• Previous Article
    Spatial propagation for a parabolic system with multiple species competing for single resource
  • DCDS-B Home
  • This Issue
  • Next Article
    Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains
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, 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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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.

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
$\underline{s}\leq s\leq s_{in}$
$0\leq x\leq s_{in}-\underline{s}$
[1]

Virginia Giorno, Serena Spina. On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 285-302. doi: 10.3934/mbe.2014.11.285

[2]

Annalisa Cesaroni, Matteo Novaga, Enrico Valdinoci. A symmetry result for the Ornstein-Uhlenbeck operator. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2451-2467. doi: 10.3934/dcds.2014.34.2451

[3]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142

[4]

Tomasz Komorowski, Lenya Ryzhik. Fluctuations of solutions to Wigner equation with an Ornstein-Uhlenbeck potential. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 871-914. doi: 10.3934/dcdsb.2012.17.871

[5]

Kai Liu. Quadratic control problem of neutral Ornstein-Uhlenbeck processes with control delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1651-1661. doi: 10.3934/dcdsb.2013.18.1651

[6]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013

[7]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701

[8]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

[9]

Antonio Avantaggiati, Paola Loreti. Hypercontractivity, Hopf-Lax type formulas, Ornstein-Uhlenbeck operators (II). Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 525-545. doi: 10.3934/dcdss.2009.2.525

[10]

Tiziana Durante, Abdelaziz Rhandi. On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 649-655. doi: 10.3934/dcdss.2013.6.649

[11]

Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049

[12]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[13]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1

[14]

Deena Schmidt, Janet Best, Mark S. Blumberg. Random graph and stochastic process contributions to network dynamics. Conference Publications, 2011, 2011 (Special) : 1279-1288. doi: 10.3934/proc.2011.2011.1279

[15]

Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081

[16]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829

[17]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639

[18]

Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093

[19]

Yuncheng You. Random attractor for stochastic reversible Schnackenberg equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1347-1362. doi: 10.3934/dcdss.2014.7.1347

[20]

Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523

2017 Impact Factor: 0.972

Article outline

Figures and Tables

[Back to Top]