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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 939187, 12 pages
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.


We consider a nutrient-phytoplankton model with a Holling type II functional response and a time delay. By selecting the time delay used as a bifurcation parameter, we prove that the system is stable if the delay value is lower than the critical value but unstable when it is above this value. First, we investigate the existence and stability of the equilibria, as well as the existence of Hopf bifurcations. Second, we consider the direction, stability, and period of the periodic solutions from the steady state based on the normal form and the center manifold theory, thereby deriving explicit formulas. Finally, some numerical simulations are given to illustrate the main theoretical results.