August  2019, 24(8): 3591-3614. doi: 10.3934/dcdsb.2018280

Modeling and analysis of random and stochastic input flows in the chemostat model

1. 

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, Sevilla 41012, Spain

2. 

MISTEA, Univ. Montpellier, Inra, Montpellier SupAgro, 2, place Pierre Viala, 34060 Montpellier, France

* Corresponding author: Alain Rapaport

Received  April 2018 Revised  June 2018 Published  October 2018

Fund Project: Partially supported by FEDER and Ministerio de Economía y Competitividad under grant MTM2015-63723-P, Junta de Andalucía under the Proyecto de Excelencia P12-FQM-1492 and Ⅵ Plan Propio de Investigación y Transferencia de la Universidad de Sevilla

In this paper we study a new way to model noisy input flows in the chemostat model, based on the Ornstein-Uhlenbeck process. We introduce a parameter β as drift in the Langevin equation, that allows to bridge a gap between a pure Wiener process, which is a common way to model random disturbances, and no noise at all. The value of the parameter β is related to the amplitude of the deviations observed on the realizations. We show that this modeling approach is well suited to represent noise on an input variable that has to take non-negative values for almost any time.

Citation: Tomás Caraballo, Maria-José Garrido-Atienza, Javier López-de-la-Cruz, Alain Rapaport. Modeling and analysis of random and stochastic input flows in the chemostat model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3591-3614. doi: 10.3934/dcdsb.2018280
References:
[1]

S. Al-azzawiJ. Liu and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 227-245.  doi: 10.3934/dcdsb.2017012.  Google Scholar

[2]

L.Arnold, Random Dynamical Systems, Springer Berlin Heidelberg, 1998.  Google Scholar

[3]

J. Barlow, W. Schaffner, F. de Noyelles, B. Peterson and J. Peterson, Continuous Flow Nutrient Bioassays with Natural Phytoplankton Populations, G. Glass (Editor): Bioassay Techniques and Environmental Chemistry, John Wiley & Sons Ltd., 1973. Google Scholar

[4]

F. CampilloM. Joannides and I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecological Modelling, 222 (2011), 2676-2689.  doi: 10.1016/j.ecolmodel.2011.04.027.  Google Scholar

[5]

F. CampilloM. Joannides and I. Larramendy-Valverde, Approximation of the Fokker-Planck equation of the stochastic chemostat, Mathematics and Computers in Simulation, 99 (2014), 37-53.  doi: 10.1016/j.matcom.2013.04.012.  Google Scholar

[6]

F. CampilloM. Joannides and I. Larramendy-Valverde, Analysis and approximation of a stochastic growth model with extinction, Methodology and Computing in Applied Probability, 18 (2016), 499-515.  doi: 10.1007/s11009-015-9438-7.  Google Scholar

[7]

T. Caraballo, M. J. Garrido-Atienza and J. López-de-la-Cruz, Some aspects concerning the dynamics of stochastic chemostats, Advances in Dynamical Systems and Control, 227–246, Stud. Syst. Decis. Control, 69, Springer, [Cham], 2016.  Google Scholar

[8]

T. CaraballoM.J. Garrido-uppercaseatienza and J. López-de-la-uppercasecruz, Dynamics of some stochastic chemostat models with multiplicative noise, Communications on Pure and Applied Analysis, 16 (2017), 1893-1914.  doi: 10.3934/cpaa.2017092.  Google Scholar

[9]

T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz and A. Rapaport, Corrigendum to "Some aspects concerning the dynamics of stochastic chemostats", 2017, arXiv: 1710.00774 [math.DS]. Google Scholar

[10]

T. CaraballoM.J. Garrido-uppercaseatienzaB. Schmalfuss and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete and Continuous Dynamical Systems - Series B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[11]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems, Springer International Publishing, 2016. doi: 10.1007/978-3-319-49247-6.  Google Scholar

[12]

T. CaraballoP.E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Applied Mathematics and Optimization, 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar

[13]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.  Google Scholar

[14]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[15]

I.F. CreedD.M. McKnightB.A. PellerinM.B. GreenB.A. BergamaschiG.R. AikenD.A. BurnsS.E.G. FindlayJ.B. ShanleyR.G. StrieglB.T. AulenbachD.W. ClowH. LaudonB.L. McGlynnK.J. McGuireR.A. Smith and S.M. Stackpoole, The river as a chemostat: Fresh perspectives on dissolved organic matter flowing down the river continuum, Canadian Journal of Fisheries and Aquatic Sciences, 72 (2015), 1272-1285.  doi: 10.1139/cjfas-2014-0400.  Google Scholar

[16]

G. D'AnsP. Kokotovic and D. Gottlieb, A nonlinear regulator problem for a model of biological waste treatment, IEEE Transactions on Automatic Control, 16 (1971), 341-347.   Google Scholar

[17]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic navier-stokes equation with multiplicative white noise, Stochastics and Stochastic Reports, 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[18]

J. GrasmanM.D. Gee and O.A.V. Herwaarden, Breakdown of a chemostat exposed to stochastic noise, Journal of Engineering Mathematics, 53 (2005), 291-300.  doi: 10.1007/s10665-005-9004-3.  Google Scholar

[19]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Micro-organisms Cultures, Wiley, Chemical Engineering Series, John Wiley & Sons, Inc., 2017.  Google Scholar

[20]

D.J. Higham., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[21]

S.B. HsuS. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[22]

L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, Journal of Differential Equations, 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017.  Google Scholar

[23]

H.W. Jannasch, Steady state and the chemostat in ecology, Limnology and Oceanography, 19 (1974), 716-720.   Google Scholar

[24]

J. Kalff and R. Knoechel, Phytoplankton and their dynamics in oligotrophic and eutrophic lakes, Annual Review of Ecology and Systematics, 9 (1978), 475-495.  doi: 10.1146/annurev.es.09.110178.002355.  Google Scholar

[25]

J. W. M. La Rivière, Microbial ecology of liquid waste treatment, in Advances in Microbial Ecology, vol. 1, Springer US, 1977, 215–259. Google Scholar

[26]

E. Rurangwa and M.C.J. Verdegem, Microorganisms in recirculating aquaculture systems and their management, Reviews in Aquaculture, 7 (2015), 117-130.  doi: 10.1111/raq.12057.  Google Scholar

[27] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[28]

L. Wang and D. Jiang, Periodic solution for the stochastic chemostat with general response function, Physica A: Statistical Mechanics and its Applications, 486 (2017), 378-385.  doi: 10.1016/j.physa.2017.05.097.  Google Scholar

[29]

L. WangD. Jiang and D. O'Regan, The periodic solutions of a stochastic chemostat model with periodic washout rate, Communications in Nonlinear Science and Numerical Simulation, 37 (2016), 1-13.  doi: 10.1016/j.cnsns.2016.01.002.  Google Scholar

[30]

C. Xu and S. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Applied Mathematics Letters, 48 (2015), 62-68.  doi: 10.1016/j.aml.2015.03.012.  Google Scholar

[31]

C. Xu, S. Yuan and T. Zhang, Asymptotic behavior of a chemostat model with stochastic perturbation on the dilution rate, Abstract and Applied Analysis, 2013 (2013), Art. ID 423154, 11 pp.  Google Scholar

[32]

D. Zhao and S. Yuan, Critical result on the break-even concentration in a single-species stochastic chemostat model, Journal of Mathematical Analysis and Applications, 434 (2016), 1336-1345.  doi: 10.1016/j.jmaa.2015.09.070.  Google Scholar

[33]

D. Zhao and S. Yuan, Break-even concentration and periodic behavior of a stochastic chemostat model with seasonal fluctuation, Communications in Nonlinear Science and Numerical Simulation, 46 (2017), 62-73.  doi: 10.1016/j.cnsns.2016.10.014.  Google Scholar

show all references

References:
[1]

S. Al-azzawiJ. Liu and X. Liu, Convergence rate of synchronization of systems with additive noise, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 227-245.  doi: 10.3934/dcdsb.2017012.  Google Scholar

[2]

L.Arnold, Random Dynamical Systems, Springer Berlin Heidelberg, 1998.  Google Scholar

[3]

J. Barlow, W. Schaffner, F. de Noyelles, B. Peterson and J. Peterson, Continuous Flow Nutrient Bioassays with Natural Phytoplankton Populations, G. Glass (Editor): Bioassay Techniques and Environmental Chemistry, John Wiley & Sons Ltd., 1973. Google Scholar

[4]

F. CampilloM. Joannides and I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecological Modelling, 222 (2011), 2676-2689.  doi: 10.1016/j.ecolmodel.2011.04.027.  Google Scholar

[5]

F. CampilloM. Joannides and I. Larramendy-Valverde, Approximation of the Fokker-Planck equation of the stochastic chemostat, Mathematics and Computers in Simulation, 99 (2014), 37-53.  doi: 10.1016/j.matcom.2013.04.012.  Google Scholar

[6]

F. CampilloM. Joannides and I. Larramendy-Valverde, Analysis and approximation of a stochastic growth model with extinction, Methodology and Computing in Applied Probability, 18 (2016), 499-515.  doi: 10.1007/s11009-015-9438-7.  Google Scholar

[7]

T. Caraballo, M. J. Garrido-Atienza and J. López-de-la-Cruz, Some aspects concerning the dynamics of stochastic chemostats, Advances in Dynamical Systems and Control, 227–246, Stud. Syst. Decis. Control, 69, Springer, [Cham], 2016.  Google Scholar

[8]

T. CaraballoM.J. Garrido-uppercaseatienza and J. López-de-la-uppercasecruz, Dynamics of some stochastic chemostat models with multiplicative noise, Communications on Pure and Applied Analysis, 16 (2017), 1893-1914.  doi: 10.3934/cpaa.2017092.  Google Scholar

[9]

T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz and A. Rapaport, Corrigendum to "Some aspects concerning the dynamics of stochastic chemostats", 2017, arXiv: 1710.00774 [math.DS]. Google Scholar

[10]

T. CaraballoM.J. Garrido-uppercaseatienzaB. Schmalfuss and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete and Continuous Dynamical Systems - Series B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[11]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems, Springer International Publishing, 2016. doi: 10.1007/978-3-319-49247-6.  Google Scholar

[12]

T. CaraballoP.E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Applied Mathematics and Optimization, 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar

[13]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.  Google Scholar

[14]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[15]

I.F. CreedD.M. McKnightB.A. PellerinM.B. GreenB.A. BergamaschiG.R. AikenD.A. BurnsS.E.G. FindlayJ.B. ShanleyR.G. StrieglB.T. AulenbachD.W. ClowH. LaudonB.L. McGlynnK.J. McGuireR.A. Smith and S.M. Stackpoole, The river as a chemostat: Fresh perspectives on dissolved organic matter flowing down the river continuum, Canadian Journal of Fisheries and Aquatic Sciences, 72 (2015), 1272-1285.  doi: 10.1139/cjfas-2014-0400.  Google Scholar

[16]

G. D'AnsP. Kokotovic and D. Gottlieb, A nonlinear regulator problem for a model of biological waste treatment, IEEE Transactions on Automatic Control, 16 (1971), 341-347.   Google Scholar

[17]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic navier-stokes equation with multiplicative white noise, Stochastics and Stochastic Reports, 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[18]

J. GrasmanM.D. Gee and O.A.V. Herwaarden, Breakdown of a chemostat exposed to stochastic noise, Journal of Engineering Mathematics, 53 (2005), 291-300.  doi: 10.1007/s10665-005-9004-3.  Google Scholar

[19]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Micro-organisms Cultures, Wiley, Chemical Engineering Series, John Wiley & Sons, Inc., 2017.  Google Scholar

[20]

D.J. Higham., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[21]

S.B. HsuS. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM Journal on Applied Mathematics, 32 (1977), 366-383.  doi: 10.1137/0132030.  Google Scholar

[22]

L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, Journal of Differential Equations, 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017.  Google Scholar

[23]

H.W. Jannasch, Steady state and the chemostat in ecology, Limnology and Oceanography, 19 (1974), 716-720.   Google Scholar

[24]

J. Kalff and R. Knoechel, Phytoplankton and their dynamics in oligotrophic and eutrophic lakes, Annual Review of Ecology and Systematics, 9 (1978), 475-495.  doi: 10.1146/annurev.es.09.110178.002355.  Google Scholar

[25]

J. W. M. La Rivière, Microbial ecology of liquid waste treatment, in Advances in Microbial Ecology, vol. 1, Springer US, 1977, 215–259. Google Scholar

[26]

E. Rurangwa and M.C.J. Verdegem, Microorganisms in recirculating aquaculture systems and their management, Reviews in Aquaculture, 7 (2015), 117-130.  doi: 10.1111/raq.12057.  Google Scholar

[27] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[28]

L. Wang and D. Jiang, Periodic solution for the stochastic chemostat with general response function, Physica A: Statistical Mechanics and its Applications, 486 (2017), 378-385.  doi: 10.1016/j.physa.2017.05.097.  Google Scholar

[29]

L. WangD. Jiang and D. O'Regan, The periodic solutions of a stochastic chemostat model with periodic washout rate, Communications in Nonlinear Science and Numerical Simulation, 37 (2016), 1-13.  doi: 10.1016/j.cnsns.2016.01.002.  Google Scholar

[30]

C. Xu and S. Yuan, An analogue of break-even concentration in a simple stochastic chemostat model, Applied Mathematics Letters, 48 (2015), 62-68.  doi: 10.1016/j.aml.2015.03.012.  Google Scholar

[31]

C. Xu, S. Yuan and T. Zhang, Asymptotic behavior of a chemostat model with stochastic perturbation on the dilution rate, Abstract and Applied Analysis, 2013 (2013), Art. ID 423154, 11 pp.  Google Scholar

[32]

D. Zhao and S. Yuan, Critical result on the break-even concentration in a single-species stochastic chemostat model, Journal of Mathematical Analysis and Applications, 434 (2016), 1336-1345.  doi: 10.1016/j.jmaa.2015.09.070.  Google Scholar

[33]

D. Zhao and S. Yuan, Break-even concentration and periodic behavior of a stochastic chemostat model with seasonal fluctuation, Communications in Nonlinear Science and Numerical Simulation, 46 (2017), 62-73.  doi: 10.1016/j.cnsns.2016.10.014.  Google Scholar

Figure 1.  Realizations of the perturbed dilution rate with $D = 2$, $\alpha = 0.8$ and $\beta = 2$
Figure 2.  Realizations of the perturbed dilution rate with $D = 2$, $\alpha = 0.8$ and $\nu = 0.5$
Figure 3.  Realizations of the perturbed dilution rate, $\underline{s}$ and $\bar{s}$
Figure 4.  Attracting set $\widehat{B}_0$
Figure 5.  $\mbox{Absorbing set } B_\varepsilon(\omega)$
Figure 6.  $\mbox{Absorbing set }{\color{blue}{ B_0(\omega)}}$
Figure 7.  Persistence of the species in the random chemostat model
Figure 8.  Extinction of the species in the random chemostat model
Figure 9.  Stochastic chemostat model. Extinction (left) and persistence (right)
Figure 10.  Comparison in case of extinction
Figure 11.  Comparison in case of persistence
Table 1.  Internal structure of the attracting set $\widehat{B}_0$
$s_{in}>\bar{s}$ $s_{in}=\bar{s}$ $s_{in}< \bar{s}$
$D>\mu(s_{in})$ impossible impossible Extinction
Proposition 2.2
$\{(s_{in}, 0)\}$
$D=\mu(s_{in})$ Persistence
Theorem 2.3
$\underline{s}\leq s\leq \bar{s}$
$s_{in}-\bar{s}\leq x\leq s_{in}-\underline{s}$
$s+x=s_{in}$
(2.14) not fulfilled
$\underline{s}\leq s\leq s_{in}$
$0\leq x\leq s_{in}-\underline{s}$
(2.14) not fulfilled
$\underline{s}\leq s\leq s_{in}$
$0\leq x\leq s_{in}-\underline{s}$
$D< \mu(s_{in})$ impossible impossible (2.14) not fulfilled
$\underline{s}\leq s\leq s_{in}$
$0\leq x\leq s_{in}-\underline{s}$
$s_{in}>\bar{s}$ $s_{in}=\bar{s}$ $s_{in}< \bar{s}$
$D>\mu(s_{in})$ impossible impossible Extinction
Proposition 2.2
$\{(s_{in}, 0)\}$
$D=\mu(s_{in})$ Persistence
Theorem 2.3
$\underline{s}\leq s\leq \bar{s}$
$s_{in}-\bar{s}\leq x\leq s_{in}-\underline{s}$
$s+x=s_{in}$
(2.14) not fulfilled
$\underline{s}\leq s\leq s_{in}$
$0\leq x\leq s_{in}-\underline{s}$
(2.14) not fulfilled
$\underline{s}\leq s\leq s_{in}$
$0\leq x\leq s_{in}-\underline{s}$
$D< \mu(s_{in})$ impossible impossible (2.14) not fulfilled
$\underline{s}\leq s\leq s_{in}$
$0\leq x\leq s_{in}-\underline{s}$
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