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September  2017, 16(5): 1893-1914. doi: 10.3934/cpaa.2017092

## Dynamics of some stochastic chemostat models with multiplicative noise

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

* Corresponding author: caraball@us.es

Received  August 2016 Revised  March 2017 Published  May 2017

Fund Project: Partially supported by FEDER and Ministerio de Economía y Competitividad under grant MTM2015-63723-P and Junta de Andalucía under Proyecto de Excelencia P12-FQM-1492.

In this paper we study two stochastic chemostat models, with and without wall growth, driven by a white noise. Specifically, we analyze the existence and uniqueness of solutions for these models, as well as the existence of the random attractor associated to the random dynamical system generated by the solution. The analysis will be carried out by means of the well-known Ornstein-Uhlenbeck process, that allows us to transform our stochastic chemostat models into random ones.

Citation: T. Caraballo, M. J. Garrido-Atienza, J. López-de-la-Cruz. Dynamics of some stochastic chemostat models with multiplicative noise. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1893-1914. doi: 10.3934/cpaa.2017092
##### References:
 [1] L. Arnold, Random Dynamical Systems, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] H. R. Bungay and M. L. Bungay, Microbial interactions in continuous culture, Advances in Applied Microbiology, 10 (1968), 269-290. [3] T. Caraballo, Recent results on stabilization of PDEs by noise, Bol. Soc. Esp. Mat. Apl., 37 (2006), 47-70. [4] 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, Ⅱ, Studies in Systems, Decision and Control, vol. 69, Springer International Publishing, Cham, (2016), 227-246. [5] T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behaviour 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. [6] T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems, Springer, 2016. doi: 10.1007/978-3-319-49247-6. [7] T. Caraballo, X. Han and P. E. Kloeden, Chemostats with time-dependent inputs and wall growth, Applied Mathematics and Information Sciences, 9 (2015), 2283-2296. [8] T. Caraballo, X. Han and P. E. Kloeden, Chemostats with random inputs and wall growth, Math. Methods Appl. Sci., 38 (2015), 3538-3550. doi: 10.1002/mma.3437. [9] T. Caraballo, P. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Applied Mathematics & Optimization, 50 (2004), 183-207. doi: 10.1007/s00245-004-0802-1. [10] 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. [11] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis TMA, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [12] A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous cultures, Mathematics in Microbiology, Academic Press, London, (1983), 77-103. [13] G. D'ans, P. V. Kokotovic and D. Gottlieb, A nonlinear regulator problem for a model of biological waste treatment, IEEE Transactions on Automatic Control, AC-16 (1971), 341-347. [14] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics Stochastics Rep., 59 (1996), 21-45. [15] D. Foster and P. Young, Stochastic evolutionary game dynamics, Theor. Pop. Biol., 38 (1990), 219-232. doi: 10.1016/0040-5809(90)90011-J. [16] A. G. Fredrickson and G. Stephanopoulos, Microbial competition, Science, 213 (1981), 972-979. doi: 10.1126/science.7268409. [17] R. Freter, Mechanisms that control the microflora in the large intestine, Human Intestinal microflora in Health and Disease, J. Hentges, ed. , Academic Press, New York, (1983), 33-54. [18] R. Freter, An understanding of colonization of the large intestine requires mathematical analysis, Microecology and Therapy, 16 (1986), 147-155. [19] D. Fudenberg and C. Harris, Evolutionary dynamics with aggregate shocks, J. Econom. Theory, 57 (1992), 420-441. doi: 10.1016/0022-0531(92)90044-Ⅰ. [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] J. Hofbauer and L. A. Imhof, Time averages, recurrence and transience in the stochastic replicator dynamics, Ann. Appl. Probab., 19 (2009), 1347-1368. doi: 10.1214/08-AAP577. [22] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217 (2005), 26-53. doi: 10.1016/j.jde.2005.06.017. [23] H. W. Jannash and R. T. Mateles, Experimental bacterial ecology studies in continuous culture, Advances in Microbial Physiology, 11 (1974), 165-212. [24] R. Khasminskii and N. Potsepun, On the replicator dynamics behavior under Stratonovich type random perturbations, Stoch. Dyn., 6 (2006), 197-211. doi: 10.1142/S0219493706001712. [25] J. W. M. La Riviere, Microbial ecology of liquid waste, Advances in Microbial Ecology, 1 (1977), 215-259. [26] H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, 41 (1995). American Mathematical Society, Providence, RI [27] H. L. Smith and P. Waltman, The theory of the chemostat: dynamics of microbial competition, Cambridge University Press, Cambridge, UK, 1995. doi: 10.1017/CBO9780511530043. [28] V. Sree Hari Rao and P. Raja Sekhara Rao, Dynamic Models and Control of Biological Systems, Springer-Verlag, Heidelberg, 2009. [29] P. A. Taylor and J. L. Williams, Theoretical studies on the coexistence of competing species under contunous flow conditions, Canadian Journal of Microbiology, 21 (1975), 90-98. [30] M. Turelli, Random environments and stochastic calculus, Theoret. Population Biology, 12 (1977), 140-178. [31] H. Veldcamp, Ecological studies with the chemostat, Advances in Microbial Ecology, 1 (1977), 59-95. [32] P. Waltman, Competition models in population biology, CBMS-NSF Regional Conference Series in Applied Mathematics, 45 Society for Industrial and Applied Mathematics, Philadelphia, 1983. doi: 10.1137/1.9781611970258. [33] P. Waltman, Coexistence in chemostat-like model, Rocky Mountain Journal of Mathematics, 20 (1990), 777-807. doi: 10.1216/rmjm/1181073042. [34] P. Waltman, S. P. Hubbel and S. B. Hsu, Theoretical and experimental investigations of microbial competition in continuous culture, Modeling and Differential Equations in Biology (Conf. , southern Illinois Univ. Carbonadle, Ⅲ. , 1978), (1980) pp. 107-152. Lecture Notes in Pure and Appl. Math. , 58, Dekker, New York.

show all references

##### References:
 [1] L. Arnold, Random Dynamical Systems, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [2] H. R. Bungay and M. L. Bungay, Microbial interactions in continuous culture, Advances in Applied Microbiology, 10 (1968), 269-290. [3] T. Caraballo, Recent results on stabilization of PDEs by noise, Bol. Soc. Esp. Mat. Apl., 37 (2006), 47-70. [4] 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, Ⅱ, Studies in Systems, Decision and Control, vol. 69, Springer International Publishing, Cham, (2016), 227-246. [5] T. Caraballo, M.J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behaviour 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. [6] T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied Dynamical Systems, Springer, 2016. doi: 10.1007/978-3-319-49247-6. [7] T. Caraballo, X. Han and P. E. Kloeden, Chemostats with time-dependent inputs and wall growth, Applied Mathematics and Information Sciences, 9 (2015), 2283-2296. [8] T. Caraballo, X. Han and P. E. Kloeden, Chemostats with random inputs and wall growth, Math. Methods Appl. Sci., 38 (2015), 3538-3550. doi: 10.1002/mma.3437. [9] T. Caraballo, P. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Applied Mathematics & Optimization, 50 (2004), 183-207. doi: 10.1007/s00245-004-0802-1. [10] 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. [11] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis TMA, 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111. [12] A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous cultures, Mathematics in Microbiology, Academic Press, London, (1983), 77-103. [13] G. D'ans, P. V. Kokotovic and D. Gottlieb, A nonlinear regulator problem for a model of biological waste treatment, IEEE Transactions on Automatic Control, AC-16 (1971), 341-347. [14] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics Stochastics Rep., 59 (1996), 21-45. [15] D. Foster and P. Young, Stochastic evolutionary game dynamics, Theor. Pop. Biol., 38 (1990), 219-232. doi: 10.1016/0040-5809(90)90011-J. [16] A. G. Fredrickson and G. Stephanopoulos, Microbial competition, Science, 213 (1981), 972-979. doi: 10.1126/science.7268409. [17] R. Freter, Mechanisms that control the microflora in the large intestine, Human Intestinal microflora in Health and Disease, J. Hentges, ed. , Academic Press, New York, (1983), 33-54. [18] R. Freter, An understanding of colonization of the large intestine requires mathematical analysis, Microecology and Therapy, 16 (1986), 147-155. [19] D. Fudenberg and C. Harris, Evolutionary dynamics with aggregate shocks, J. Econom. Theory, 57 (1992), 420-441. doi: 10.1016/0022-0531(92)90044-Ⅰ. [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] J. Hofbauer and L. A. Imhof, Time averages, recurrence and transience in the stochastic replicator dynamics, Ann. Appl. Probab., 19 (2009), 1347-1368. doi: 10.1214/08-AAP577. [22] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217 (2005), 26-53. doi: 10.1016/j.jde.2005.06.017. [23] H. W. Jannash and R. T. Mateles, Experimental bacterial ecology studies in continuous culture, Advances in Microbial Physiology, 11 (1974), 165-212. [24] R. Khasminskii and N. Potsepun, On the replicator dynamics behavior under Stratonovich type random perturbations, Stoch. Dyn., 6 (2006), 197-211. doi: 10.1142/S0219493706001712. [25] J. W. M. La Riviere, Microbial ecology of liquid waste, Advances in Microbial Ecology, 1 (1977), 215-259. [26] H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, 41 (1995). American Mathematical Society, Providence, RI [27] H. L. Smith and P. Waltman, The theory of the chemostat: dynamics of microbial competition, Cambridge University Press, Cambridge, UK, 1995. doi: 10.1017/CBO9780511530043. [28] V. Sree Hari Rao and P. Raja Sekhara Rao, Dynamic Models and Control of Biological Systems, Springer-Verlag, Heidelberg, 2009. [29] P. A. Taylor and J. L. Williams, Theoretical studies on the coexistence of competing species under contunous flow conditions, Canadian Journal of Microbiology, 21 (1975), 90-98. [30] M. Turelli, Random environments and stochastic calculus, Theoret. Population Biology, 12 (1977), 140-178. [31] H. Veldcamp, Ecological studies with the chemostat, Advances in Microbial Ecology, 1 (1977), 59-95. [32] P. Waltman, Competition models in population biology, CBMS-NSF Regional Conference Series in Applied Mathematics, 45 Society for Industrial and Applied Mathematics, Philadelphia, 1983. doi: 10.1137/1.9781611970258. [33] P. Waltman, Coexistence in chemostat-like model, Rocky Mountain Journal of Mathematics, 20 (1990), 777-807. doi: 10.1216/rmjm/1181073042. [34] P. Waltman, S. P. Hubbel and S. B. Hsu, Theoretical and experimental investigations of microbial competition in continuous culture, Modeling and Differential Equations in Biology (Conf. , southern Illinois Univ. Carbonadle, Ⅲ. , 1978), (1980) pp. 107-152. Lecture Notes in Pure and Appl. Math. , 58, Dekker, New York.
Stochastic chemostat without wall growth. Values of parameters: $S_0=5$, $x_0=10$, $S^0=1$, $D=2$, $a=0.6$, $m=1$, $\alpha=0.2$ (left) and $\alpha= 0.5$ (right)
Stochastic chemostat without wall growth. Values of parameters: $S_0=5$, $x_0=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $\alpha=0.2$ (left) and $\alpha= 0.5$ (right)
Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=1.2$, $c=1$, $\alpha=0.2$
Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=1.2$, $c=1$, $\alpha=0.5$
Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=0.3$, $c=3$, $\alpha=0.2$
Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=0.3$, $c=3$, $\alpha=0.5$
Internal structure of the random attractor -Random chemostat model with wall growth
 ASYMPTOTIC BOUNDS ATTRACTOR INTERNAL STRUCTURE Case A: $b\nu c_\xi-m\geq 0$ (A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$ $\displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$ $\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$ (A-2) $\,\, \nu+\frac{\alpha^2}{2}c$ $\displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$ $\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$ (B-2) $\,\, \nu+\frac{\alpha^2}{2}  ASYMPTOTIC BOUNDS ATTRACTOR INTERNAL STRUCTURE Case A:$ b\nu c_\xi-m\geq 0$(A-1)$\,\,\nu+\frac{\alpha^2}{2}>c \displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$(A-2)$\,\, \nu+\frac{\alpha^2}{2}c \displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$(B-2)$\,\, \nu+\frac{\alpha^2}{2}
Internal structure of the random attractor -Stochastic chemostat model with wall growth
 ASYMPTOTIC BOUNDS ATTRACTOR INTERNAL STRUCTURE Case A: $b\nu c_\xi-m\geq 0$ (A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$ $\displaystyle{\lim_{t\to\infty}S(t)\geq S^0D\rho^*_\sigma(\omega)e^{-\alpha z^*(\omega)}-\varepsilon }$ $\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$ (A-2) $\,\, \nu+\frac{\alpha^2}{2}c$ $\displaystyle{\lim_{t\to\infty}S(t)\geq S^0D\rho^*_\sigma(\omega)e^{-\alpha z^*(\omega)}-\varepsilon }$ $\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$ (B-2) $\,\, \nu+\frac{\alpha^2}{2}  ASYMPTOTIC BOUNDS ATTRACTOR INTERNAL STRUCTURE Case A:$ b\nu c_\xi-m\geq 0$(A-1)$\,\,\nu+\frac{\alpha^2}{2}>c \displaystyle{\lim_{t\to\infty}S(t)\geq S^0D\rho^*_\sigma(\omega)e^{-\alpha z^*(\omega)}-\varepsilon }\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$(A-2)$\,\, \nu+\frac{\alpha^2}{2}c \displaystyle{\lim_{t\to\infty}S(t)\geq S^0D\rho^*_\sigma(\omega)e^{-\alpha z^*(\omega)}-\varepsilon }\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$(B-2)$\,\, \nu+\frac{\alpha^2}{2}
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