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December  2014, 19(10): 3169-3189. doi: 10.3934/dcdsb.2014.19.3169

Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage

Sze-Bi Hsu 1, , Junping Shi 2, and  Feng-Bin Wang 3,

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

Received  June 2013 Revised  August 2013 Published  October 2014

The dynamics of a reaction-diffusion system for two species of microorganism in an unstirred chemostat with internal storage is studied. It is shown that the diffusion coefficient is a key parameter of determining the asymptotic dynamics, and there exists a threshold diffusion coefficient above which both species become extinct. On the other hand, for diffusion coefficient below the threshold, either one species or both species persist, and in the asymptotic limit, a steady state showing competition exclusion or coexistence is reached.
Keywords: Unstirred chemostat, internal storage, reaction-diffusion system, steady state..
Mathematics Subject Classification: 35K51, 35K57, 35B35, 92D2.
Citation: Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169
References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion, American Naturalist, 115 (1980), 151-170. doi: 10.1086/283553.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

M. R. Droop, Some thoughts on nutrient limitation in algae, Journal of Phycology, 9 (1973), 264-272. Google Scholar

[5]

J. P. Grover, Dynamics of competition among microalgae in variable environments: experimental tests of alternative models, Oikos, 62 (1991), 231-243. Google Scholar

[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. Google Scholar

[7]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, American Naturalist, 178 (2011), 124-148. Google Scholar

[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.  Google Scholar

[9]

S.-B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

C. A. Klausmeier, E. Litchman, T. Daufresne and S. A. Levin, Phytoplankton stoichiometry, Ecological Research, 23 (2008), 479-485. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. A. Armstrong and R. McGehee, Competitive exclusion, American Naturalist, 115 (1980), 151-170. doi: 10.1086/283553.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

M. R. Droop, Some thoughts on nutrient limitation in algae, Journal of Phycology, 9 (1973), 264-272. Google Scholar

[5]

J. P. Grover, Dynamics of competition among microalgae in variable environments: experimental tests of alternative models, Oikos, 62 (1991), 231-243. Google Scholar

[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. Google Scholar

[7]

J. P. Grover, Resource storage and competition with spatial and temporal variation in resource availability, American Naturalist, 178 (2011), 124-148. Google Scholar

[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.  Google Scholar

[9]

S.-B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763. doi: 10.1137/0134064.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

C. A. Klausmeier, E. Litchman, T. Daufresne and S. A. Levin, Phytoplankton stoichiometry, Ecological Research, 23 (2008), 479-485. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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