American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1966. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1966. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. L. Smith and P. Waltman, "The Theory of the Chemostat," Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2021, 26 (8) : 4459-4477. doi: 10.3934/dcdsb.2020295
[8]

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
[9]

Germain Gendron. Uniqueness results in the inverse spectral Steklov problem. Inverse Problems & Imaging, 2020, 14 (4) : 631-664. doi: 10.3934/ipi.2020029
[10]

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, 2019, 24 (8) : 3591-3614. doi: 10.3934/dcdsb.2018280
[11]

Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098
[12]

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
[13]

Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif, Tewfik Sari. Mathematical analysis of a three-tiered food-web in the chemostat. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5601-5625. doi: 10.3934/dcdsb.2020369
[14]

Frédéric Grognard, Frédéric Mazenc, Alain Rapaport. Polytopic Lyapunov functions for persistence analysis of competing species. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 73-93. doi: 10.3934/dcdsb.2007.8.73
[15]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751
[16]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1
[17]

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
[18]

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
[19]

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
[20]

Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429-460. doi: 10.3934/jmd.2013.7.429

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences