Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor

Previous Article
An anisotropic regularity criterion for the 3D Navier-Stokes equations
CPAA Home
This Issue
Next Article
On vector solutions for coupled nonlinear Schrödinger equations with critical exponents

May 2013, 12(3): 1279-1297. doi: 10.3934/cpaa.2013.12.1279

Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor

Hua Nie ^1, , Wenhao Xie ^2, and Jianhua Wu ^3,

1.	College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710062
2.	School of Science, Xi'an Shiyou University, Xi'an, Shaanxi 710065, China
3.	College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119

Received January 2012 Revised August 2012 Published September 2012

Abstract
Related Papers

A competition model of two organisms is considered in the un-stirred chemostat-type system in the presence of an external inhibitor. Asymptotic stability properties of the trivial and semi-trivial steady state solutions are established by spectral analysis. The stability and uniqueness of positive steady state solutions are also given by Lyapunov Schmidt procedure and perturbation technique.

Keywords: external inhibitor, spectral analysis, Chemostat, Lyapunov-Schmidt technique., uniqueness.

Mathematics Subject Classification: Primary: 35K55, 35K57; Secondary: 92A1.

Citation: Hua Nie, Wenhao Xie, Jianhua Wu. Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1279-1297. doi: 10.3934/cpaa.2013.12.1279

References:

[1]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.
[2]	Y. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differtial Equations, 144 (1998), 390-440. doi: 10.1006/jdeq.1997.3394.
[3]	D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346. doi: 10.5269/bspm.v30i2.14502.
[4]	S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM J. Appl. Math., 52 (1992), 528-540. doi: 10.1137/0152029.
[5]	S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an un-stirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044. doi: 10.1137/0153051.
[6]	S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan J. Indust. Appl. Math., 15 (1998), 471-490. doi: 10.1007/BF03167323.
[7]	S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.
[8]	T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1966. doi: 10.1007/978-3-642-66282-9.
[9]	R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: a chemostat model based on bacteria and antibiotics, J. Theoret. Biol., 122 (1988), 83-93. doi: 10.1016/S0022-5193(86)80226-0.
[10]	H. Nie and J. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 989-1009. doi: 10.1142/S0218127406015246.
[11]	H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.
[12]	J. W. H. So and P. Waltman, A nonlinear boundary value problem arising from competition in the chemostat, Appl. Math. Comput., 32 (1989), 169-183. doi: 10.1016/0096-3003(89)90092-1.
[13]	J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.
[14]	J. Wu, H. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat, SIAM J. Appl. Math., 65 (2004), 209-229. doi: 10.1137/S0036139903423285.
[15]	J. Wu, H. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885. doi: 10.1137/050627514.
[16]	J. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat, J. Differential Equations, 172 (2001), 300-332. doi: 10.1006/jdeq.2000.3870.

show all references

References:

[1]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325.
[2]	Y. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differtial Equations, 144 (1998), 390-440. doi: 10.1006/jdeq.1997.3394.
[3]	D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346. doi: 10.5269/bspm.v30i2.14502.
[4]	S. B. Hsu and P. Waltman, Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM J. Appl. Math., 52 (1992), 528-540. doi: 10.1137/0152029.
[5]	S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an un-stirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044. doi: 10.1137/0153051.
[6]	S. B. Hsu and P. Waltman, Competition in the chemostat when one competitor produces a toxin, Japan J. Indust. Appl. Math., 15 (1998), 471-490. doi: 10.1007/BF03167323.
[7]	S. B. Hsu and P. Waltman, A survey of mathematical models of competition with an inhibitor, Math. Biosci., 187 (2004), 53-91. doi: 10.1016/j.mbs.2003.07.004.
[8]	T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1966. doi: 10.1007/978-3-642-66282-9.
[9]	R. E. Lenski and S. Hattingh, Coexistence of two competitors on one resource and one inhibitor: a chemostat model based on bacteria and antibiotics, J. Theoret. Biol., 122 (1988), 83-93. doi: 10.1016/S0022-5193(86)80226-0.
[10]	H. Nie and J. Wu, A system of reaction-diffusion equations in the unstirred chemostat with an inhibitor, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 989-1009. doi: 10.1142/S0218127406015246.
[11]	H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.
[12]	J. W. H. So and P. Waltman, A nonlinear boundary value problem arising from competition in the chemostat, Appl. Math. Comput., 32 (1989), 169-183. doi: 10.1016/0096-3003(89)90092-1.
[13]	J. Wu, Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835. doi: 10.1016/S0362-546X(98)00250-8.
[14]	J. Wu, H. Nie and G. S. K. Wolkowicz, A mathematical model of competition for two essential resources in the unstirred chemostat, SIAM J. Appl. Math., 65 (2004), 209-229. doi: 10.1137/S0036139903423285.
[15]	J. Wu, H. Nie and G. S. K. Wolkowicz, The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat, SIAM J. Math. Anal., 38 (2007), 1860-1885. doi: 10.1137/050627514.
[16]	J. Wu and G. S. K. Wolkowicz, A system of resource-based growth models with two resources in the un-stirred chemostat, J. Differential Equations, 172 (2001), 300-332. doi: 10.1006/jdeq.2000.3870.

[1]	Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201
[2]	Jianquan Li, Zuren Feng, Juan Zhang, Jie Lou. A competition model of the chemostat with an external inhibitor. Mathematical Biosciences & Engineering, 2006, 3 (1) : 111-123. doi: 10.3934/mbe.2006.3.111
[3]	Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
[4]	Bachir Bar, Tewfik Sari. The operating diagram for a model of competition in a chemostat with an external lethal inhibitor. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2093-2120. doi: 10.3934/dcdsb.2019203
[5]	Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156
[6]	Edoardo Beretta, Fortunata Solimano, Yanbin Tang. Analysis of a chemostat model for bacteria and virulent bacteriophage. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 495-520. doi: 10.3934/dcdsb.2002.2.495
[7]	Xiaoli Wang, Guohong Zhang. Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4459-4477. doi: 10.3934/dcdsb.2020295
[8]	Victor Ogesa Juma, Leif Dehmelt, Stéphanie Portet, Anotida Madzvamuse. A mathematical analysis of an activator-inhibitor Rho GTPase model. Journal of Computational Dynamics, 2022, 9 (2) : 133-158. doi: 10.3934/jcd.2021024
[9]	Marion Weedermann. Analysis of a model for the effects of an external toxin on anaerobic digestion. Mathematical Biosciences & Engineering, 2012, 9 (2) : 445-459. doi: 10.3934/mbe.2012.9.445
[10]	Germain Gendron. Uniqueness results in the inverse spectral Steklov problem. Inverse Problems and Imaging, 2020, 14 (4) : 631-664. doi: 10.3934/ipi.2020029
[11]	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 and Continuous Dynamical Systems - B, 2019, 24 (8) : 3591-3614. doi: 10.3934/dcdsb.2018280
[12]	Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098
[13]	Tewfik Sari, Miled El Hajji, Jérôme Harmand. The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat. Mathematical Biosciences & Engineering, 2012, 9 (3) : 627-645. doi: 10.3934/mbe.2012.9.627
[14]	Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif, Tewfik Sari. Mathematical analysis of a three-tiered food-web in the chemostat. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5601-5625. doi: 10.3934/dcdsb.2020369
[15]	Frédéric Grognard, Frédéric Mazenc, Alain Rapaport. Polytopic Lyapunov functions for persistence analysis of competing species. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 73-93. doi: 10.3934/dcdsb.2007.8.73
[16]	Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks and Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751
[17]	Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure and Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1
[18]	Vu Hoang Linh, Volker Mehrmann. Spectral analysis for linear differential-algebraic equations. Conference Publications, 2011, 2011 (Special) : 991-1000. doi: 10.3934/proc.2011.2011.991
[19]	Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
[20]	Rafael Tiedra De Aldecoa. Spectral analysis of time changes of horocycle flows. Journal of Modern Dynamics, 2012, 6 (2) : 275-285. doi: 10.3934/jmd.2012.6.275

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Hua Nie Wenhao Xie Jianhua Wu
on Google Scholar
Hua Nie Wenhao Xie Jianhua Wu

[Back to Top]