Stability and bifurcation of stochastic chemostat model

Document Type : Research Article

Authors

Department of Mathematics, Yazd University, Yazd, Iran

Abstract

The main purpose of this paper is to study dynamics of stochastic chemostat model. In this order, Taylor expansions, polar coordinate transformation and stochastic averaging method are our main tools. The stability and bifurcation of the stochastic chemostat model are considered. Some theorems provide sufficient conditions to investigate  stochastic stability, $D$-bifurcation and $P$-bifurcation of the  model. As a final point, to show the effects of the  noise intensity and illustrate our theoretical results, some numerical simulations are presented.

Keywords

Main Subjects


[1] L. Arnold, Random Dynamical System, Springer, 2007.
[2] H.R. Bungay, Microbial interactions in continuous culture, Adv. Appl. Microbiol. 10 (1968) 269–
290.
[3] F. Campillo, M. Joannides, I. Larramendy-Valverde, Analysis and approximation of a stochastic
growth model with extinction, Methodol. Comput. Appl. Probab. 18 (2016) 499–515.
[4] F. Campillo, M. Joannides, I. Larramendy-Valverde, Stochastic modeling of the chemostat, Ecol.
Model. 222 (2011) 2676–2689.
[5] T. Caraballo, X. Han, P. Kloeden, Chemostats with time-dependent inputs and wall growth, Appl.
Math. Inf. Sci. 9 (2015) 2283–2296.
[6] T. Caraballo, X. Han, P.E. Kloeden, Chemostats with random inputs and wall growth, Math. Meth-
ods Appl. Sci. 38 (2015) 3538–3550.
[7] T. Caraballo, M.J. Goarrid-Atienza, J.L. Cruz, Dynamics of some stochastic chemostat models with
multiplicative noise, Commun. Pure. Appl. Anal. 16 (2017) 1893–1914
[8] S. Chakraborty, S. Pal, N. Bairagi, Predator-prey fishery model under deterministic and stochastic
environments: a mathematical perspective, Intl. J. Dyn. Sys. Diff. Eq. 4 (2012) 215–241.
[9] A. Cunningham, R.M. Nisbet, Transients and oscillations in continuous culture, in: Mathematics
in Microbiology, Academic Press, London, (1983) 77–103.
[10] G. Dans, P.V. Kokotovic, D. Gottlieb, A nonlinear regulator problem for a model of biological
waste treatment, IEEE Trans. Automat. Control AC. 16 (1971) 341–347.
[11] S.R. Dtchetgnia Djeundam, R. Yampi, T.C. Kofane, M.A. Aziz-Alaoui, Deterministic and stochas-
tic bifurcations in the Hindmarsh-Rose neuronal Model, Chaos 23 (2013) 033125.
[12] M. Fatehi Nia, M.H. Akrami, Stability and bifurcation in a stochastic vocal folds model, Commun.
Nonlinear Sci. Numer. Simul. 79 (2019) 104898.
[13] R. Freter, Mechanisms that control the microflora in the large intestine, in: D. Hentges (Ed.),
Human Intestinal Microflora in Health and Disease, Academic Press, (1983) 33–54.
[14] R. Freter, H. Brickner, S. Temme, An understanding of colonization resistance of the mammalian
large intestine requires mathematical analysis, Microecol. Therapy 16 (1986) 147–155.
[15] A.G. Fredrickson, G. Stephanopoulos, Microbial competition, Science 213 (1981) 972–979.
[16] J. Grasman, M.D. Gee, O.A.V. Herwaarden, Breakdown of a chemostat exposed to stochastic noise,
J. Eng. Math. 53 (2005) 291–300.
[17] H. Jannasch, R. Mateles, Experimental bacterial ecology studied in continuous culture, Adv. Mi-
crob. Physiol. 11 (1974), 165–212.
[18] J.W.M. La Riviere, Microbial ecology of liquid waste treatment, in: M. Alexander (Ed.), Adv.
Microb. Ecol. (1977) 215–259.
[19] M. Lin, H.-F. Huo, Y.-N. Li, A competitive model in a chemostat with nutrient recycling and an-
tibiotic treatment, Nonlinear Anal. Real World Appl. 13 (2012), 2540–2555.
[20] C. Lue, S.Gue, Stability and bifurcation of two-dimensional stochastic differential equations with
multiplicative excitations, Bull. Malays. Math. Sci. Soc. 40 (2017) 795–817.
[21] F.S. Mousavinejad, M. Fatehi Nia, A. Ebrahimi, P-bifurcation of Stochastic van der Pol Model as
a Dynamical System in Neuroscience, Commun. Appl. Math. Comput. 4, (2022) 1293–1312.
[22] G. Robledo, F. Grognard, J.-L. Gouze, Global stability for a model of competition in the chemostat
with microbial inputs, Nonlinear Anal. Real World Appl. 13 (2012) 582–598.
[23] V.A. Sadovnichiy, M.Z. Zgurovsky, Advances in Dynamical Systems and Control, Springer, 2016.
[24] S. Sun, X. Zhang, Asymptotic behavior of a stochastic delayed chemostat model with nonmonotone
uptake function Physica A. 512 (2018) 38–56.
[25] L.Wang, D. Jiang, Ergodic property of the chemostat: A stochastic model under regime switching
and with general response function, Nonlinear Anal. Hybrid Syst. 27 (2018) 341–352
[26] C. Xu, S. Yuan, T. Zhang, Asymptotic Behavior of a Chemostat Model with Stochastic Perturbation
on the Dilution Rate, Abstr. Appl. Anal. 2013 (2013) 1–11.
[27] J.H. Yang, M.A. Sanjuan, H.G. Liu, G. Litak, X. Li, Stochastic P-bifurcation and stochastic res-
onance in a noisy bistable fractional-order system, Commun. Nonlinear Sci. Numer. Simul. 41
(2016) 104–117.