Dynamics of some stochastic chemostat models with multiplicative noise

T. Caraballo; M. J. Garrido-Atienza; J. López-de-la-Cruz

doi:10.3934/cpaa.2017092

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

Dynamics of some stochastic chemostat models with multiplicative noise

T. Caraballo ^, , M. J. Garrido-Atienza and J. López-de-la-Cruz

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.

Keywords: Chemostat, stochastic differential equations, multiplicative noise, random dynamical systems, random attractors.

Mathematics Subject Classification: Primary: 34C11, 34F05; Secondary: 60H10.

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. Google Scholar
[2]	H. R. Bungay and M. L. Bungay, Microbial interactions in continuous culture, Advances in Applied Microbiology, 10 (1968), 269-290. Google Scholar
[3]	T. Caraballo, Recent results on stabilization of PDEs by noise, Bol. Soc. Esp. Mat. Apl., 37 (2006), 47-70. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous cultures, Mathematics in Microbiology, Academic Press, London, (1983), 77-103. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	A. G. Fredrickson and G. Stephanopoulos, Microbial competition, Science, 213 (1981), 972-979. doi: 10.1126/science.7268409. Google Scholar
[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. Google Scholar
[18]	R. Freter, An understanding of colonization of the large intestine requires mathematical analysis, Microecology and Therapy, 16 (1986), 147-155. Google Scholar
[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. 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]	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. Google Scholar
[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. Google Scholar
[23]	H. W. Jannash and R. T. Mateles, Experimental bacterial ecology studies in continuous culture, Advances in Microbial Physiology, 11 (1974), 165-212. Google Scholar
[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. Google Scholar
[25]	J. W. M. La Riviere, Microbial ecology of liquid waste, Advances in Microbial Ecology, 1 (1977), 215-259. Google Scholar
[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 Google Scholar
[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. Google Scholar
[28]	V. Sree Hari Rao and P. Raja Sekhara Rao, Dynamic Models and Control of Biological Systems, Springer-Verlag, Heidelberg, 2009. Google Scholar
[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. Google Scholar
[30]	M. Turelli, Random environments and stochastic calculus, Theoret. Population Biology, 12 (1977), 140-178. Google Scholar
[31]	H. Veldcamp, Ecological studies with the chemostat, Advances in Microbial Ecology, 1 (1977), 59-95. Google Scholar
[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. Google Scholar
[33]	P. Waltman, Coexistence in chemostat-like model, Rocky Mountain Journal of Mathematics, 20 (1990), 777-807. doi: 10.1216/rmjm/1181073042. Google Scholar
[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. Google Scholar

show all references

References:

[1]	L. Arnold, Random Dynamical Systems, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar
[2]	H. R. Bungay and M. L. Bungay, Microbial interactions in continuous culture, Advances in Applied Microbiology, 10 (1968), 269-290. Google Scholar
[3]	T. Caraballo, Recent results on stabilization of PDEs by noise, Bol. Soc. Esp. Mat. Apl., 37 (2006), 47-70. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous cultures, Mathematics in Microbiology, Academic Press, London, (1983), 77-103. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	A. G. Fredrickson and G. Stephanopoulos, Microbial competition, Science, 213 (1981), 972-979. doi: 10.1126/science.7268409. Google Scholar
[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. Google Scholar
[18]	R. Freter, An understanding of colonization of the large intestine requires mathematical analysis, Microecology and Therapy, 16 (1986), 147-155. Google Scholar
[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. 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]	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. Google Scholar
[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. Google Scholar
[23]	H. W. Jannash and R. T. Mateles, Experimental bacterial ecology studies in continuous culture, Advances in Microbial Physiology, 11 (1974), 165-212. Google Scholar
[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. Google Scholar
[25]	J. W. M. La Riviere, Microbial ecology of liquid waste, Advances in Microbial Ecology, 1 (1977), 215-259. Google Scholar
[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 Google Scholar
[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. Google Scholar
[28]	V. Sree Hari Rao and P. Raja Sekhara Rao, Dynamic Models and Control of Biological Systems, Springer-Verlag, Heidelberg, 2009. Google Scholar
[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. Google Scholar
[30]	M. Turelli, Random environments and stochastic calculus, Theoret. Population Biology, 12 (1977), 140-178. Google Scholar
[31]	H. Veldcamp, Ecological studies with the chemostat, Advances in Microbial Ecology, 1 (1977), 59-95. Google Scholar
[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. Google Scholar
[33]	P. Waltman, Coexistence in chemostat-like model, Rocky Mountain Journal of Mathematics, 20 (1990), 777-807. doi: 10.1216/rmjm/1181073042. Google Scholar
[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. Google Scholar

Figure 2. 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)

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Figure 1. 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)

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Figure 5. Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=1.2$, $c=1$, $\alpha=0.2$

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Figure 6. Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=1.2$, $c=1$, $\alpha=0.5$

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Figure 3. Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=0.3$, $c=3$, $\alpha=0.2$

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Figure 4. Stochastic chemostat with wall growth. Values of parameters: $S_0=5$, $x_{01}=10$, $x_{02}=10$, $S^0=1$, $D=2$, $a=0.6$, $m=5$, $b=0.5$, $r_1=0.2$, $r_2=0.8$, $\nu=0.3$, $c=3$, $\alpha=0.5$

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Table 1. 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 }$
	(A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$ \displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\kappa(t)$ does not provide any extra information
Case B: $ b\nu c_\xi-m< 0$	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$ \displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$
	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\sigma(t)$ does not provide any extra information
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\kappa(t)$ does not provide any extra information

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		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 }$
	(A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$ \displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\kappa(t)$ does not provide any extra information
Case B: $ b\nu c_\xi-m< 0$	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$ \displaystyle{\lim_{t\to\infty}\sigma(t)\geq S^0D\rho^*_\sigma(\omega)-\varepsilon }$
	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\kappa(t)\leq \varepsilon }$
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\sigma(t)$ does not provide any extra information
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$\kappa(t)$ does not provide any extra information

Table 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 }$
	(A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$
	(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 }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$x_1+x_2$ does not provide any extra information
Case B: $ b\nu c_\xi-m< 0$	(B-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 }$
	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$S$ does not provide any extra information
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$x_1+x_2$ does not provide any extra information

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		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 }$
	(A-1) $\,\,\nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$
	(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 }$
	(A-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$x_1+x_2$ does not provide any extra information
Case B: $ b\nu c_\xi-m< 0$	(B-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 }$
	(B-1) $\,\, \nu+\frac{\alpha^2}{2}>c$	$\displaystyle{\lim_{t\to\infty}\left[x_1(t)+x_2(t)\right]\leq \varepsilon }$
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$S$ does not provide any extra information
	(B-2) $\,\, \nu+\frac{\alpha^2}{2}<c$	$x_1+x_2$ does not provide any extra information

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2018 Impact Factor: 0.925

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2018 Impact Factor: 0.925

American Institute of Mathematical Sciences

Dynamics of some stochastic chemostat models with multiplicative noise

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Dynamics of some stochastic chemostat models with multiplicative noise

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