Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

Pan, Xinhong; Zhao, Min; Dai, Chuanjun; Wang, Yapei

doi:10.1155/2014/471507

Cited by 3 publications

(1 citation statement)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The dynamics of zooplankton-phytoplankton systems have been discussed by many authors [2][3][4][5][6][7][8]. A wealth of studies have shown that a time delay may have a complex effect on the dynamics of such systems, with effects that include stability switches for equilibria, the existence of Hopf bifurcation periodic solutions, and the direction and stability of Hopf bifurcation [9][10][11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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

Wang

Zhao

Dai

et al. 2016

Discrete Dynamics in Nature and Society

Self Cite

View full text Add to dashboard Cite

We analyze a nutrient-plankton system with a time delay. We choose the time delay as a bifurcation parameter and investigate the stability of a positive equilibrium and the existence of Hopf bifurcations. By using the center manifold theorem and the normal form theory, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are researched. The theoretical results indicate that the time delay can induce a positive equilibrium to switch from a stable to an unstable to a stable state and so on. Numerical simulations show that the theoretical results are correct and feasible, and the system exhibits rich complex dynamics.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Wang

Zhao

Dai

et al. 2016

Discrete Dynamics in Nature and Society

Self Cite

View full text Add to dashboard Cite

show abstract

Nonlinear Dynamics of a Nutrient-Phytoplankton Model with Time Delay

Mei

Zhao

et al. 2015

Discrete Dynamics in Nature and Society

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Dynamic Analysis of a Time Delayed Nutrient-Phytoplankton System

尚¹

2017

AAM

View full text Add to dashboard Cite

The paper mainly considered the time lag problem caused by the susceptible phytoplankton absorbing nutrient and translating into energy and susceptible phytoplankton turning into infected phytoplankton, and then, a time delayed nutrient-phytoplankton system will be established. On this basis, the stability of the equilibrium point and the Hopf bifurcation of the system have been analyzed by using the Routh-Hurwitz criterion and the Lyapunov-Krasovskii function, and then the critical conditions of these particular dynamical behaviors have been obtained. Finally, these results have some guiding significance for the prevention and control of large area outbreaks of phytoplankton.

show abstract

Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

Cited by 3 publications

References 22 publications

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

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

Nonlinear Dynamics of a Nutrient-Phytoplankton Model with Time Delay

Dynamic Analysis of a Time Delayed Nutrient-Phytoplankton System

Contact Info

Product

Resources

About