193(74)90186-6 doi: 10.1016/0022-5193(74)90186-6
[13]	E. Beretta, Y. Kuang, Global stability in a well known delayed chemostat model, Commun. Appl. Anal., 4 (2000), 147–155.
[14]	J. Kato, J. Pan, Stability domain of a chemostat system with delay, in Differential Equations with Applications to Biology (eds. S. Ruan, G. S. K. Wolkowicz, J. Wu), Fields Institute Communications, 21 (1999), 307–315. https://doi.org/10.1090/fic/021
[15]	J. Pan, Parameter analysis of a chemostat equation with delay, Funckialaj Ekvacioj, 41 (1998), 347–361.
[16]	H. Xia, G. S. K. Wolkowicz, L. Wang, Transient oscillations induced by delayed growth response in the chemostat, J. Math. Biol., 50 (2005), 489–530. https://doi.org/10.1007/s00285-004-0311-5 doi: 10.1007/s00285-004-0311-5
[17]	T. Zhao, Global periodic–solutions for a differential delay system modeling a microbial population in the chemostat, J. Math. Anal. Appl., 193 (1995), 329–352. https://doi.org/10.1006/jmaa.1995.1239 doi: 10.1006/jmaa.1995.1239
[18]	P. Gajardo, F. Mazenc, H. Ramirez, Competitive exclusion principle in a model of chemostat with delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 16 (2009), 253–272.
[19]	F. Mazenc, S. I. Niculescu, G. Robledo, Stability analysis of mathematical model of competition in a chain of chemostats in series with delay, Appl. Math. Model., 76 (2019), 311–329. https://doi.org/10.1016/j.apm.2019.06.006 doi: 10.1016/j.apm.2019.06.006
[20]	S. B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol., 9 (1980), 115–132. https://doi.org/10.1007/BF00275917 doi: 10.1007/BF00275917
[21]	J. K. Hale, A. S. Somolinos, Competition for fluctuating nutrient, J. Math. Biol., 18 (1983), 255–280. https://doi.org/10.1007/BF00276091 doi: 10.1007/BF00276091
[22]	G. S. K. Wolkowicz, X. Q. Zhao, $N$-species competition in a periodic chemostat, Differ. Integr. Equations, 11 (1998), 465–491. https://doi.org/10.57262/die/1367341063 doi: 10.57262/die/1367341063
[23]	X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. https://doi.org/10.1007/978-3-319-56433-3
[24]	M. Malisoff, F. Mazenc, Constructions of Strict Lyapunov Functions, Springer series: Communications and Control Engineering, London, 2009. https://doi.org/10.1007/978-1-84882-535-2
[25]	J. R. Graef, J. Henderson, L. Kong, X. S. Liu, Ordinary Differential Equations and Boundary Value Problems, World Scientific, Singapore, 2018. https://doi.org/10.1142/10888
[26]	H. Khalil, Nonlinear Systems, Prentice Hall, Upper Saddle River, 1996.
[27]	E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34 (1989), 435–443. https://doi.org/10.1109/9.28018 doi: 10.1109/9.28018
[28]	A. Mironchenko, Input-to-State Stability. Theory and Applications, Springer serie: Communications and Control Engineering, Cham, 2023. https://doi.org/10.1007/978-3-031-14674-9
[29]	O. Bernard, G. Malara, A. Sciandra, The effects of a controlled fluctuating nutrient environment on continuous cultures of phytoplankton monitored by computers, J. Exp. Mar. Biol. Ecol., 197 (1996), 263–278. https://doi.org/10.1016/0022-0981(95)00161-1 doi: 10.1016/0022-0981(95)00161-1
[30]	G. Malara, A. Sciandra, A multiparameter phytoplankton culture system driven by microcomputer, J. Appl. Phycol., 3 (1991), 235–241. https://doi.org/10.1007/BF00003581 doi: 10.1007/BF00003581
[31]	I. Vatcheva, H. de Jong, O. Bernard, N. J. Mars, Experiment selection for the discrimination of semi-quantitative models of dynamical systems, Artif. Intell., 170 (2006), 472–506. https://doi.org/10.1016/j.artint.2005.11.001 doi: 10.1016/j.artint.2005.11.001
[32]	O. Bernard, Étude Expérimentale et Théorique de la Croissance de Dunaliella Tertiolecta (Chlorophyceae) Soumise à une Limitation Variable de Nitrate, PhD. thesis, Université Pierre & Marie-Curie, Paris, France, 1995.
[33]	S. F. Ellermeyer, Delayed Growth Response in Models of Microbial Growth and Competition in Continuous Culture, PhD. thesis, Emory University, Atlanta, 1991.
[34]	S. F. Ellermeyer, Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth, SIAM J. Appl. Math., 54 (1994), 456–465. https://doi.org/10.1137/S003613999222522X doi: 10.1137/S003613999222522X
[35]	S. Ellermeyer, J. Hendrix, N. Ghoochan, A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria, J. Theoret. Biol., 222 (2003), 485–494. https://doi.org/10.1016/S0022-5193(03)00063-8 doi: 10.1016/S0022-5193(03)00063-8
[36]	H. I. Freedman, J. W. H. So, P. Waltman, Chemostat competition with time delays, in IMACS 1988 — 12th World Congress on Scientific Computing — Proceedings (eds. R. Vichnevetsky, P. Borne, J. Vignes), Gerfidn Cite Scientifique, Paris, (1988), 102–104.
[37]	P. Amster, G. Robledo, D. Sepúlveda, Existence of $\omega$-periodic solutions for a delayed chemostat with periodic inputs, Nonlinear Anal. Real World Appl., 55 (2020), 103134. https://doi.org/10.1016/j.nonrwa.2020.103134 doi: 10.1016/j.nonrwa.2020.103134
[38]	M. Rodriguez Cartabia, Persistence criteria for a chemostat with variable nutrient input and variable washout with delayed response in growth, Chaos Solitons Fractals, 172 (2023), 113514. https://doi.org/10.1016/j.chaos.2023.113514 doi: 10.1016/j.chaos.2023.113514
[39]	X. Zhang, Ultimate boundedness of a stochastic chemostat model with periodic nutrient inputs and discrete delay, Chaos Solitons Fractals, 175 (2023), 113956. https://doi.org/10.1016/j.chaos.2023.113956 doi: 10.1016/j.chaos.2023.113956
[40]	H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-7646-8

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