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Multiple periodic solutions for a discrete time model of plankton allelopathy

Abstract

We study a discrete time model of the growth of two species of plankton with competitive and allelopathic effects on each other N1(k+1) = N1(k)exp{r1(k)-a11(k)N1(k)-a12(k)N2(k)-b1(k)N1(k)N2(k)}, N2(k+1) = N2(k)exp{r2(k)-a21(k)N2(k)-b2(k)N1(k)N1(k)N2(k)}. A set of sufficient conditions is obtained for the existence of multiple positive periodic solutions for this model. The approach is based on Mawhin's continuation theorem of coincidence degree theory as well as some a priori estimates. Some new results are obtained.

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References

  1. 1.

    Arditi R, Ginzburg LR, Akcakaya HR: Variation in plankton densities among lakes: a case for ratio-dependent predation models. The American Naturalist 1991, 138: 1287–1296. 10.1086/285286

    Article  Google Scholar 

  2. 2.

    Chattopadhyay J: Effect of toxic substances on a two-species competitive system. Ecological Modelling 1996,84(1–3):287–289.

    Article  Google Scholar 

  3. 3.

    Chen Y: Multiple periodic solutions of delayed predator-prey systems with type IV functional responses. Nonlinear Analysis: Real World Applications 2004,5(1):45–53. 10.1016/S1468-1218(03)00014-2

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Fan M, Wang K: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Mathematical and Computer Modelling 2002,35(9–10):951–961. 10.1016/S0895-7177(02)00062-6

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Freedman HI, Wu J: Periodic solutions of single-species models with periodic delay. SIAM Journal on Mathematical Analysis 1992,23(3):689–701. 10.1137/0523035

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Gaines RE, Mawhin JL: Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics. Volume 568. Springer, Berlin; 1977:i+262.

    Google Scholar 

  7. 7.

    Hellebust JA: Extracellular Products in Algal Physiology and Biochemistry, edited by W. D. P. Stewart. University of California Press, California; 1974.

    Google Scholar 

  8. 8.

    Kuang Y: Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering. Volume 191. Academic Press, Massachusetts; 1993:xii+398.

    Google Scholar 

  9. 9.

    Maynard-Smith J: Models in Ecology. Cambridge University Press, Cambridge, UK; 1974.

    Google Scholar 

  10. 10.

    Mukhopadhyay A, Chattopadhyay J, Tapaswi PK: A delay differential equations model of plankton allelopathy. Mathematical Biosciences 1998,149(2):167–189. 10.1016/S0025-5564(98)00005-4

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Rice EL: Allelopathy. Academic Press, New York; 1984.

    Google Scholar 

  12. 12.

    Zhang RY, Wang ZC, Chen Y, Wu J: Periodic solutions of a single species discrete population model with periodic harvest/stock. Computers & Mathematics with Applications 2000,39(1–2):77–90. 10.1016/S0898-1221(99)00315-6

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Zhen J, Ma Z: Periodic solutions for delay differential equations model of plankton allelopathy. Computers & Mathematics with Applications 2002,44(3–4):491–500. 10.1016/S0898-1221(02)00163-3

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Jianbao Zhang.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zhang, J., Fang, H. Multiple periodic solutions for a discrete time model of plankton allelopathy. Adv Differ Equ 2006, 090479 (2006). https://doi.org/10.1155/ADE/2006/90479

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Periodic Solution