Skip to content


  • Research Article
  • Open Access

Multiple periodic solutions for a discrete time model of plankton allelopathy

Advances in Difference Equations20062006:090479

  • Received: 19 May 2005
  • Accepted: 27 September 2005
  • Published:


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.


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


Authors’ Affiliations

Center for Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan, 650093, China


  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. 10.1086/285286View ArticleGoogle Scholar
  2. Chattopadhyay J: Effect of toxic substances on a two-species competitive system. Ecological Modelling 1996,84(1–3):287–289.View ArticleGoogle Scholar
  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)00014-2MathSciNetView ArticleMATHGoogle Scholar
  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)00062-6MathSciNetView ArticleMATHGoogle Scholar
  5. Freedman HI, Wu J: Periodic solutions of single-species models with periodic delay. SIAM Journal on Mathematical Analysis 1992,23(3):689–701. 10.1137/0523035MathSciNetView ArticleMATHGoogle Scholar
  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. Hellebust JA: Extracellular Products in Algal Physiology and Biochemistry, edited by W. D. P. Stewart. University of California Press, California; 1974.Google Scholar
  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. Maynard-Smith J: Models in Ecology. Cambridge University Press, Cambridge, UK; 1974.Google Scholar
  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)00005-4MathSciNetView ArticleMATHGoogle Scholar
  11. Rice EL: Allelopathy. Academic Press, New York; 1984.Google Scholar
  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)00315-6MathSciNetView ArticleMATHGoogle Scholar
  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)00163-3MathSciNetView ArticleMATHGoogle Scholar


© J. Zhang and H. Fang 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.