Stability and global Hopf bifurcation in toxic phytoplankton–zooplankton model with delay and selective harvesting

Wang, Yong; Jiang, Weihua; Wang, Hongbin

doi:10.1007/s11071-013-0839-2

Cited by 56 publications

(24 citation statements)

References 25 publications

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“…Thus, all the nontrivial solutions of system (2) Proof. In order to explain that system (2) has no nonconstant periodic solution of periodic , we only prove that system (4) has no nonconstant periodic solution (see [36], Lemma 5.5).…”

Section: Discrete Dynamics In Nature and Societymentioning

confidence: 99%

“…In the past decade, the issues of phytoplanktonzooplankton models have been investigated by many experts [2][3][4][5][6][7][8][9][10][11][12][13]. Chattopadhy et al [2] showed the effect of the delay forming the period of toxic substances on blooms.…”

Section: Introductionmentioning

confidence: 99%

“…Chattopadhy et al [2] showed the effect of the delay forming the period of toxic substances on blooms. Wang et al [4] considered a toxic phytoplankton-zooplankton model with delay-dependent coefficient and investigated influence of delay and harvesting for the system. Yang et al [8] discussed the impact of reaction-diffusion on the dynamic behaviours of plankton.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical Behaviour of a Nutrient-Plankton Model with Holling Type IV, Delay, and Harvesting

Meng

Jiao-guo

Huo

2018

Discrete Dynamics in Nature and Society

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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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Section: Discrete Dynamics In Nature and Societymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamical Behaviour of a Nutrient-Plankton Model with Holling Type IV, Delay, and Harvesting

Meng

Jiao-guo

Huo

2018

Discrete Dynamics in Nature and Society

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“…And due to the seasonal and economic reasons, periodic harvesting is an effective harvesting strategy for the infrequent harvesting. This periodic harvesting can be described by impulsive differential equations [9][10][11][12][13][14][15][16]. There are some papers studying the effects of periodic impulse harvesting strategy to the species resource.…”

Section: Introductionmentioning

confidence: 99%

Periodic solution and control optimization of a prey-predator model with two types of harvesting

et al. 2018

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In this work, a prey-predator model with both state-dependent impulsive harvesting and constant rate harvesting is investigated, where the replenishment rate of prey and the harvesting rate are linearly related with the selected threshold. By first using the successor function method and differential equation geometry theory, the existence, uniqueness and asymptotic stability of the order-1 periodic solution are discussed. And then numerical simulations with an example are given to illustrate the feasibility of the theorem-related results. Moreover, in order to increase the total profit, the optimization strategy is presented and the optimal threshold is obtained.

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“…Extending the work of [43], Saha and Bandyopadhyay [44] and Rahim and Imran [45] have discussed the stability of bifurcating periodic orbits and switching phenomenon of the plankton system with respect to time delay "τ " respectively. Wang et al [46] recently considered the similar model in [44,45] with the idea of selective harvesting of zooplankton population and discussed the effect of harvesting on the delayed phytoplankton-zooplankton system. The dynamics of the plankton system with single delay has been studied by many authors but the effect of multiple delays on the plankton system is rare.…”

mentioning

confidence: 99%

Bifurcation behaviors analysis of a plankton model with multiple delays

Sharma¹,

Sharma

Agnihotri³

2016

Int. J. Biomath.

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A mathematical model describing the dynamics of toxin producing phytoplanktonzooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multiple delays; one due to gestation period in the growth of phytoplankton population and second due to the delay in toxin liberated by TPP. It is established that a sequence of Hopf bifurcations occurs at the interior equilibrium as the delay increases through its critical value. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined using the theory of normal form and center manifold. Meanwhile, effect of toxin on the stability of delayed plankton system is also established numerically. Finally, numerical simulations are carried out to support and supplement the analytical findings. less affected by human activities, and often cause economic losses, such as those in aquaculture and tourism. In the last decades of the past century, a global increase of toxin producing phytoplankton (TPP) blooms has been observed. Number of HABs, economic losses from them, the types of resources affected and number of toxic species have also increased dramatically in recent years [1][2][3]. The adverse effect of HABs is clear, but the control of such a problem is under investigation. HABs is known to have a negative impact on the zooplankton and on the species feeding on them such as large fish in the ocean which ultimately affects the human food chain. The different modes and mechanisms by which harmful phytoplankton species can cause mortality, physiological impairment, or other negative in situ effects have been discussed in [2]. The results presented by Webb et al. [8], Gakkhar and Naji [9], Henson and coworkers [10] and Upadhyay et al. [11] have also contributed to the clarification of the mechanism of HAB's. Several researchers have tried to explain the dynamics of plankton blooms by focusing on different factors such as nutrient upwelling [4,5], spatial patchiness [6] and species diversity [7]. The impact of zooplankton grazing on HAB is an important aspect of the plankton ecology. If zooplankton community's grazing impact on initial stages of a HAB is sufficiently high, then a bloom does not develop [12]. A number of field studies [13-15] as well as laboratory experiments have established that toxicity has a great impact on phytoplankton-zooplankton interaction dynamics. Some studies reveal that many copepods experience decreased grazing and fecundity in the presence of HAB species [16]. It is also well known that zooplankton population tries to avoid the areas where the concentration of phytoplankton is very large. The reason may be either dense concentration of phytoplankton or the effect of toxic substances released by them. Buskey and Hyatt [17] have shown in their field study that macroand meso-zooplankton populations are reduced during the blooms of chrysophyte Aureococcus anophagefferens. Many researchers have proposed different deterministic models on HAB...

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Stability and global Hopf bifurcation in toxic phytoplankton–zooplankton model with delay and selective harvesting

Cited by 56 publications

References 25 publications

Dynamical Behaviour of a Nutrient-Plankton Model with Holling Type IV, Delay, and Harvesting

Dynamical Behaviour of a Nutrient-Plankton Model with Holling Type IV, Delay, and Harvesting

Periodic solution and control optimization of a prey-predator model with two types of harvesting

Bifurcation behaviors analysis of a plankton model with multiple delays

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