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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

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