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Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays
Coexistence solutions of a competition model with two species in a water column
1. | College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China |
2. | Department of Mathematics, National Tsing Hua University, National Center of Theoretical Science, Hsinchu 300 |
References:
[1] |
M. Ballyk, L. Dung, D. A. Jones and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596.
doi: 10.1137/S0036139997325345. |
[2] |
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953. |
[3] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.
doi: 10.1016/0022-247X(83)90098-7. |
[5] |
E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.
doi: 10.1090/S0002-9947-1984-0743741-4. |
[6] |
Y. Du and L. F. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.
doi: 10.1088/0951-7715/24/1/016. |
[7] |
J. P. Grover, Resource Competition, Chapman and Hall, London, 1997.
doi: 10.1007/978-1-4615-6397-6. |
[8] |
S. B. Hsu, Steady states of a system of partial differential equations modeling microbial ecology, SIAM J. Math. Anal., 14 (1983), 1130-1138.
doi: 10.1137/0514087. |
[9] |
S. B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.
doi: 10.1137/100782358. |
[10] |
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. |
[11] |
J. López-Gómez and R. Parda, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case, Differential Integral Equations, 6 (1993), 1025-1031. |
[12] |
P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[13] |
J. P. Mellard, K. Yoshiyama, E. Litchman and C. A. Klausmeier, The vertical distribution of phytoplankton in stratified water columns, J. Theoret. Biol., 269 (2011), 16-30.
doi: 10.1016/j.jtbi.2010.09.041. |
[14] |
H. Nie and J. Wu, Multiplicity results for the unstirred chemostat model with general response functions, Sci. China Math., 56 (2013), 2035-2050.
doi: 10.1007/s11425-012-4550-4. |
[15] |
H. Nie and J. Wu, Positive solutions of a competition model for two resources in the unstirred chemostat, J. Math. Anal. Appl., 355 (2009), 231-242.
doi: 10.1016/j.jmaa.2009.01.045. |
[16] |
H. Nie and J. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model, Appl. Anal., 89 (2010), 1141-1159.
doi: 10.1080/00036811003717954. |
[17] |
J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[18] |
H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.
doi: 10.1016/S0362-546X(01)00678-2. |
[19] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[20] |
D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, 1982. |
[21] |
M. X. Wang, Nonlinear Elliptic Equations, (in Chinese) Science Press, Beijing, 2010. |
[22] |
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. |
[23] |
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. |
[24] |
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. |
[25] |
K. Yoshiyama and H. Nakajima, Catastrophic transition in vertical distributions of phytoplankton: alternative equilibria in a water column, J. Theoret. Biol., 216 (2002), 397-408.
doi: 10.1006/jtbi.2002.3007. |
[26] |
K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, Am. Nat., 174 (2009), 190-203.
doi: 10.1086/600113. |
show all references
References:
[1] |
M. Ballyk, L. Dung, D. A. Jones and H. L. Smith, Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math., 59 (1999), 573-596.
doi: 10.1137/S0036139997325345. |
[2] |
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1953. |
[3] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.
doi: 10.1016/0022-247X(83)90098-7. |
[5] |
E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.
doi: 10.1090/S0002-9947-1984-0743741-4. |
[6] |
Y. Du and L. F. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 24 (2011), 319-349.
doi: 10.1088/0951-7715/24/1/016. |
[7] |
J. P. Grover, Resource Competition, Chapman and Hall, London, 1997.
doi: 10.1007/978-1-4615-6397-6. |
[8] |
S. B. Hsu, Steady states of a system of partial differential equations modeling microbial ecology, SIAM J. Math. Anal., 14 (1983), 1130-1138.
doi: 10.1137/0514087. |
[9] |
S. B. Hsu and Y. Lou, Single phytoplankton species growth with light and advection in a water column, SIAM J. Appl. Math., 70 (2010), 2942-2974.
doi: 10.1137/100782358. |
[10] |
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. |
[11] |
J. López-Gómez and R. Parda, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case, Differential Integral Equations, 6 (1993), 1025-1031. |
[12] |
P. Magal and X. Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[13] |
J. P. Mellard, K. Yoshiyama, E. Litchman and C. A. Klausmeier, The vertical distribution of phytoplankton in stratified water columns, J. Theoret. Biol., 269 (2011), 16-30.
doi: 10.1016/j.jtbi.2010.09.041. |
[14] |
H. Nie and J. Wu, Multiplicity results for the unstirred chemostat model with general response functions, Sci. China Math., 56 (2013), 2035-2050.
doi: 10.1007/s11425-012-4550-4. |
[15] |
H. Nie and J. Wu, Positive solutions of a competition model for two resources in the unstirred chemostat, J. Math. Anal. Appl., 355 (2009), 231-242.
doi: 10.1016/j.jmaa.2009.01.045. |
[16] |
H. Nie and J. Wu, Uniqueness and stability for coexistence solutions of the unstirred chemostat model, Appl. Anal., 89 (2010), 1141-1159.
doi: 10.1080/00036811003717954. |
[17] |
J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.
doi: 10.1016/j.jde.2008.09.009. |
[18] |
H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.
doi: 10.1016/S0362-546X(01)00678-2. |
[19] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, $2^{nd}$ edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[20] |
D. Tilman, Resource Competition and Community Structure, Princeton University Press, Princeton, 1982. |
[21] |
M. X. Wang, Nonlinear Elliptic Equations, (in Chinese) Science Press, Beijing, 2010. |
[22] |
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. |
[23] |
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. |
[24] |
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. |
[25] |
K. Yoshiyama and H. Nakajima, Catastrophic transition in vertical distributions of phytoplankton: alternative equilibria in a water column, J. Theoret. Biol., 216 (2002), 397-408.
doi: 10.1006/jtbi.2002.3007. |
[26] |
K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a stratified water column, Am. Nat., 174 (2009), 190-203.
doi: 10.1086/600113. |
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