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

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


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  • [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.


    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.


    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.


    M. Droop, Some thoughts on nutrient limitation in algae, J. Phycol., 9 (1973), 264-272.


    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.


    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.


    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.


    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.


    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.


    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.


    L. C. Evans, "Partial Differential Equations," American Mathematical Society, 1998.


    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.


    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.


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


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


    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.


    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.


    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.


    J. Keener, "Principles of Applied Mathematics," Addison-Wesley, Reading, MA, 1987.


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


    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.


    H. H. Schaefer, "Topological Vector Spaces," Macmillan, New York, 1966.


    Junping Shi, Persistence and bifurcation of degenerate solutions, Jour. Funct. Anal., 169 (1999), 494-531.doi: 10.1006/jfan.1999.3483.


    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.


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


    M. X. Wang, "Nonlinear Parabolic Equations," Science Press, Beijing, 1993 (in Chinese).

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