Advanced Search
Article Contents
Article Contents

Dynamics of harmful algae with seasonal temperature variations in the cove-main lake

Abstract Related Papers Cited by
  • In this paper, we investigate two-vessel gradostat models describing the dynamics of harmful algae with seasonal temperature variations, in which one vessel represents a small cove connected to a larger lake. We first define the basic reproduction number for the model system, and then show that the trivial periodic state is globally asymptotically stable, and algae is washed out eventually if the basic reproduction number is less than unity, while there exists at least one positive periodic state and algal blooms occur when it is greater than unity. There are several types of productions for dissolved toxins, related to the algal growth rate, and nutrient limitation, respectively. For the system with a specific toxin production, the global attractivity of positive periodic steady-state solution can be established. Numerical simulations from the basic reproduction number show that the factor of seasonality plays an important role in the persistence of harmful algae.
    Mathematics Subject Classification: Primary: 34C12, 34D20; Secondary: 34D23.


    \begin{equation} \\ \end{equation}
  • [1]

    G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Math. Biosci., 38 (1978), 113-122.doi: 10.1016/0025-5564(78)90021-4.


    N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco, J. Math. Biol., 53 (2006), 421-436.doi: 10.1007/s00285-006-0015-0.


    S. Chakraborty, S. Roy and J. Chattopadhyay, Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: a mathematical model, Ecol. Model., 213 (2008), 191-201.doi: 10.1016/j.ecolmodel.2007.12.008.


    O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.doi: 10.1007/BF00178324.


    P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold en- demic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.doi: 10.1016/S0025-5564(02)00108-6.


    E. I. R. Falconer and A. R. Humpage, Cyanobacterial (bluegreen algal) toxins in water supplies: cylindrospermopsins, Environ. Toxicol., 21 (2006), 299-304.


    J. P. Grover, S.-B. Hsu and F.-B. Wang, Competition and coexistence in flowing habitats with a hydraulic storage zone, Math. Biosci., 222 (2009), 42-52.doi: 10.1016/j.mbs.2009.08.006.


    E. Graneĺi and N. Johansson, Increase in the production of allelopathic substances by Prymnesium parvum cells grown under N- or P-deficient conditions, Harmful Algae, 2 (2003), 135-145.


    J. P. Grover, K. W. Crane, J. W. Baker, B. W. Brooks and D. L. Roelke, Spatial variation of harmful algae and their toxins in flowing-water habitats: a theoretical exploration, Journal of Plankton Research, 33 (2011), 211-227.doi: 10.1093/plankt/fbq070.


    M. W. Hirsch, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439.doi: 10.1137/0516030.


    J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society Providence, RI, 1988.


    P. R. Hawkins, E. Putt and I. Falconer, et al, Phenotypical variation in a toxic strain of the phytoplankter, Cylindrospermopsis raciborskii (Nostocales, Cyanophyceae) during batch culture, Environ. Toxicol., 16 (2001), 460-476.


    S. B. Hsu, F. B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Diff. Eqns., 255 (2013), 265-297.doi: 10.1016/j.jde.2013.04.006.


    J. Jiang, On the global stability of cooperative systems, Bull London Math. Soc., 26 (1994), 455-458.doi: 10.1112/blms/26.5.455.


    N. Johansson and E. Graneĺi, Cell density, chemical composition and toxicity of Chrysochromulina polylepis (Haptophyta) in relation to different N:P supply ratios, Mar. Biol., 135 (1999), 209-217.doi: 10.1007/s002270050618.


    D. Lekan and C. R. Tomas, The brevetoxin and brevenal composition of three Karenia brevis clones at different salinities and nutrient conditions, Harmful Algae, 9 (2010), 39-47.doi: 10.1016/j.hal.2009.07.004.


    C. G. R. Maier, M. D. Burch and M. Bormans, Flow management strategies to control blooms of the cyanobacterium, Anabaena circinalis, in the river Murray at Morgan, South Australia, Regul. Rivers Res. Mgmt., 17 (2001), 637-650.doi: 10.1002/rrr.623.


    S. M. Mitrovic, L. Hardwick and R. Oliver, et. al., Use of flow management to control saxitoxin producing cyanobacterial blooms in the Lower Darling River, Australia, J. Plankton Res., 33 (2011), 229-241.


    C. S. Reynolds, Potamoplankton: Paradigms, Paradoxes and Prognoses, in Algae and the Aquatic Environment, F. E. Round, ed., Biopress, Bristol, UK, 1990.


    D. L. Roelke, G. M. Gable and T. W. Valenti, Hydraulic flushing as a Prymnesium parvum bloom terminating mechanism in a subtropical lake, Harmful Algae, 9 (2010), 323-332.doi: 10.1016/j.hal.2009.12.003.


    D. L. Roelke, J. P. Grover and B. W. Brooks et al, A decade of fishkilling Prymnesium parvum blooms in Texas: Roles of inflow and salinity, J. Plankton Res., 33 (2011), 243-253.


    H. L. Smith, Microbial growth in periodic gradostats, Rocky Mountain J. Math., 20 (1990), 1173-1194.doi: 10.1216/rmjm/1181073069.


    H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr 41, American Mathematical Society Providence, RI, 1995.


    G. M. Southard, L. T. Fries and A. Barkoh, Prymnesium parvum: the Texas experience, J. Am. Water Resources Assoc., 46 (2010), 14-23.doi: 10.1111/j.1752-1688.2009.00387.x.


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


    H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.doi: 10.1016/S0362-546X(01)00678-2.


    W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717.doi: 10.1007/s10884-008-9111-8.


    K.F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516.doi: 10.1016/j.jmaa.2006.01.085.


    X.-Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Commun. Appl. Nonlinear Anal., 3 (1996), 43-66.


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

  • 加载中

Article Metrics

HTML views() PDF downloads(164) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint