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.  Google Scholar

[2]

T. CaraballoX. Han and P. Kloeden, Nonautonomous chemostats with variable delays, SIAM Journal on Mathematical Analysis, 47 (2015), 2178-2199.  doi: 10.1137/14099930X.  Google Scholar

[3]

P. ColletS. MartinezS. 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.  Google Scholar

[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.  Google Scholar

[5]

S. DikshituluB. BaltzisG. 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.  Google Scholar

[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.  Google Scholar

[7]

C. FritschJ. Harmand and F. Campillo, A Modeling approach of the chemostat, Ecological Modelling, 299 (2015), 1-13.  doi: 10.1016/j.ecolmodel.2014.11.021.  Google Scholar

[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.  Google Scholar

[9]

F. GrognardF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. Hardin, Competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[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.  Google Scholar

[14]

S.-B. Hsu, A competition model for a seasonally fluctuating nutrient, Journal of Mathematical Biology, 9 (1980), 115-132.  doi: 10.1007/BF00275917.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[18] H. Khalil, Nonlinear Systems, Third Edition, Prentice-Hall, Englewood Cliffs, NJ, 2002.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[21]

S. LiuX. WangL. 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.  Google Scholar

[22]

C. LobryF. 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.  Google Scholar

[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.  Google Scholar

[24] M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag London Ltd., London, UK, 2009.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

F. MazencM. 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.  Google Scholar

[30]

F. MazencM. 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.   Google Scholar

[31]

X. MengQ. 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.  Google Scholar

[32]

X. MengZ. 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.  Google Scholar

[33]

G. MeszénaM. GyllenbergL. Pásztor and J. Metz, Competitive exclusion and limiting similarity: A unified theory, Theoretical Population Biology, 69 (2006), 68-87.   Google Scholar

[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.  Google Scholar

[35]

C. NeillT. Daufresne and C. Jones, A competitive coexistence principle?, Oikos, 118 (2009), 1570-1578.  doi: 10.1111/j.1600-0706.2009.17522.x.  Google Scholar

[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.  Google Scholar

[37]

A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), 715-716.  doi: 10.1126/science.112.2920.715.  Google Scholar

[38]

S. Pavlou, Microbial competition in bioreactors, Chemical Industry and Chemical Engineering Quarterly, 12 (2006), 71-81.  doi: 10.2298/CICEQ0601071P.  Google Scholar

[39]

G. RobledoF. 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.  Google Scholar

[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.  Google Scholar

[41]

H. Smith, Competitive coexistence in an oscillating chemostat, SIAM Journal on Applied Mathematics, 40 (1981), 498-522.  doi: 10.1137/0140042.  Google Scholar

[42] H. Smith and P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995.   Google Scholar
[43]

W. Mathematica, The world's definitive system for modern technical computing, Wolfram Research, Accessed August 11, 2016, http://www.wolfram.com/mathematica/. Google Scholar

[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.  Google Scholar

[45]

G. Wolkowicz and X.-Q. Zhao, N-species competition in a periodic chemostat, Differential Integral Equations, 11 (1998), 465-491.   Google Scholar

[46]

H. ZhangL. 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.  Google Scholar

[47]

H. ZhangP. GeorgescuJ.-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.  Google Scholar

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.  Google Scholar

[2]

T. CaraballoX. Han and P. Kloeden, Nonautonomous chemostats with variable delays, SIAM Journal on Mathematical Analysis, 47 (2015), 2178-2199.  doi: 10.1137/14099930X.  Google Scholar

[3]

P. ColletS. MartinezS. 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.  Google Scholar

[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.  Google Scholar

[5]

S. DikshituluB. BaltzisG. 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.  Google Scholar

[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.  Google Scholar

[7]

C. FritschJ. Harmand and F. Campillo, A Modeling approach of the chemostat, Ecological Modelling, 299 (2015), 1-13.  doi: 10.1016/j.ecolmodel.2014.11.021.  Google Scholar

[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.  Google Scholar

[9]

F. GrognardF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. Hardin, Competitive exclusion principle, Science, 131 (1960), 1292-1297.  doi: 10.1126/science.131.3409.1292.  Google Scholar

[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.  Google Scholar

[14]

S.-B. Hsu, A competition model for a seasonally fluctuating nutrient, Journal of Mathematical Biology, 9 (1980), 115-132.  doi: 10.1007/BF00275917.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[18] H. Khalil, Nonlinear Systems, Third Edition, Prentice-Hall, Englewood Cliffs, NJ, 2002.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[21]

S. LiuX. WangL. 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.  Google Scholar

[22]

C. LobryF. 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.  Google Scholar

[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.  Google Scholar

[24] M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer-Verlag London Ltd., London, UK, 2009.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

F. MazencM. 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.  Google Scholar

[30]

F. MazencM. 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.   Google Scholar

[31]

X. MengQ. 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.  Google Scholar

[32]

X. MengZ. 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.  Google Scholar

[33]

G. MeszénaM. GyllenbergL. Pásztor and J. Metz, Competitive exclusion and limiting similarity: A unified theory, Theoretical Population Biology, 69 (2006), 68-87.   Google Scholar

[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.  Google Scholar

[35]

C. NeillT. Daufresne and C. Jones, A competitive coexistence principle?, Oikos, 118 (2009), 1570-1578.  doi: 10.1111/j.1600-0706.2009.17522.x.  Google Scholar

[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.  Google Scholar

[37]

A. Novick and L. Szilard, Description of the chemostat, Science, 112 (1950), 715-716.  doi: 10.1126/science.112.2920.715.  Google Scholar

[38]

S. Pavlou, Microbial competition in bioreactors, Chemical Industry and Chemical Engineering Quarterly, 12 (2006), 71-81.  doi: 10.2298/CICEQ0601071P.  Google Scholar

[39]

G. RobledoF. 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.  Google Scholar

[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.  Google Scholar

[41]

H. Smith, Competitive coexistence in an oscillating chemostat, SIAM Journal on Applied Mathematics, 40 (1981), 498-522.  doi: 10.1137/0140042.  Google Scholar

[42] H. Smith and P. Waltman, The Theory of the Chemostat. Dynamics of Microbial Competition, Cambridge University Press, Cambridge, UK, 1995.   Google Scholar
[43]

W. Mathematica, The world's definitive system for modern technical computing, Wolfram Research, Accessed August 11, 2016, http://www.wolfram.com/mathematica/. Google Scholar

[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.  Google Scholar

[45]

G. Wolkowicz and X.-Q. Zhao, N-species competition in a periodic chemostat, Differential Integral Equations, 11 (1998), 465-491.   Google Scholar

[46]

H. ZhangL. 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.  Google Scholar

[47]

H. ZhangP. GeorgescuJ.-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.  Google Scholar

Figure 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: $(s(0), x_1(0), x_2(0)) = (0.2, 0.1, 1)$.
Figure 2.  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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