Stability analysis of coexistence of three species prey–predator model

Panja, Prabir; Mondal, Shyamal Kumar

doi:10.1007/s11071-015-1997-1

Cited by 59 publications

(19 citation statements)

References 22 publications

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“…The dynamical behaviors of three species such as toxin‐producing Phytoplankton, Zooplankton, and Fish in a fishery system is studied with Holling type II functional response (Panja & Mondal, 2015). Further they extended predator–prey interaction among Phytoplankton, Zooplankton, and Fish in the presence of Environmental toxin and includes harvesting on the Fish (Panja et al, 2017).…”

Section: Introductionmentioning

confidence: 99%

Coexistence and harvesting optimal policy in three species food chain model with general Holling type functional response

Dawed

Kebedow

2021

Natural Resource Modeling

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In this paper, we have discussed harvesting of prey and intermediate predator species. Both are subjected to Holling type I–V functional response. Conditions for local and global stability of the nonnegative equilibria are verified. The permanent coexistence criterion of the model system and existence of optimal equilibrium solution of the control problem are demonstrated. Maximum sustainable yield and maximal net present revenue are determined. To confirm analytical results, numerical solution has been carried out using the Matlab™ ODE solver ODE45 and the simulations show the model system reveals complex behavior (such as oscillations), which reflects the real situation. Recommendations for Resource Managers From our investigation of this study, we recommend to the management the following points. Coexistence of the three species with harvesting, or persistence of the model system is possible provided that good management(treatment) of some factors (such as harvesting rate, growth rate of species, etc.) are performed. The dynamics reveals complex behavior (such as oscillations), which reflects the real situation and it is sensitive to the above factors, especially the growth rate of the intermediate predator. The policy makers should recommend the optimal effort h * to be applied and the optimal stock ( x * , y * , z * ) to harvest. This indicates that maximum profit will attain while securing sustainability of the three species in the ecosystem.

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Section: Introductionmentioning

confidence: 99%

Coexistence and harvesting optimal policy in three species food chain model with general Holling type functional response

Dawed

Kebedow

2021

Natural Resource Modeling

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“…So, the study of the dynamics of Fish in the presence of Phytoplankton and Zooplankton is very much important research topics. There exists very few research articles [4,14,21,22,27,36] in the literature on Phytoplankton-Zooplankton-Fish dynamics. But till now, several unexplored dynamics of Phytoplankton-Zooplankton-Fish to be investigated.…”

Section: Introductionmentioning

confidence: 99%

A Novel Prey-Predator Quadratic Harvesting Model via Optimal Control Theory and Hopf Bifurcation

Panja¹,

Jana²

2020

Nep J. Math. Sci.

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In this investigation, a predator-prey interaction model among Phytoplankton, Zooplankton and Fish has been developed. In the absence of Zooplankton and Fish, it is assumed that Phytoplankton grows logistically. It is assumed that Zooplankton consumes Phytoplankton and Fish consumes Phytoplankton as well as Zooplankton. Holling type I & II functional responses have been considered to formulate the our proposed model. It is considered that Phytoplankton releases some toxin in the aquatic environment which makes some death in Zooplankton population. Quadratic harvesting is considered on Fish species. Boundedness of the solution of our proposed model has also been studied. Local stability of the system around each equilibrium point has been investigated. Also, the global stability of the interior equilibrium point has been studied. Existence condition of Hopf bifurcation of our proposed system has been studied. It is found that half saturation constant (α) can change the system dynamics. It is also found that the harvesting rate of Fish (E) and consumption rate of Zooplankton (γ1) has a significant role in the stability of the system. Again, it is found that the harvesting of Fish species will be increased if the selling price of Fish (p) and the annual discount (δ1) of Fish production cost increases. It is also found that the optimal harvesting rate of Fish decreases due to the increase of cost (c) of harvesting of Fish. Finally, some numerical simulation results have been presented to verify our analytical findings.

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“…As well known, mathematical models describing the plankton dynamics have played an important role in understanding the various mechanisms involved in toxin-producing phytoplankton. ere are many scienti c works have been carried out to investigate the e ects of toxin-producing phytoplankton on plankton ecosystems [1][2][3][4][5][6][7]. For example, Upadhyay and Craniopathy [1] proposed three species food chain model with di erent functional forms to describe the liberation of toxin.…”

Section: Introductionmentioning

confidence: 99%

Stationary Distribution and Periodic Solution of Stochastic Toxin-Producing Phytoplankton–Zooplankton Systems

Wei

2020

Complexity

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In this paper, we investigate the dynamics of autonomous and nonautonomous stochastic toxin-producing phytoplankton–zooplankton system. For the autonomous system, we establish the sufficient conditions for the existence of the globally positive solution as well as the solution of population extinction and persistence in the mean. Furthermore, by constructing some suitable Lyapunov functions, we also prove that there exists a single stationary distribution which is ergodic, what is more important is that Lyapunov function does not depend on existence and stability of equilibrium. For the nonautonomous periodic system, we prove that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, some numerical simulations are introduced to illustrate our theoretical results. The results show that weaker white noise and/or toxicity will strengthen the stability of system, while stronger white noise and/or toxicity will result in the extinction of one or two populations.

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Stability analysis of coexistence of three species prey–predator model

Cited by 59 publications

References 22 publications

Coexistence and harvesting optimal policy in three species food chain model with general Holling type functional response

Coexistence and harvesting optimal policy in three species food chain model with general Holling type functional response

A Novel Prey-Predator Quadratic Harvesting Model via Optimal Control Theory and Hopf Bifurcation

Stationary Distribution and Periodic Solution of Stochastic Toxin-Producing Phytoplankton–Zooplankton Systems

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