Dynamics of a nutrient-plankton model with delay and toxicity

In this paper, a Holling type IV nutrient-plankton model with time delay and linear plankton harvesting is investigated. The existence and local stability of all equilibria of model without time delay are given. Regarding time delay as bifurcation parameter, such system around the interior equilibrium loses its local stability, and Hopf bifurcation occurs when time delay crosses its critical value. In addition, the properties of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem. What is more, the global continuation of the local Hopf bifurcation is discussed by using a global Hopf bifurcation result. Furthermore, the optimal harvesting is obtained by the Pontryagin’s Maximum Principle. Finally, some numerical simulations are given to confirm our theoretical analysis.

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Dynamics in a Plankton Model with Toxic Substances and Phytoplankton Harvesting

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420500352 ◽

2020 ◽

Vol 30 (02) ◽

pp. 2050035 ◽

Cited By ~ 1

Author(s):

Hua Zhang ◽

Ben Niu

Keyword(s):

Harmful Algal Blooms ◽

Algal Blooms ◽

Initial Population ◽

Toxic Substances ◽

Environmental Capacity ◽

Existence And Stability ◽

Small Phytoplankton ◽

Normal Form Method ◽

Bautin Bifurcation ◽

Plankton Model

In this paper, a phytoplankton–zooplankton model incorporating toxic substances and nonlinear phytoplankton harvesting is established. The existence and stability of the equilibrium of this model are first investigated. The occurrence of transcritical, saddle-node, Hopf and Bautin bifurcations at different equilibria is then verified. In addition, the properties of Hopf bifurcation and Bautin bifurcation are discussed by using normal form method. These results demonstrate that phytoplankton and zooplankton populations will oscillate periodically when the harvesting level is high. More interestingly, it is found that the oscillations are always unstable for small phytoplankton carrying capacity, while the dynamics have close relations with the initial population densities for a large environmental capacity. The existence of Bautin bifurcation theoretically indicates that toxic phytoplankton can cause extinction once there exist harmful algal blooms for some time. These results are numerically illustrated for the model with spatial diffusion, which shows that local phytoplankton blooms will lead to global populations extinction.

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Chaotic dynamics in a simple dynamical green ocean plankton model

Journal of Marine Systems ◽

10.1016/j.jmarsys.2014.08.002 ◽

2014 ◽

Vol 139 ◽

pp. 483-495 ◽

Cited By ~ 1

Author(s):

Roger Cropp ◽

Irene M. Moroz ◽

John Norbury

Keyword(s):

Chaotic Dynamics ◽

Plankton Model

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A simple plankton model for the Oregon upwelling ecosystem: Sensitivity and validation against time-series ocean data

Ecological Modelling ◽

10.1016/j.ecolmodel.2011.01.001 ◽

2011 ◽

Vol 222 (6) ◽

pp. 1222-1235 ◽

Cited By ~ 5

Author(s):

James J. Ruzicka ◽

Thomas C. Wainwright ◽

William T. Peterson

Keyword(s):

Time Series ◽

Ecosystem Sensitivity ◽

Plankton Model

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Propagation of Turing patterns in a plankton model

Journal of Biological Dynamics ◽

10.1080/17513758.2012.655327 ◽

2012 ◽

Vol 6 (2) ◽

pp. 524-538 ◽

Cited By ~ 10

Author(s):

R. K. Upadhyay ◽

V. Volpert ◽

N. K. Thakur

Keyword(s):

Turing Patterns ◽

Plankton Model

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Permanence and extinction of a nonautonomous impulsive plankton model with help

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.4521 ◽

2017 ◽

Vol 40 (18) ◽

pp. 7175-7184

Author(s):

Wen Wang ◽

Shutang Liu ◽

Dadong Tian ◽

Qiuyue Zhao

Keyword(s):

Plankton Model

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MODELING FLUCTUATIONS IN A MINIMAL PLANKTON MODEL: ROLE OF SPATIAL HETEROGENEITY AND STOCHASTICITY

Advances in Complex Systems ◽

10.1142/s021952590700101x ◽

2007 ◽

Vol 10 (02) ◽

pp. 197-216 ◽

Cited By ~ 1

Author(s):

R. BHATTACHARYYA ◽

B. MUKHOPADHYAY

Keyword(s):

Driving Forces ◽

Reaction Diffusion ◽

High Amplitude ◽

Equation Model ◽

Reaction Diffusion Equation ◽

Variable Diffusivity ◽

Basic Model ◽

Population Concentration ◽

Constant Diffusivity ◽

Plankton Model

The present paper studies the naturally observed phenomenon of population fluctuation in the context of a minimal plankton model. The analysis of the basic model reveals asymptotic stable behavior that is unable to explain any kind of population outburst. We introduce the nonuniform spatial distribution of plankton by considering physical diffusion of the species concerned. The resulting reaction-diffusion equation model is first analyzed with constant diffusivity and then with variable diffusivity for the zooplankton. The model with variable diffusivity is analyzed by using Floquet theory. In both cases, the model is seen to exhibit stable behavior. Finally, we study another characteristic feature of any ecological population, namely, environmental fluctuations. We achieve this by perturbing the growth rate of phytoplankton and death rate of zooplankton by using colored noise. The resulting model is analyzed by evaluating the spectral density functions. It is observed that the stochastic model can generate a large fluctuation in population concentration for high amplitude driving forces.

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