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On vector solutions for coupled nonlinear Schrödinger equations with critical exponents
Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor
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 |
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. |
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