September  2011, 10(5): 1479-1501. doi: 10.3934/cpaa.2011.10.1479

On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis

1. 

Department of Mathematics, National Tsing Hua University, Hsinchu 300

2. 

Department of Mathematics, National Tsing-Hua University, Hsinchu 300, Taiwan

Received  August 2009 Revised  June 2010 Published  April 2011

In this paper we construct a mathematical model of two microbial populations competing for a single-limited nutrient with internal storage in an unstirred chemostat. First we establish the existence and uniqueness of steady-state solutions for the single population. The conditions for the coexistence of steady states are determined. Techniques include the maximum principle, theory of bifurcation and degree theory in cones.
Citation: Sze-Bi Hsu, Feng-Bin Wang. On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1479-1501. doi: 10.3934/cpaa.2011.10.1479
References:
[1]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae,, J. Theoret. Biol., 84 (1980), 189.   Google Scholar

[2]

A. Cunningham and R. M. Nisbet, "Transient and Oscillation in Continuous Culture,", in Mathematics in Microbiology, (1983).   Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[4]

M. Droop, Some thoughts on nutrient limitation in algae,, J. Phycol., 9 (1973), 264.   Google Scholar

[5]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications,, J. Math. Anal. Appl., 91 (1983), 131.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

[6]

E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Am. Math. Soc., 284 (1984), 729.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar

[7]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion I. General existence results,, Nonlinear Anal., 254 (1995), 337.  doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar

[8]

Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment,, Nonlinear Dynamics and Evolution Equations, (2006), 95.   Google Scholar

[9]

Y. Du and J. Shi, Spatially heterogeneous predator-prey model with protect zone for prey,, Journal of Differential Equations, 229 (2006), 63.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[10]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model,, Trans. Am. Math. Soc., 359 (2007), 4557.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations,", American Mathematical Society, (1998).   Google Scholar

[12]

D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Commun. Part. Diff. Eq., 17 (1992), 339.  doi: 10.1080/03605309208820844.  Google Scholar

[13]

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.   Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983).   Google Scholar

[15]

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

[16]

S. B. Hsu, Steady states of a system of partial differential equations modeling microbial ecology,, SIAM J. Math. Anal., 14 (1983), 1130.  doi: 10.1137/0514087.  Google Scholar

[17]

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.  doi: 10.1137/0132030.  Google Scholar

[18]

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.  doi: 10.1137/0153051.  Google Scholar

[19]

J. Keener, "Principles of Applied Mathematics,", Addison-Wesley, (1987).   Google Scholar

[20]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984).   Google Scholar

[21]

W. Ruan and W. Feng, On the fixed point index and multiple steady-state solutions of reaction-diffusion systems,, Differential Integral Equations, 8 (1995), 371.   Google Scholar

[22]

H. H. Schaefer, "Topological Vector Spaces,", Macmillan, (1966).   Google Scholar

[23]

Junping Shi, Persistence and bifurcation of degenerate solutions,, Jour. Funct. Anal., 169 (1999), 494.  doi: 10.1006/jfan.1999.3483.  Google Scholar

[24]

H. L. Smith and P. E. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model,, SIAM J. Appl. Math., 34 (1994), 1113.  doi: 10.1137/S0036139993245344.  Google Scholar

[25]

H. L. Smith and P. E. Waltman, "The Theory of the Chemostat,", Cambridge Univ. Press, (1995).   Google Scholar

[26]

M. X. Wang, "Nonlinear Parabolic Equations,", Science Press, (1993).   Google Scholar

show all references

References:
[1]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae,, J. Theoret. Biol., 84 (1980), 189.   Google Scholar

[2]

A. Cunningham and R. M. Nisbet, "Transient and Oscillation in Continuous Culture,", in Mathematics in Microbiology, (1983).   Google Scholar

[3]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[4]

M. Droop, Some thoughts on nutrient limitation in algae,, J. Phycol., 9 (1973), 264.   Google Scholar

[5]

E. N. Dancer, On the indices of fixed points of mappings in cones and applications,, J. Math. Anal. Appl., 91 (1983), 131.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

[6]

E. N. Dancer, On positive solutions of some pairs of differential equations,, Trans. Am. Math. Soc., 284 (1984), 729.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar

[7]

E. N. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion I. General existence results,, Nonlinear Anal., 254 (1995), 337.  doi: 10.1016/0362-546X(94)E0063-M.  Google Scholar

[8]

Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment,, Nonlinear Dynamics and Evolution Equations, (2006), 95.   Google Scholar

[9]

Y. Du and J. Shi, Spatially heterogeneous predator-prey model with protect zone for prey,, Journal of Differential Equations, 229 (2006), 63.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[10]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model,, Trans. Am. Math. Soc., 359 (2007), 4557.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations,", American Mathematical Society, (1998).   Google Scholar

[12]

D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Commun. Part. Diff. Eq., 17 (1992), 339.  doi: 10.1080/03605309208820844.  Google Scholar

[13]

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.   Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag, (1983).   Google Scholar

[15]

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

[16]

S. B. Hsu, Steady states of a system of partial differential equations modeling microbial ecology,, SIAM J. Math. Anal., 14 (1983), 1130.  doi: 10.1137/0514087.  Google Scholar

[17]

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.  doi: 10.1137/0132030.  Google Scholar

[18]

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.  doi: 10.1137/0153051.  Google Scholar

[19]

J. Keener, "Principles of Applied Mathematics,", Addison-Wesley, (1987).   Google Scholar

[20]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (1984).   Google Scholar

[21]

W. Ruan and W. Feng, On the fixed point index and multiple steady-state solutions of reaction-diffusion systems,, Differential Integral Equations, 8 (1995), 371.   Google Scholar

[22]

H. H. Schaefer, "Topological Vector Spaces,", Macmillan, (1966).   Google Scholar

[23]

Junping Shi, Persistence and bifurcation of degenerate solutions,, Jour. Funct. Anal., 169 (1999), 494.  doi: 10.1006/jfan.1999.3483.  Google Scholar

[24]

H. L. Smith and P. E. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model,, SIAM J. Appl. Math., 34 (1994), 1113.  doi: 10.1137/S0036139993245344.  Google Scholar

[25]

H. L. Smith and P. E. Waltman, "The Theory of the Chemostat,", Cambridge Univ. Press, (1995).   Google Scholar

[26]

M. X. Wang, "Nonlinear Parabolic Equations,", Science Press, (1993).   Google Scholar

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