# American Institute of Mathematical Sciences

June  2018, 23(4): 1851-1872. doi: 10.3934/dcdsb.2018098

## Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties

 1 EPI DISCO Inria-Saclay, CNRS, CentraleSupélec, Université Paris-Sud, 91192, Gif-sur-Yvette, France 2 Departamento de Matemáticas, Universidad de Chile, Casilla 653, Santiago, Chile 3 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA

* Corresponding author: Michael Malisoff

Received  January 2017 Revised  October 2017 Published  March 2018

Fund Project: The first and second authors were supported by the MATHAMSUD Cooperation Program (16 MATH-04 STADE). The third author was supported by NSF grant 1408295. A summary of some of this work that was confined to the case where the delays are zero was presented at the 2017 American Control Conference

We study a chemostat model with an arbitrary number of competing species, one substrate, and constant dilution rates. We allow delays in the growth rates and additive uncertainties. Using constant inputs of certain species, we derive bounds on the sizes of the delays that ensure asymptotic stability of an equilibrium when the uncertainties are zero, which can allow persistence of multiple species. Under delays and uncertainties, we provide bounds on the delays and on the uncertainties that ensure input-to-state stability with respect to uncertainties.

Citation: Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098
##### References:
 [1] A. Bush and A. Cool, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, Journal of Theoretical Biology, 63 (1975), 385-395. doi: 10.1016/0022-5193(76)90041-2. [2] T. Caraballo, X. Han and P. Kloeden, Nonautonomous chemostats with variable delays, SIAM Journal on Mathematical Analysis, 47 (2015), 2178-2199. doi: 10.1137/14099930X. [3] P. Collet, S. Martinez, S. Meleard and J. San Martin, Stochastic models for a chemostat and long time behavior, Advances in Applied Probability, 45 (2013), 822-836. doi: 10.1017/S0001867800006595. [4] Y. Collos, Time-lag algal growth dynamics: Biological constraints on primary production in aquatic environments, Marine Ecology Progress Series, 33 (1986), 193-206. doi: 10.3354/meps033193. [5] S. Dikshitulu, B. Baltzis, G. Lewandowski and S. Pavlou, Competition between two microbial populations in a sequencing fed-batch reactor: theory, experimental verification, and implications for waste treatment applications, Biotechnology and Bioengineering, 42 (1993), 643-656. doi: 10.1002/bit.260420513. [6] S. Ellermeyer, Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth, SIAM Journal on Applied Mathematics, 54 (1994), 456-465. doi: 10.1137/S003613999222522X. [7] C. Fritsch, J. Harmand and F. Campillo, A Modeling approach of the chemostat, Ecological Modelling, 299 (2015), 1-13. doi: 10.1016/j.ecolmodel.2014.11.021. [8] J-L. Gouzé and G. Robledo, Feedback control for nonmonotone competition models in the chemostat, Nonlinear Analysis: Real World Applications, 6 (2005), 671-690. doi: 10.1016/j.nonrwa.2004.12.003. [9] F. Grognard, F. Mazenc and A. Rapaport, Polytopic Lyapunov functions for persistence analysis of competing species, Discrete and Continuous Dynamical Systems Series B, 8 (2007), 73-93. doi: 10.3934/dcdsb.2007.8.73. [10] H. Guo and S. Zheng, A competition model for two resources in un-stirred chemostat, Applied Mathematics and Computation, 217 (2011), 6934-6949. doi: 10.1016/j.amc.2011.01.102. [11] B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, Journal of Biological Dynamics, 2 (2008), 1-13. doi: 10.1080/17513750801942537. [12] G. Hardin, Competitive exclusion principle, Science, 131 (1960), 1292-1297. doi: 10.1126/science.131.3409.1292. [13] X.-Z. He and S. Ruan, Global stability in chemostat-type plankton models with delayed nutrient recycling, Journal of Mathematical Biology, 37 (1998), 253-271. doi: 10.1007/s002850050128. [14] S.-B. Hsu, A competition model for a seasonally fluctuating nutrient, Journal of Mathematical Biology, 9 (1980), 115-132. doi: 10.1007/BF00275917. [15] S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM Journal on Applied Mathematics, 68 (2008), 1600-1617. doi: 10.1137/070700784. [16] S.-B. Hsu and P. Waltman, On a system of reaction-diffus ion equations arising from competition in an unstirred chemostat, SIAM Journal on Applied Mathematics, 53 (1993), 1026-1044. doi: 10.1137/0153051. [17] J. Jia and H. Zhang, Existence and global attractivity of periodic solutions for chemostat model with delayed nutrients recycling, Differential Equations and Applications, 6 (2014), 275-286. [18] H. Khalil, Nonlinear Systems, Third Edition, Prentice-Hall, Englewood Cliffs, NJ, 2002. [19] P. Lenas and S. Pavlou, Coexistence of three competing microbial populations in a chemostat with periodically varying dilution rate, Mathematical Biosciences, 129 (1995), 111-142. doi: 10.1016/0025-5564(94)00056-6. [20] B. Li and H. Smith, Competition for essential resources: A brief review, in Dynamical Systems and its Applications in Biology, (eds. S. Ruan, G. Wolkowicz, and J. Wu), American Mathematical Society, Providence, RI, 36 (2003), 213-227. [21] S. Liu, X. Wang, L. Wang and H. Song, Competitive exclusion in delayed chemostat models with differential removal rates, SIAM Journal on Applied Mathematics, 74 (2014), 634-648. doi: 10.1137/130921386. [22] C. Lobry, F. Mazenc and A. Rapaport, Persistence in ecological models of competition for a single resource, Comptes Rendus Mathematique, 340 (2005), 199-204. doi: 10.1016/j.crma.2004.12.021. [23] Z. Lu, Global stability for a chemostat-type model with delayed nutrient recycling, Discrete and Continuous Dynamical Systems Series B, 4 (2004), 663-670. doi: 10.3934/dcdsb.2004.4.663. [24] M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag London Ltd., London, UK, 2009. [25] F. Mazenc, J. Harmand and M. Malisoff, Stabilization in a chemostat with sampled and delayed measurements, in Proceedings of the 2016 American Control Conference, Boston, MA, (2016), 1857-1862. doi: 10.1109/ACC.2016.7525189. [26] F. Mazenc and Z.-P. Jiang, Global output feedback stabilization of a chemostat with an arbitrary number of species, IEEE Transactions on Automatic Control, 55 (2010), 2570-2575. doi: 10.1109/TAC.2010.2060246. [27] F. Mazenc and M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurement, Automatica, 46 (2010), 1428-1436. doi: 10.1016/j.automatica.2010.06.012. [28] F. Mazenc and M. Malisoff, Stability and stabilization for models of chemostats with multiple limiting substrates, Journal of Biological Dynamics, 6 (2012), 612-627. doi: 10.1080/17513758.2012.663795. [29] F. Mazenc, M. Malisoff and P. De Leenheer, On the stability of periodic solutions in the perturbed chemostat, Mathematical Biosciences and Engineering, 4 (2007), 319-338. doi: 10.3934/mbe.2007.4.319. [30] F. Mazenc, M. Malisoff and J. Harmand, Further results on stabilization of periodic trajectories for a chemostat with two species, IEEE Transactions on Automatic Control, 53 (2008), 66-74. [31] X. Meng, Q. Gao and Z. Li, The effects of delayed growth response on the dynamic behaviors of the Monod type chemostat model with impulsive input nutrient concentration, Nonlinear Analysis: Real World Applications, 11 (2010), 4476-4486. doi: 10.1016/j.nonrwa.2010.05.030. [32] X. Meng, Z. Li and J.-J. Nieto, Dynamic analysis of Michaelis-Menten chemostat-type competition models with time delay and pulse in a polluted environment, Journal of Mathematical Chemistry, 47 (2010), 123-144. doi: 10.1007/s10910-009-9536-2. [33] G. Meszéna, M. Gyllenberg, L. Pásztor and J. Metz, Competitive exclusion and limiting similarity: A unified theory, Theoretical Population Biology, 69 (2006), 68-87. [34] J. Monod, La technique de culture continue, théorie et applications, Selected Papers in Molecular Biology by Jacques Monod, (1978), 184-204. doi: 10.1016/B978-0-12-460482-7.50023-3. [35] C. Neill, T. Daufresne and C. Jones, A competitive coexistence principle?, Oikos, 118 (2009), 1570-1578. doi: 10.1111/j.1600-0706.2009.17522.x. [36] H. Nie and J. Wu, Coexistence of an unstirred chemostat model with Beddington-De Angelis functional response and inhibitor, Nonlinear Analysis: Real World Applications, 11 (2010), 3639-3652. doi: 10.1016/j.nonrwa.2010.01.010. [37] A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), 715-716. doi: 10.1126/science.112.2920.715. [38] S. Pavlou, Microbial competition in bioreactors, Chemical Industry and Chemical Engineering Quarterly, 12 (2006), 71-81. doi: 10.2298/CICEQ0601071P. [39] G. Robledo, F. Grognard and J.-L. Gouzé, Global stability for a model of competition in the chemostat with microbial inputs, Nonlinear Analysis: Real World Applications, 13 (2012), 582-598. doi: 10.1016/j.nonrwa.2011.07.049. [40] S. Ruan, The effect of delays on stability and persistence in plankton models, Nonlinear Analysis: Theory, Methods, and Applications, 24 (1995), 575-585. doi: 10.1016/0362-546X(95)93092-I. [41] H. Smith, Competitive coexistence in an oscillating chemostat, SIAM Journal on Applied Mathematics, 40 (1981), 498-522. doi: 10.1137/0140042. [42] H. Smith and P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995. [43] W. Mathematica, The world's definitive system for modern technical computing, Wolfram Research, Accessed August 11, 2016, http://www.wolfram.com/mathematica/. [44] G. Wolkowicz and H. Xia, Global asymptotic behavior of a chemostat model with discrete delays, SIAM Journal on Applied Mathematics, 57 (1997), 1019-1043. doi: 10.1137/S0036139995287314. [45] G. Wolkowicz and X.-Q. Zhao, N-species competition in a periodic chemostat, Differential Integral Equations, 11 (1998), 465-491. [46] H. Zhang, L. Chen and J.-J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Analysis: Real World Applications, 9 (2008), 1714-1726. doi: 10.1016/j.nonrwa.2007.05.004. [47] H. Zhang, P. Georgescu, J.-J. Nieto and L. Chen, Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield, Applied Mathematics and Mechanics, 30 (2009), 933-944. doi: 10.1007/s10483-009-0712-x.

show all references

##### References:
 [1] A. Bush and A. Cool, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, Journal of Theoretical Biology, 63 (1975), 385-395. doi: 10.1016/0022-5193(76)90041-2. [2] T. Caraballo, X. Han and P. Kloeden, Nonautonomous chemostats with variable delays, SIAM Journal on Mathematical Analysis, 47 (2015), 2178-2199. doi: 10.1137/14099930X. [3] P. Collet, S. Martinez, S. Meleard and J. San Martin, Stochastic models for a chemostat and long time behavior, Advances in Applied Probability, 45 (2013), 822-836. doi: 10.1017/S0001867800006595. [4] Y. Collos, Time-lag algal growth dynamics: Biological constraints on primary production in aquatic environments, Marine Ecology Progress Series, 33 (1986), 193-206. doi: 10.3354/meps033193. [5] S. Dikshitulu, B. Baltzis, G. Lewandowski and S. Pavlou, Competition between two microbial populations in a sequencing fed-batch reactor: theory, experimental verification, and implications for waste treatment applications, Biotechnology and Bioengineering, 42 (1993), 643-656. doi: 10.1002/bit.260420513. [6] S. Ellermeyer, Competition in the chemostat: global asymptotic behavior of a model with delayed response in growth, SIAM Journal on Applied Mathematics, 54 (1994), 456-465. doi: 10.1137/S003613999222522X. [7] C. Fritsch, J. Harmand and F. Campillo, A Modeling approach of the chemostat, Ecological Modelling, 299 (2015), 1-13. doi: 10.1016/j.ecolmodel.2014.11.021. [8] J-L. Gouzé and G. Robledo, Feedback control for nonmonotone competition models in the chemostat, Nonlinear Analysis: Real World Applications, 6 (2005), 671-690. doi: 10.1016/j.nonrwa.2004.12.003. [9] F. Grognard, F. Mazenc and A. Rapaport, Polytopic Lyapunov functions for persistence analysis of competing species, Discrete and Continuous Dynamical Systems Series B, 8 (2007), 73-93. doi: 10.3934/dcdsb.2007.8.73. [10] H. Guo and S. Zheng, A competition model for two resources in un-stirred chemostat, Applied Mathematics and Computation, 217 (2011), 6934-6949. doi: 10.1016/j.amc.2011.01.102. [11] B. Haegeman and A. Rapaport, How flocculation can explain coexistence in the chemostat, Journal of Biological Dynamics, 2 (2008), 1-13. doi: 10.1080/17513750801942537. [12] G. Hardin, Competitive exclusion principle, Science, 131 (1960), 1292-1297. doi: 10.1126/science.131.3409.1292. [13] X.-Z. He and S. Ruan, Global stability in chemostat-type plankton models with delayed nutrient recycling, Journal of Mathematical Biology, 37 (1998), 253-271. doi: 10.1007/s002850050128. [14] S.-B. Hsu, A competition model for a seasonally fluctuating nutrient, Journal of Mathematical Biology, 9 (1980), 115-132. doi: 10.1007/BF00275917. [15] S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM Journal on Applied Mathematics, 68 (2008), 1600-1617. doi: 10.1137/070700784. [16] S.-B. Hsu and P. Waltman, On a system of reaction-diffus ion equations arising from competition in an unstirred chemostat, SIAM Journal on Applied Mathematics, 53 (1993), 1026-1044. doi: 10.1137/0153051. [17] J. Jia and H. Zhang, Existence and global attractivity of periodic solutions for chemostat model with delayed nutrients recycling, Differential Equations and Applications, 6 (2014), 275-286. [18] H. Khalil, Nonlinear Systems, Third Edition, Prentice-Hall, Englewood Cliffs, NJ, 2002. [19] P. Lenas and S. Pavlou, Coexistence of three competing microbial populations in a chemostat with periodically varying dilution rate, Mathematical Biosciences, 129 (1995), 111-142. doi: 10.1016/0025-5564(94)00056-6. [20] B. Li and H. Smith, Competition for essential resources: A brief review, in Dynamical Systems and its Applications in Biology, (eds. S. Ruan, G. Wolkowicz, and J. Wu), American Mathematical Society, Providence, RI, 36 (2003), 213-227. [21] S. Liu, X. Wang, L. Wang and H. Song, Competitive exclusion in delayed chemostat models with differential removal rates, SIAM Journal on Applied Mathematics, 74 (2014), 634-648. doi: 10.1137/130921386. [22] C. Lobry, F. Mazenc and A. Rapaport, Persistence in ecological models of competition for a single resource, Comptes Rendus Mathematique, 340 (2005), 199-204. doi: 10.1016/j.crma.2004.12.021. [23] Z. Lu, Global stability for a chemostat-type model with delayed nutrient recycling, Discrete and Continuous Dynamical Systems Series B, 4 (2004), 663-670. doi: 10.3934/dcdsb.2004.4.663. [24] M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag London Ltd., London, UK, 2009. [25] F. Mazenc, J. Harmand and M. Malisoff, Stabilization in a chemostat with sampled and delayed measurements, in Proceedings of the 2016 American Control Conference, Boston, MA, (2016), 1857-1862. doi: 10.1109/ACC.2016.7525189. [26] F. Mazenc and Z.-P. Jiang, Global output feedback stabilization of a chemostat with an arbitrary number of species, IEEE Transactions on Automatic Control, 55 (2010), 2570-2575. doi: 10.1109/TAC.2010.2060246. [27] F. Mazenc and M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurement, Automatica, 46 (2010), 1428-1436. doi: 10.1016/j.automatica.2010.06.012. [28] F. Mazenc and M. Malisoff, Stability and stabilization for models of chemostats with multiple limiting substrates, Journal of Biological Dynamics, 6 (2012), 612-627. doi: 10.1080/17513758.2012.663795. [29] F. Mazenc, M. Malisoff and P. De Leenheer, On the stability of periodic solutions in the perturbed chemostat, Mathematical Biosciences and Engineering, 4 (2007), 319-338. doi: 10.3934/mbe.2007.4.319. [30] F. Mazenc, M. Malisoff and J. Harmand, Further results on stabilization of periodic trajectories for a chemostat with two species, IEEE Transactions on Automatic Control, 53 (2008), 66-74. [31] X. Meng, Q. Gao and Z. Li, The effects of delayed growth response on the dynamic behaviors of the Monod type chemostat model with impulsive input nutrient concentration, Nonlinear Analysis: Real World Applications, 11 (2010), 4476-4486. doi: 10.1016/j.nonrwa.2010.05.030. [32] X. Meng, Z. Li and J.-J. Nieto, Dynamic analysis of Michaelis-Menten chemostat-type competition models with time delay and pulse in a polluted environment, Journal of Mathematical Chemistry, 47 (2010), 123-144. doi: 10.1007/s10910-009-9536-2. [33] G. Meszéna, M. Gyllenberg, L. Pásztor and J. Metz, Competitive exclusion and limiting similarity: A unified theory, Theoretical Population Biology, 69 (2006), 68-87. [34] J. Monod, La technique de culture continue, théorie et applications, Selected Papers in Molecular Biology by Jacques Monod, (1978), 184-204. doi: 10.1016/B978-0-12-460482-7.50023-3. [35] C. Neill, T. Daufresne and C. Jones, A competitive coexistence principle?, Oikos, 118 (2009), 1570-1578. doi: 10.1111/j.1600-0706.2009.17522.x. [36] H. Nie and J. Wu, Coexistence of an unstirred chemostat model with Beddington-De Angelis functional response and inhibitor, Nonlinear Analysis: Real World Applications, 11 (2010), 3639-3652. doi: 10.1016/j.nonrwa.2010.01.010. [37] A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), 715-716. doi: 10.1126/science.112.2920.715. [38] S. Pavlou, Microbial competition in bioreactors, Chemical Industry and Chemical Engineering Quarterly, 12 (2006), 71-81. doi: 10.2298/CICEQ0601071P. [39] G. Robledo, F. Grognard and J.-L. Gouzé, Global stability for a model of competition in the chemostat with microbial inputs, Nonlinear Analysis: Real World Applications, 13 (2012), 582-598. doi: 10.1016/j.nonrwa.2011.07.049. [40] S. Ruan, The effect of delays on stability and persistence in plankton models, Nonlinear Analysis: Theory, Methods, and Applications, 24 (1995), 575-585. doi: 10.1016/0362-546X(95)93092-I. [41] H. Smith, Competitive coexistence in an oscillating chemostat, SIAM Journal on Applied Mathematics, 40 (1981), 498-522. doi: 10.1137/0140042. [42] H. Smith and P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995. [43] W. Mathematica, The world's definitive system for modern technical computing, Wolfram Research, Accessed August 11, 2016, http://www.wolfram.com/mathematica/. [44] G. Wolkowicz and H. Xia, Global asymptotic behavior of a chemostat model with discrete delays, SIAM Journal on Applied Mathematics, 57 (1997), 1019-1043. doi: 10.1137/S0036139995287314. [45] G. Wolkowicz and X.-Q. Zhao, N-species competition in a periodic chemostat, Differential Integral Equations, 11 (1998), 465-491. [46] H. Zhang, L. Chen and J.-J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Analysis: Real World Applications, 9 (2008), 1714-1726. doi: 10.1016/j.nonrwa.2007.05.004. [47] H. Zhang, P. Georgescu, J.-J. Nieto and L. Chen, Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield, Applied Mathematics and Mechanics, 30 (2009), 933-944. doi: 10.1007/s10483-009-0712-x.
Solution Components of (10) Plotted on Time Interval $[0, 25]$. Species $x_{1}(t)$ and $x_{2}(t)$ and Substrate $s(t)$. Initial State: $(s(0), x_1(0), x_2(0)) = (0.2, 0.1, 1)$.
Solution Components of (10) Plotted on Time Interval $[0, 25]$. Species $x_{1}(t)$ and $x_{2}(t)$ and Substrate $s(t)$. Initial State: $(s(0), x_1(0), x_2(0)) = (1.3, 0.2, 0.1)$.
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