On the Dynamical Behavior of Toxic-Phytoplankton-Zooplankton Model with Delay

Rehim, Mehbuba; Wu, Weixin; Muhammadhaji, Ahmadjan

doi:10.1155/2015/756315

Cited by 10 publications

(12 citation statements)

References 36 publications

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“…Thus, we hope that our work could be helpful for studying harmful phytoplankton blooms. In addition, it is noteworthy that system should consider the gestation delay of zooplankton . Because it takes time for zooplankton to digest foods, so the number of zooplankton does not immediately increase when zooplankton feed on phytoplankton.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

Zhao

Yan

Huang

et al. 2019

Math Methods in App Sciences

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This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms.

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Section: Conclusion and Discussionmentioning

confidence: 99%

“…In addition, it is noteworthy that system (1.2) should consider the gestation delay of zooplankton. 5,[23][24][25] Because it takes time for zooplankton to digest foods, so the number of zooplankton does not immediately increase when zooplankton feed on phytoplankton. Unfortunately, we do not take it into account in the model.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

Zhao

Yan

Huang

et al. 2019

Math Methods in App Sciences

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“…From 2002, many authors [2,3] have studied the complex dynamic behaviors dynamics of phytoplankton-zooplankton models, such as stability of equilibrium, Hopf bifurcation, global Hopf-bifurcation and so on. Some references [4,5,6] have discussed persistence, periodic resonance and chaos of phytoplankton-zooplankton models.…”

Section: Introductionmentioning

confidence: 99%

Simple-zero singularity in toxic phytoplankton-zooplankton model

Wang¹

2018

Proceedings of the 2018 2nd International Conference on Advances in Energy, Environment and Chemical Science (AEECS 2018)

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In this paper, we have considered a phytoplankton-zooplankton system with toxic liberation delay. Firstly, we give the critical values of transcritical bifurcation. Then, we derive the normal forms and their unfolding with original parameters in the system near the bifurcation point by the normal form method and center manifold theory. Finally, we show that the plankton system undergoes a transcritial bifurcation, and the equilibrium is unstable.

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“…Those references showed that the growth of the algae satisfies the linear function or Monod equation [6][7][8][9][10][11][12][13][14][15][16][17], which is related to the nutrient concentration. The interactions of the algae and the fish are the relationships of predator and prey [18,19].…”

Section: Introductionmentioning

confidence: 99%

A kind of non-traditional biomanipulation model with constant releasing fish

Guo

Chen

Song

2017

Math. Meth. Appl. Sci.

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On the basis of the ideas of non‐traditional biomanipulation control in fresh water body, a kind of nutrient–algae fish model is presented to investigate the effects of constant releasing fish on the nutrient and the algae. The threshold conditions for the extinction of the algae are obtained by discussing the stability of boundary equilibrium. The conditions for the coexistence of the algae and the fish are obtained by discussing the existence and stability of positive equilibrium. Besides, Hopf bifurcation is also analyzed by considering the parameter about the amount of the released fish. Furthermore, a kind of optimal problem is presented, and the necessary condition for the existence of the optimal solution is given by Pontryagin maximum principle. Finally, the mathematical results are verified by numerical simulations. The mathematical results show that there is a threshold amount of the released fish, above which the activity of releasing fish can control the growth of the algae and further reduce the probability of the algae bloom, but can not decrease the eutrophication level of the water body. Copyright © 2017 John Wiley & Sons, Ltd.

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On the Dynamical Behavior of Toxic-Phytoplankton-Zooplankton Model with Delay

Cited by 10 publications

References 36 publications

Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

Simple-zero singularity in toxic phytoplankton-zooplankton model

A kind of non-traditional biomanipulation model with constant releasing fish

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