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Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage
1. | Department of Mathematics and The National Center for Theoretical Science, National Tsing-Hua University, Hsinchu 30013 |
2. | Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795 |
3. | Department of Natural Science in the Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan 333 |
References:
[1] |
R. A. Armstrong and R. McGehee, Competitive exclusion, American Naturalist, 115 (1980), 151-170.
doi: 10.1086/283553. |
[2] |
A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous culture, Mathematics in Microbiology, (eds. M. J. Bazin,), Academic Press, (1983), 77-103. |
[3] |
M. R. Droop, Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth and inhibition in Monochrysis lutheri, J. Mar. Biol. Assoc. UK, 48 (1968), 689-733. |
[4] |
M. R. Droop, Some thoughts on nutrient limitation in algae, Journal of Phycology, 9 (1973), 264-272. |
[5] |
J. P. Grover, Dynamics of competition among microalgae in variable environments: experimental tests of alternative models, Oikos, 62 (1991), 231-243. |
[6] |
J. P. Grover, Resource competition in a variable environment: Phytoplankton growing according to the variable-internal-stores model, American Naturalist, 138 (1991), 811-835. |
[7] |
J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, American Naturalist, 178 (2011), 124-148. |
[8] |
J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, J. Math. Biol., 64 (2012), 713-743.
doi: 10.1007/s00285-011-0426-4. |
[9] |
S.-B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
[10] |
S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.
doi: 10.1137/070700784. |
[11] |
S.-B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.
doi: 10.1137/0132030. |
[12] |
S.-B. Hsu, J.-F. Jiang and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations, 248 (2010), 2470-2496.
doi: 10.1016/j.jde.2009.12.014. |
[13] |
S.-B. Hsu, J.-F. Jiang and F.-B. Wang, Reaction-diffusion equations of two species competing for two complementary resources with internal storage, J. Differential Equations, 251 (2011), 918-940.
doi: 10.1016/j.jde.2011.05.003. |
[14] |
S.-B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.
doi: 10.1090/S0002-9947-96-01724-2. |
[15] |
S.-B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.
doi: 10.1137/0153051. |
[16] |
S.-B. Hsu and F.-B. Wang, On a mathematical model arising from competition of phytoplankton species for a single nutrient with internal storage: steady state analysis, Commun. Pure Appl. Anal., 10 (2011), 1479-1501.
doi: 10.3934/cpaa.2011.10.1479. |
[17] |
C. A. Klausmeier, E. Litchman, T. Daufresne and S. A. Levin, Phytoplankton stoichiometry, Ecological Research, 23 (2008), 479-485. |
[18] |
B. S. Leadbeater, The 'Droop Equation'-Michael Droop and the legacy of the 'Cell-Quota Model' of phytoplankton growth, Protist, 157 (2006), 345-358. |
[19] |
H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995. |
[20] |
H. L. Smith, The periodically forced Droop model for phytoplankton growth in a chemostat, J. Math. Biol., 35 (1997), 545-556.
doi: 10.1007/s002850050065. |
[21] |
H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.
doi: 10.1137/S0036139993245344. |
[22] |
M. C. White and X.-Q. Zhao, A periodic Droop model for two species competition in a chemostat, Bull. Math. Biol., 71 (2009), 145-161.
doi: 10.1007/s11538-008-9357-7. |
show all references
References:
[1] |
R. A. Armstrong and R. McGehee, Competitive exclusion, American Naturalist, 115 (1980), 151-170.
doi: 10.1086/283553. |
[2] |
A. Cunningham and R. M. Nisbet, Transients and oscillations in continuous culture, Mathematics in Microbiology, (eds. M. J. Bazin,), Academic Press, (1983), 77-103. |
[3] |
M. R. Droop, Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth and inhibition in Monochrysis lutheri, J. Mar. Biol. Assoc. UK, 48 (1968), 689-733. |
[4] |
M. R. Droop, Some thoughts on nutrient limitation in algae, Journal of Phycology, 9 (1973), 264-272. |
[5] |
J. P. Grover, Dynamics of competition among microalgae in variable environments: experimental tests of alternative models, Oikos, 62 (1991), 231-243. |
[6] |
J. P. Grover, Resource competition in a variable environment: Phytoplankton growing according to the variable-internal-stores model, American Naturalist, 138 (1991), 811-835. |
[7] |
J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, American Naturalist, 178 (2011), 124-148. |
[8] |
J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, J. Math. Biol., 64 (2012), 713-743.
doi: 10.1007/s00285-011-0426-4. |
[9] |
S.-B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
[10] |
S.-B. Hsu and T.-H. Hsu, Competitive exclusion of microbial species for a single nutrient with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.
doi: 10.1137/070700784. |
[11] |
S.-B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383.
doi: 10.1137/0132030. |
[12] |
S.-B. Hsu, J.-F. Jiang and F.-B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Differential Equations, 248 (2010), 2470-2496.
doi: 10.1016/j.jde.2009.12.014. |
[13] |
S.-B. Hsu, J.-F. Jiang and F.-B. Wang, Reaction-diffusion equations of two species competing for two complementary resources with internal storage, J. Differential Equations, 251 (2011), 918-940.
doi: 10.1016/j.jde.2011.05.003. |
[14] |
S.-B. Hsu, H. L. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094.
doi: 10.1090/S0002-9947-96-01724-2. |
[15] |
S.-B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.
doi: 10.1137/0153051. |
[16] |
S.-B. Hsu and F.-B. Wang, On a mathematical model arising from competition of phytoplankton species for a single nutrient with internal storage: steady state analysis, Commun. Pure Appl. Anal., 10 (2011), 1479-1501.
doi: 10.3934/cpaa.2011.10.1479. |
[17] |
C. A. Klausmeier, E. Litchman, T. Daufresne and S. A. Levin, Phytoplankton stoichiometry, Ecological Research, 23 (2008), 479-485. |
[18] |
B. S. Leadbeater, The 'Droop Equation'-Michael Droop and the legacy of the 'Cell-Quota Model' of phytoplankton growth, Protist, 157 (2006), 345-358. |
[19] |
H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995. |
[20] |
H. L. Smith, The periodically forced Droop model for phytoplankton growth in a chemostat, J. Math. Biol., 35 (1997), 545-556.
doi: 10.1007/s002850050065. |
[21] |
H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.
doi: 10.1137/S0036139993245344. |
[22] |
M. C. White and X.-Q. Zhao, A periodic Droop model for two species competition in a chemostat, Bull. Math. Biol., 71 (2009), 145-161.
doi: 10.1007/s11538-008-9357-7. |
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