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

Sze-Bi Hsu; Feng-Bin Wang

doi:10.3934/cpaa.2011.10.1479

Article Contents

2011, Volume 10, Issue 5: 1479-1501. Doi: 10.3934/cpaa.2011.10.1479

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

Sze-Bi Hsu^1, and
Feng-Bin Wang^2,

1.
Department of Mathematics, National Tsing Hua University, Hsinchu 300
2.
Department of Mathematics, National Tsing-Hua University, Hsinchu 300

Received: July 31, 2009

Revised: May 31, 2010

Published: August 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J55, 35J65; Secondary: 92D25.

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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), 189-203.
[2]	A. Cunningham and R. M. Nisbet, "Transient and Oscillation in Continuous Culture," in Mathematics in Microbiology, M. J. Bazin, ed., Academic ress, New York, 1983.
[3]	M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.doi: 10.1016/0022-1236(71)90015-2.
[4]	M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.
[5]	E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.doi: 10.1016/0022-247X(83)90098-7.
[6]	E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Am. Math. Soc., 284 (1984), 729-743.doi: 10.1090/S0002-9947-1984-0743741-4.
[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), 337-357.doi: 10.1016/0362-546X(94)E0063-M.
[8]	Y. Du and J. Shi, Some recent results on diffusive predator-prey models in spatially heterogeneous environment, Nonlinear Dynamics and Evolution Equations, Fields Institute Communications, Vol. 48, American Mathematical Society, (2006), 95-135.
[9]	Y. Du and J. Shi, Spatially heterogeneous predator-prey model with protect zone for prey, Journal of Differential Equations, 229 (2006), 63-91.doi: 10.1016/j.jde.2006.01.013.
[10]	Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Am. Math. Soc., 359 (2007), 4557-4593.doi: 10.1090/S0002-9947-07-04262-6.
[11]	L. C. Evans, "Partial Differential Equations," American Mathematical Society, 1998.
[12]	D. G. Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Commun. Part. Diff. Eq., 17 (1992), 339-346.doi: 10.1080/03605309208820844.
[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), 409-428.
[14]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.
[15]	S. B. Hsu, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.doi: 10.1137/0134064.
[16]	S. B. Hsu, Steady states of a system of partial differential equations modeling microbial ecology, SIAM J. Math. Anal., 14 (1983), 1130-1138.doi: 10.1137/0514087.
[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), 366-383.doi: 10.1137/0132030.
[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), 1026-1044.doi: 10.1137/0153051.
[19]	J. Keener, "Principles of Applied Mathematics," Addison-Wesley, Reading, MA, 1987.
[20]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, 1984.
[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), 371-392.
[22]	H. H. Schaefer, "Topological Vector Spaces," Macmillan, New York, 1966.
[23]	Junping Shi, Persistence and bifurcation of degenerate solutions, Jour. Funct. Anal., 169 (1999), 494-531.doi: 10.1006/jfan.1999.3483.
[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), 1113-1131.doi: 10.1137/S0036139993245344.
[25]	H. L. Smith and P. E. Waltman, "The Theory of the Chemostat," Cambridge Univ. Press, 1995.
[26]	M. X. Wang, "Nonlinear Parabolic Equations," Science Press, Beijing, 1993 (in Chinese).