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

Tomás Caraballo; Maria-José Garrido-Atienza; Javier López-de-la-Cruz; Alain Rapaport

doi:10.3934/dcdsb.2018280

Article Contents

2019, Volume 24, Issue 8: 3591-3614. Doi: 10.3934/dcdsb.2018280

This issue Previous Article Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping Next Article Smoothness of density for stochastic differential equations with Markovian switching

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
* Corresponding author: Alain Rapaport

Received: April 2018

Revised: June 2018

Early access: October 2018

Published: August 2019

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

Abstract / Introduction Full Text(HTML) Figure(11) / Table(1) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 37N25, 92D25, 34F05, 37H10; Secondary: 37B55, 60H10.

Citation:

Full Text(HTML)

Figure 1. Realizations of the perturbed dilution rate with $D = 2$, $\alpha = 0.8$ and $\beta = 2$

Download: Full-size image PowerPoint slide

Figure 2. Realizations of the perturbed dilution rate with $D = 2$, $\alpha = 0.8$ and $\nu = 0.5$

Download: Full-size image PowerPoint slide

Figure 3. Realizations of the perturbed dilution rate, $\underline{s}$ and $\bar{s}$

Download: Full-size image PowerPoint slide

Figure 4. Attracting set $\widehat{B}_0$

Download: Full-size image PowerPoint slide

Figure 5. $\mbox{Absorbing set } B_\varepsilon(\omega)$

Download: Full-size image PowerPoint slide

Figure 6. $\mbox{Absorbing set }{\color{blue}{ B_0(\omega)}}$

Download: Full-size image PowerPoint slide

Figure 7. Persistence of the species in the random chemostat model

Download: Full-size image PowerPoint slide

Figure 8. Extinction of the species in the random chemostat model

Download: Full-size image PowerPoint slide

Figure 9. Stochastic chemostat model. Extinction (left) and persistence (right)

Download: Full-size image PowerPoint slide

Figure 10. Comparison in case of extinction

Download: Full-size image PowerPoint slide

Figure 11. Comparison in case of persistence

Download: Full-size image PowerPoint slide

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 $\underline{s}\leq s\leq s_{in}$ $0\leq x\leq s_{in}-\underline{s}$

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	S. Al-azzawi, J. 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.
[2]	L.Arnold, Random Dynamical Systems, Springer Berlin Heidelberg, 1998.
[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.
[4]	F. Campillo, M. Joannides and I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecological Modelling, 222 (2011), 2676-2689. doi: 10.1016/j.ecolmodel.2011.04.027.
[5]	F. Campillo, M. 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.
[6]	F. Campillo, M. 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.
[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.
[8]	T. Caraballo, M.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.
[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].
[10]	T. Caraballo, M.J. Garrido-uppercaseatienza, B. 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.
[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.
[12]	T. Caraballo, P.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.
[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.
[14]	T. Caraballo, G. 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.
[15]	I.F. Creed, D.M. McKnight, B.A. Pellerin, M.B. Green, B.A. Bergamaschi, G.R. Aiken, D.A. Burns, S.E.G. Findlay, J.B. Shanley, R.G. Striegl, B.T. Aulenbach, D.W. Clow, H. Laudon, B.L. McGlynn, K.J. McGuire, R.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.
[16]	G. D'Ans, P. Kokotovic and D. Gottlieb, A nonlinear regulator problem for a model of biological waste treatment, IEEE Transactions on Automatic Control, 16 (1971), 341-347.
[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.
[18]	J. Grasman, M.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.
[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.
[20]	D.J. Higham., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302.
[21]	S.B. Hsu, S. 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.
[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.
[23]	H.W. Jannasch, Steady state and the chemostat in ecology, Limnology and Oceanography, 19 (1974), 716-720.
[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.
[25]	J. W. M. La Rivière, Microbial ecology of liquid waste treatment, in Advances in Microbial Ecology, vol. 1, Springer US, 1977, 215–259.
[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.
[27]	H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.
[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.
[29]	L. Wang, D. 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.
[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.
[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.
[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.
[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.