American Institute of Mathematical Sciences

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December  2014, 7(6): 1259-1285. doi: 10.3934/dcdss.2014.7.1259

Dynamics of two phytoplankton species competing for light and nutrient with internal storage

Sze-Bi Hsu 1, and  Chiu-Ju Lin 2,

1. 

Department of Mathematics, National Tsing Hua University, National Center of Theoretical Science, Hsinchu 300, Taiwan
2. 

Department of Mathematics, National Tsing Hua University, Hsinchu 300

Received  March 2013 Revised  November 2013 Published  June 2014

We analyze a competition model of two phytoplankton species for a single nutrient with internal storage and light in a well mixed aquatic environment. We apply the theory of monotone dynamical system to determine the outcomes of competition: extinction of two species, competitive exclusion, stable coexistence and bistability of two species. We also present the graphical presentation to classify the competition outcomes and to compare outcome of models with and without internal storage.
Keywords: Two species competition, competitive exclusion, well-mixed water column, Droop model, Tilman's graph presentation., bistability, stable coexistence.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Sze-Bi Hsu, Chiu-Ju Lin. Dynamics of two phytoplankton species competing for light and nutrient with internal storage. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1259-1285. doi: 10.3934/dcdss.2014.7.1259
References:
[1]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272. doi: 10.1111/j.1529-8817.1973.tb04092.x.  Google Scholar

[2]

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. doi: 10.1016/S0022-5193(05)80707-6.  Google Scholar

[3]

S.-B. Hsu, K.-S. Cheng and S. P. Hubbell, Exploitative competition of microorganism for two complementary nutrients in continuous cultures, SIAM J. Appl. Math., 41 (1981), 422-444. doi: 10.1137/0141036.  Google Scholar

[4]

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

[5]

S.-B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous culture of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383. doi: 10.1137/0132030.  Google Scholar

[6]

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

[7]

J. Huisman and F. J. Weissing, Light limited growth and competition for light in well-mixed aquatic environments: An elementary, Ecology, 75 (1994), 507-520. doi: 10.2307/1939554.  Google Scholar

[8]

J. Huisman and F. J. Weissing, Competition for nutrients and light in a mixed water column: A theoretical analysis, Am. Nat., 146 (1995), 536-564. doi: 10.1086/285814.  Google Scholar

[9]

J. Jiang, X. Liang and X. Zhao, Saddle-point behavior for monotone semifolws and reaction-diffusion models, J. Diff. Equations, 203 (2004), 313-330. doi: 10.1016/j.jde.2004.05.002.  Google Scholar

[10]

B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage, J. Math. Biol., 55 (2007), 481-515. doi: 10.1007/s00285-007-0092-8.  Google Scholar

[11]

J. Passarge, S. Hol, M. Escher and J. Huisman, Competition for nutrients and light: Stable coexistence, alternative stable states, or competitive exclusion?, Ecological Monographs, 76 (2006), 57-72. doi: 10.1890/04-1824.  Google Scholar

[12]

H. L. Smith and H. R. Thieme, Stable coexistence and bi-stablility for competitive systems on ordered Banach spaces, J. Diff. Equations, 176 (2001), 195-222. doi: 10.1006/jdeq.2001.3981.  Google Scholar

[13]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[14]

D. Tilman, Resource Competition and Community Structure, Princeton University Press, New Jersey, 1982. Google Scholar

[15]

K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a statified water column, Am. Nat., 174 (2009), 190-203. doi: 10.1086/600113.  Google Scholar

[16]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272. doi: 10.1111/j.1529-8817.1973.tb04092.x.  Google Scholar

[2]

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. doi: 10.1016/S0022-5193(05)80707-6.  Google Scholar

[3]

S.-B. Hsu, K.-S. Cheng and S. P. Hubbell, Exploitative competition of microorganism for two complementary nutrients in continuous cultures, SIAM J. Appl. Math., 41 (1981), 422-444. doi: 10.1137/0141036.  Google Scholar

[4]

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

[5]

S.-B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous culture of micro-organisms, SIAM J. Appl. Math., 32 (1977), 366-383. doi: 10.1137/0132030.  Google Scholar

[6]

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

[7]

J. Huisman and F. J. Weissing, Light limited growth and competition for light in well-mixed aquatic environments: An elementary, Ecology, 75 (1994), 507-520. doi: 10.2307/1939554.  Google Scholar

[8]

J. Huisman and F. J. Weissing, Competition for nutrients and light in a mixed water column: A theoretical analysis, Am. Nat., 146 (1995), 536-564. doi: 10.1086/285814.  Google Scholar

[9]

J. Jiang, X. Liang and X. Zhao, Saddle-point behavior for monotone semifolws and reaction-diffusion models, J. Diff. Equations, 203 (2004), 313-330. doi: 10.1016/j.jde.2004.05.002.  Google Scholar

[10]

B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage, J. Math. Biol., 55 (2007), 481-515. doi: 10.1007/s00285-007-0092-8.  Google Scholar

[11]

J. Passarge, S. Hol, M. Escher and J. Huisman, Competition for nutrients and light: Stable coexistence, alternative stable states, or competitive exclusion?, Ecological Monographs, 76 (2006), 57-72. doi: 10.1890/04-1824.  Google Scholar

[12]

H. L. Smith and H. R. Thieme, Stable coexistence and bi-stablility for competitive systems on ordered Banach spaces, J. Diff. Equations, 176 (2001), 195-222. doi: 10.1006/jdeq.2001.3981.  Google Scholar

[13]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[14]

D. Tilman, Resource Competition and Community Structure, Princeton University Press, New Jersey, 1982. Google Scholar

[15]

K. Yoshiyama, J. P. Mellard, E. Litchman and C. A. Klausmeier, Phytoplankton competition for nutrients and light in a statified water column, Am. Nat., 174 (2009), 190-203. doi: 10.1086/600113.  Google Scholar

[16]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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