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Competition between two similar species in the unstirred chemostat
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Growth of single phytoplankton species with internal storage in a water column
1. | Department of Mathematics, Henan Normal University, Xinxiang, Henan 453007 |
2. | Department of Mathematics, National Tsing Hua University, National Center of Theoretical Science, Hsinchu 300 |
3. | Department of Natural Science in the Center for General Education, Chang Gung University, Kwei-Shan, Taoyuan 333 |
References:
[1] |
J. V. Baxley and S. B. Robinson, Coexistence in the unstirred chemostat, Appl. Math. Computation, 89 (1998), 41-65.
doi: 10.1016/S0096-3003(97)81647-5. |
[2] |
A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), 189-203.
doi: 10.1016/S0022-5193(80)80003-8. |
[3] |
A. Cunningham and R. M. Nisbet, Transient and oscillation in continuous culture, Mathematics in microbiology, 77-103, Academic Press, London, 1983. |
[4] |
M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.
doi: 10.1111/j.1529-8817.1973.tb04092.x. |
[5] |
J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol., 158 (1992), 409-428. |
[6] |
J. P. Grover, Resource Competition, Chapman and Hall, London, 1997.
doi: 10.1007/978-1-4615-6397-6. |
[7] |
J. P. Grover, Is storage an adaptation to spatial variation in resource availability?, The American Naturalist, 173 (2009), E44-E61.
doi: 10.1086/595751. |
[8] |
J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, The American Naturalist, 178 (2011), E124-E148.
doi: 10.1086/662163. |
[9] |
J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: a theoretical exploration, Journal of Plankton Research, 33 (2011), 211-227.
doi: 10.1093/plankt/fbq070. |
[10] |
J. P. Grover, S. B. Hsu and F. B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone, Mathematical Biosciences, 222 (2009), 42-52.
doi: 10.1016/j.mbs.2009.08.006. |
[11] |
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, Journal of Mathematical Biology, 64 (2012), 713-743.
doi: 10.1007/s00285-011-0426-4. |
[12] |
J. P. Grover and F.-B. Wang, Dynamics of a model of microbial competition with internal nutrient storage in a flowing habitat, Applied Mathematics and Computation, 225 (2013), 747-764.
doi: 10.1016/j.amc.2013.09.054. |
[13] |
S. B. Hsu, S. Hubbell and P. Waltman, Mathematical theoy for single nutrient competition in continuous cultures of microorganisms, SIAM J. Appl. Math., 32 (1977), 366-383.
doi: 10.1137/0132030. |
[14] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Berlin, New York, Springer, 1981. |
[15] |
J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988. |
[16] |
P. Hess, Periodic-parabolic Boundary Value Problem and Positivity, Pitman Res. Notes Math., 247, Longman Scientific and Technical, 1991. |
[17] |
S. B. Hsu and T. H. Hsu, Competitive exclusion of microbial species for a single-limited resource with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.
doi: 10.1137/070700784. |
[18] |
S. B. Hsu, J. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496.
doi: 10.1016/j.jde.2009.12.014. |
[19] |
S. B. Hsu, L. Mei and F. B. Wang, On a nonlocal reaction-diffusion-advection system modelling the growth of phytoplankton with cell quota structure, J. Diff. Eqns., 259 (2015), 5353-5378.
doi: 10.1016/j.jde.2015.06.030. |
[20] |
S. B. Hsu, H. L. Smith and P. Waltman, Dynamics of competition in the unstirred chemostat, Canad. Appl. Math. Quart., 2 (1994), 461-483. |
[21] |
S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unsirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.
doi: 10.1137/0153051. |
[22] |
P. Maga 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. |
[23] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[24] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995. |
[25] |
H. L. Smith and P. E. Waltman, Competition for a single limiting resouce in continuous culture: The variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.
doi: 10.1137/S0036139993245344. |
[26] |
H. L. Smith and P. E. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995.
doi: 10.1017/CBO9780511530043. |
[27] |
H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[28] |
K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. A., 463 (2007), 1029-1043.
doi: 10.1098/rspa.2006.1806. |
[29] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
show all references
References:
[1] |
J. V. Baxley and S. B. Robinson, Coexistence in the unstirred chemostat, Appl. Math. Computation, 89 (1998), 41-65.
doi: 10.1016/S0096-3003(97)81647-5. |
[2] |
A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol., 84 (1980), 189-203.
doi: 10.1016/S0022-5193(80)80003-8. |
[3] |
A. Cunningham and R. M. Nisbet, Transient and oscillation in continuous culture, Mathematics in microbiology, 77-103, Academic Press, London, 1983. |
[4] |
M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.
doi: 10.1111/j.1529-8817.1973.tb04092.x. |
[5] |
J. P. Grover, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol., 158 (1992), 409-428. |
[6] |
J. P. Grover, Resource Competition, Chapman and Hall, London, 1997.
doi: 10.1007/978-1-4615-6397-6. |
[7] |
J. P. Grover, Is storage an adaptation to spatial variation in resource availability?, The American Naturalist, 173 (2009), E44-E61.
doi: 10.1086/595751. |
[8] |
J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, The American Naturalist, 178 (2011), E124-E148.
doi: 10.1086/662163. |
[9] |
J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: a theoretical exploration, Journal of Plankton Research, 33 (2011), 211-227.
doi: 10.1093/plankt/fbq070. |
[10] |
J. P. Grover, S. B. Hsu and F. B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone, Mathematical Biosciences, 222 (2009), 42-52.
doi: 10.1016/j.mbs.2009.08.006. |
[11] |
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, Journal of Mathematical Biology, 64 (2012), 713-743.
doi: 10.1007/s00285-011-0426-4. |
[12] |
J. P. Grover and F.-B. Wang, Dynamics of a model of microbial competition with internal nutrient storage in a flowing habitat, Applied Mathematics and Computation, 225 (2013), 747-764.
doi: 10.1016/j.amc.2013.09.054. |
[13] |
S. B. Hsu, S. Hubbell and P. Waltman, Mathematical theoy for single nutrient competition in continuous cultures of microorganisms, SIAM J. Appl. Math., 32 (1977), 366-383.
doi: 10.1137/0132030. |
[14] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Berlin, New York, Springer, 1981. |
[15] |
J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988. |
[16] |
P. Hess, Periodic-parabolic Boundary Value Problem and Positivity, Pitman Res. Notes Math., 247, Longman Scientific and Technical, 1991. |
[17] |
S. B. Hsu and T. H. Hsu, Competitive exclusion of microbial species for a single-limited resource with internal storage, SIAM J. Appl. Math., 68 (2008), 1600-1617.
doi: 10.1137/070700784. |
[18] |
S. B. Hsu, J. Jiang and F. B. Wang, On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat, J. Diff. Eqns., 248 (2010), 2470-2496.
doi: 10.1016/j.jde.2009.12.014. |
[19] |
S. B. Hsu, L. Mei and F. B. Wang, On a nonlocal reaction-diffusion-advection system modelling the growth of phytoplankton with cell quota structure, J. Diff. Eqns., 259 (2015), 5353-5378.
doi: 10.1016/j.jde.2015.06.030. |
[20] |
S. B. Hsu, H. L. Smith and P. Waltman, Dynamics of competition in the unstirred chemostat, Canad. Appl. Math. Quart., 2 (1994), 461-483. |
[21] |
S. B. Hsu and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unsirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.
doi: 10.1137/0153051. |
[22] |
P. Maga 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. |
[23] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[24] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995. |
[25] |
H. L. Smith and P. E. Waltman, Competition for a single limiting resouce in continuous culture: The variable-yield model, SIAM J. Appl. Math., 54 (1994), 1113-1131.
doi: 10.1137/S0036139993245344. |
[26] |
H. L. Smith and P. E. Waltman, The Theory of the Chemostat, Cambridge Univ. Press, 1995.
doi: 10.1017/CBO9780511530043. |
[27] |
H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[28] |
K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage, Proc. R. Soc. A., 463 (2007), 1029-1043.
doi: 10.1098/rspa.2006.1806. |
[29] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
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