Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient

Jiang, Zhi Chao; Bi, Xiao Hua; Zhang, Tongqian; Pradeep, B. G. Sampath Aruna

doi:10.3934/mbe.2019188

Cited by 21 publications

(10 citation statements)

References 33 publications

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“…In 1949, Riley et al [38] first used the mathematical model to study the nutrient-plankton dynamics, which leads to the formulation of a growing number of mathematical models to describe the nutrient-phytoplankton dynamics or nutrient-plankton dynamics, and many dynamic mechanisms of phytoplankton growth response to various factors have been revealed [14,15,22,[39][40][41][42][43][44][45][46][47]. For example, Chen et al [43] showed that the proper control of the ratio for nitrogen and phosphorus can more effectively control and eliminateblue-green algae blooms.…”

Section: Introductionmentioning

confidence: 99%

Impact of Cell Size Effect on Nutrient‐Phytoplankton Dynamics

Liao

Dai

et al. 2019

Complexity

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In this paper, a nutrient-phytoplankton model, which is described by a system of ordinary differential equations incorporating the effect of cell size, and its corresponding stochastic differential equation version are studied analytically and numerically. A key advantage of considering cell size effect is that it can more accurately reveal the intrinsic law of interaction between nutrient and phytoplankton. The main purpose of this paper is to research how cell size affects the nutrient-phytoplankton dynamics within the deterministic and stochastic environments. Mathematically, we show that the existence and stability of the equilibria in the deterministic model can be determined by cell size: the smaller or larger cell size can lead to the disappearance of the positive equilibrium, but the boundary equilibrium always exists and is globally asymptotically stable; the intermediate cell size is capable to drive the positive equilibrium to appear and be globally asymptotically stable, whereas the boundary equilibrium becomes unstable. In the case of the stochastic model, the stochastic dynamics including the stochastic extinction, persistence in the mean, and the existence of ergodic stationary distribution is found to be largely dependent on cell size and noise intensity. Ecologically, via numerical simulations, it is found that the smaller cell size or larger cell size can result in the extinction of phytoplankton, which is similar to the effect of larger random environmental fluctuations on the phytoplankton. More interestingly, it is discovered that the intermediate cell size is the optimal size for promoting the growth of phytoplankton, but increasing appropriately the cell size can rapidly reduce phytoplankton density and nutrient concentrations at the same time, which provides a possible strategy for biological control of algal blooms.

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Section: Introductionmentioning

confidence: 99%

Impact of Cell Size Effect on Nutrient‐Phytoplankton Dynamics

Liao

Dai

et al. 2019

Complexity

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“…Set Ω k � τ k ≤ T , and from inequality (10), we have P(Ω k ) ≥ ϵ. Note that, for every ω ∈ Ω k , there is at least one of x 1 (τ k , ω), x 2 (τ k , ω), or u(τ k , ω) equaling either k or (1/k), and then, we have 4 Complexity…”

Section: Existence and Uniqueness Of Globalmentioning

confidence: 99%

“…In nature, time delays exist in many ecosystems [1][2][3][4][5]. For example, maturity stage is a common phenomenon in biological population, and many diseases have a long incubation period.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Analysis of Stochastic Lotka–Volterra Predator‐Prey Model with Discrete Delays and Feedback Control

Liu

Zhao

2019

Complexity

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In this paper, a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control is studied. Firstly, the existence and uniqueness of global positive solution are proved. Further, we investigate the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model and establish sufficient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is verified by numerical simulations.

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“…How to regulate the ecosystem reasonably to ensure its balance and sustainable development is of importance for issue facing mankind [29][30][31][32][33]. Maltby et al [34] considered ecosystem management to be a control of physical, chemical, and biological processes, which linked the regulation of organisms and their nonliving environment and human activities to create an ideal ecosystem state.…”

Section: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (3)mentioning

confidence: 99%

Dynamics Analysis of a Stochastic Leslie–Gower Predator‐Prey Model with Feedback Controls

Ren

Zhao

2019

Mathematical Problems in Engineering

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This study focuses on the investigation of a stochastic Leslie–Gower predator-prey model with feedback controls and Holling type II functional responses. First, the existence and uniqueness of a global positive solution to the system under white noise interference are proved. Second, the conditions for the existence of the system’s positive recurrence are established by constructing suitable Lyapunov functions. Additionally, the persistence and extinction of prey and predator in the system are discussed, and the impacts of noise interference and feedback controls on the system are revealed. Finally, we validate the theoretical results by numerical simulations.

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Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient

Cited by 21 publications

References 33 publications

Impact of Cell Size Effect on Nutrient‐Phytoplankton Dynamics

Impact of Cell Size Effect on Nutrient‐Phytoplankton Dynamics

Dynamic Analysis of Stochastic Lotka–Volterra Predator‐Prey Model with Discrete Delays and Feedback Control

Dynamics Analysis of a Stochastic Leslie–Gower Predator‐Prey Model with Feedback Controls

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