Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 939187, 12 pages
http://dx.doi.org/10.1155/2015/939187
Research Article

Nonlinear Dynamics of a Nutrient-Phytoplankton Model with 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 325035, China

Received 1 April 2015; Accepted 9 June 2015

Academic Editor: Luca Guerrini

Copyright © 2015 DeBing Mei 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. A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530–1535, 1992. View at Publisher · View at Google Scholar · View at Scopus
  2. H.-B. Shi, W.-T. Li, and G. Lin, “Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 3711–3721, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. M. Baurmann, T. Gross, and U. Feudel, “Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations,” Journal of Theoretical Biology, vol. 245, no. 2, pp. 220–229, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. G.-Q. Sun, Z. Jin, Q.-X. Liu, and L. Li, “Dynamical complexity of a spatial predator-prey model with migration,” Ecological Modelling, vol. 219, no. 1-2, pp. 248–255, 2008. View at Publisher · View at Google Scholar · View at Scopus
  5. S. Abbas, M. Banerjee, and N. Hungerbühler, “Existence, uniqueness and stability analysis of allelopathic stimulatory phytoplankton model,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 249–259, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. Y. P. Wang, M. Zhao, C. J. Dai, and X. H. Pan, “Nonlinear dynamics of a nutrient-plankton model,” Abstract and Applied Analysis, vol. 2014, Article ID 451757, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. M. Sandulescu, C. López, E. Hernández-García, and U. Feudel, “Plankton blooms in vortices: the role of biological and hydrodynamic timescales,” Nonlinear Processes in Geophysics, vol. 14, no. 4, pp. 443–454, 2007. View at Publisher · View at Google Scholar · View at Scopus
  8. C. J. Dai, M. Zhao, and L. S. Chen, “Complex dynamic behavior of three-species ecological model with impulse perturbations and seasonal disturbances,” Mathematics and Computers in Simulation, vol. 84, pp. 83–97, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. W. Wang, Q.-X. Liu, and Z. Jin, “Spatiotemporal complexity of a ratio-dependent predator-prey system,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 75, no. 5, Article ID 051913, 9 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  10. M. R. Garvie, “Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB,” Bulletin of Mathematical Biology, vol. 69, no. 3, pp. 931–956, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. W. W. Zhang and M. Zhao, “Dynamical complexity of a spatial phytoplankton-zooplankton model with an alternative prey and refuge effect,” Journal of Applied Mathematics, vol. 2013, Article ID 608073, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  12. 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
  13. K.-G. Mäler, “Development, ecological resources and their management: a study of complex dynamic systems,” European Economic Review, vol. 44, no. 4–6, pp. 645–665, 2000. View at Publisher · View at Google Scholar · View at Scopus
  14. 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
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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 MathSciNet · View at Scopus
  21. 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
  22. 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 & Application, vol. 70, no. 2, pp. 658–670, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. 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
  24. B. D. Hassard, N. D. Kazarinoff, and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981. View at MathSciNet