Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2016, Article ID 9797624, 12 pages
http://dx.doi.org/10.1155/2016/9797624
Research Article

Dynamics Analysis of a Nutrient-Plankton Model with a Time Delay

1School of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China
2Zhejiang Provincial Key Laboratory for Water Environment and Marine Biological Resources Protection, Wenzhou University, Wenzhou, Zhejiang 325035, China
3School of Life and Environmental Science, Wenzhou University, Wenzhou, Zhejiang 325027, China

Received 26 October 2015; Revised 21 November 2015; Accepted 23 November 2015

Academic Editor: Carlo Bianca

Copyright © 2016 Beibei Wang et al. 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.

Linked References

  1. D. M. Anderson, “Turning back the harmful red tide,” Nature, vol. 338, no. 7, pp. 513–514, 1997. View at Google Scholar
  2. A. M. Edwards and J. Brindley, “Zooplankton mortality and the dynamical behaviour of plankton population models,” Bulletin of Mathematical Biology, vol. 61, no. 2, pp. 303–339, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. J. Chattopadhyay, R. R. Sarkar, and A. El-Abdllaoui, “A delay differential equation model on harmful algal blooms in the presence of toxic substances,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 19, no. 2, pp. 137–161, 2002. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Saha and M. Bandyopadhyay, “Dynamical analysis of toxin producing phytoplankton-zooplankton interactions,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 314–332, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. S. Roy, “The coevolution of two phytoplankton species on a single resource: allelopathy as a pseudo-mixotrophy,” Theoretical Population Biology, vol. 75, no. 1, pp. 68–75, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. M. Banerjee and E. Venturino, “A phytoplankton-toxic phytoplankton-zooplankton model,” Ecological Complexity, vol. 8, no. 3, pp. 239–248, 2011. View at Publisher · View at Google Scholar · View at Scopus
  7. L. F. Mei and X. Y. Zhang, “Existence and nonexistence of positive steady states in multi-species phytoplankton dynamics,” Journal of Differential Equations, vol. 253, no. 7, pp. 2025–2063, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. Y. Wang, W. Jiang, and H. Wang, “Stability and global Hopf bifurcation in toxic phytoplankton-zooplankton model with delay and selective harvesting,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 881–896, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. J. Z. Zhang, Z. Jin, J. R. Yan, and G. Q. Sun, “Stability and Hopf bifurcation in a delayed competition system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 658–670, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. X. H. Pan, M. Zhao, C. J. Dai, and Y. P. Wang, “Stability and Hopf bifurcation analysis of a nutrient-phytoplankton model with delay effect,” Abstract and Applied Analysis, vol. 2014, Article ID 471507, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. Y. Su, J. Wei, and J. Shi, “Hopf bifurcations in a reaction-diffusion population model with delay effect,” Journal of Differential Equations, vol. 247, no. 4, pp. 1156–1184, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. M. Zhao, X. Wang, H. Yu, and J. Zhu, “Dynamics of an ecological model with impulsive control strategy and distributed time delay,” Mathematics and Computers in Simulation, vol. 82, no. 8, pp. 1432–1444, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. F. Lian and Y. Xu, “Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1484–1495, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. Wang and G. Lv, “Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays,” Nonlinearity, vol. 23, no. 7, pp. 1609–1630, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. S. Chen, J. Shi, and J. Wei, “The effect of delay on a diffusive predator-prey system with Holling type-II predator functional response,” Communications on Pure and Applied Analysis, vol. 12, no. 1, pp. 481–501, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. S. S. Chen, J. P. Shi, and J. J. Wei, “Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems,” Journal of Nonlinear Science, vol. 23, no. 1, pp. 1–38, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. Y. Chen and F. Zhang, “Dynamics of a delayed predator-prey model with predator migration,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 1400–1412, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. H. Yu, M. Zhao, and R. P. Agarwal, “Stability and dynamics analysis of time delayed eutrophication ecological model based upon the Zeya reservoir,” Mathematics and Computers in Simulation, vol. 97, pp. 53–67, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. A. El Abdllaoui, J. Chattopadhyay, and O. Arino, “Comparisons, by models, of some basic mechanisms acting on the dynamics of the zooplankton-toxic phytoplankton system,” Mathematical Models and Methods in Applied Sciences, vol. 12, no. 10, pp. 1421–1451, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. A. Huppert, B. Blasius, R. Olinky, and L. Stone, “A model for seasonal phytoplankton blooms,” Journal of Theoretical Biology, vol. 236, no. 3, pp. 276–290, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. J. D. Ives, “Possible mechanisms underlying copepod grazing responses to levels of toxicity in red tide dinoflagellates,” Journal of Experimental Marine Biology and Ecology, vol. 112, no. 2, pp. 131–144, 1987. View at Publisher · View at Google Scholar · View at Scopus
  22. J. Chattopadhayay, R. R. Sarkar, and S. Mandal, “Toxin-producing plankton may act as a biological control for planktonic blooms—field study and mathematical modelling,” Journal of Theoretical Biology, vol. 215, no. 3, pp. 333–344, 2002. View at Publisher · View at Google Scholar · View at Scopus
  23. C. Bianca and L. Guerrini, “On the Dalgaard-Strulik model with logistic population growth rate and delayed-carrying capacity,” Acta Applicandae Mathematicae, vol. 128, no. 1, pp. 39–48, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. C. Bianca, M. Ferrara, and L. Guerrini, “The time delays' effects on the qualitative behavior of an economic growth model,” Abstract and Applied Analysis, vol. 2013, Article ID 901014, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  25. W. Zuo and J. Wei, “Stability and Hopf bifurcation in a diffusive predatory-prey system with delay effect,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 1998–2011, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. S. Jana, M. Chakraborty, K. Chakraborty, and T. K. Kar, “Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge,” Mathematics and Computers in Simulation, vol. 85, pp. 57–77, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. C. Bianca, M. Ferrara, and L. Guerrini, “Qualitative analysis of a retarded mathematical framework with applications to living systems,” Abstract and Applied Analysis, vol. 2013, Article ID 7360528, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. C. Bianca and S. Ferrara, “Hopf bifurcations in a delayed-energy-based model of capital accumulation,” Applied Mathematics & Information Sciences, vol. 7, no. 1, pp. 139–143, 2013. View at Google Scholar
  29. S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete & Impulsive Systems—Series A: Mathematical Analysis, vol. 10, no. 6, pp. 863–874, 2003. View at Google Scholar · View at MathSciNet
  30. J. Lü and G. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos, vol. 12, no. 3, pp. 659–661, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. C. Bianca, M. Ferrara, and L. Guerrini, “The cai model with time delay: existence of periodic solutions and asymptotic analysis,” Applied Mathematics & Information Sciences, vol. 7, no. 1, pp. 21–27, 2013. View at Publisher · View at Google Scholar · View at Scopus
  32. B. D. Hassard, N. D. Kazarinoff, and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.