Chaotic behavior of predator-prey model with group defense and non-linear harvesting in prey

Kumar, Sachin; Kharbanda, Harsha

doi:10.1016/j.chaos.2018.12.011

Cited by 48 publications

(24 citation statements)

References 25 publications

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“…In this article, we mainly focus on the existence, uniqueness and boundness of solution, the stability of equilibrium point and the existence of Hopf bifurcation of fractional order predator-prey model. The research method and theoretical findings are different from those in 11 . According to this viewpoint, the results of this paper complete the works of Kumar and Kharbanda 11 .…”

mentioning

confidence: 81%

“…The research method and theoretical findings are different from those in 11 . According to this viewpoint, the results of this paper complete the works of Kumar and Kharbanda 11 .…”

mentioning

confidence: 81%

“…All the parameters γ i (i = 1, 2, 3, 4), κ are positive. Considering that the harvesting play an important role in describing the evolution process of a population, Kumar and Kharbanda 11 introduced the Michaelis-Menten type harvesting into predator-prey model (1.1). Then they established the following predator-prey model with group defense and Michaelis-Menten type harvesting:…”

Section: Dynamics For a Fractional-order Predator-prey Model With Gromentioning

confidence: 99%

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Dynamics for a fractional-order predator-prey model with group defense

Tang

2020

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mentioning

confidence: 81%

“…The research method and theoretical findings are different from those in 11 . According to this viewpoint, the results of this paper complete the works of Kumar and Kharbanda 11 .…”

mentioning

confidence: 81%

Section: Dynamics For a Fractional-order Predator-prey Model With Gromentioning

confidence: 99%

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Dynamics for a fractional-order predator-prey model with group defense

Tang

2020

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“…To tackle that type of situation, many researchers worked on a predator-prey system with harvesting effects (15) (16) (17) (18). Basically researchers used three types of harvesting and the classification is constant harvesting (19), proportional harvesting (20) and non-linear harvesting (21). In this paper we consider harvested predator and prey population.…”

Section: Introductionmentioning

confidence: 99%

Mathematical study on a dynamical predator-prey model with constant prey harvesting and proportional harvesting in predator

Mortuja¹,

Chaube²,

Kumar³

2021

Preprint

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A dynamical predator-prey model with constant prey harvesting, proportional harvesting in predator has been studied. The square root func- tional response also has been incorporated in the system to describe the prey herd behaviour, assuming the average handling time is zero. The existence and the local stability of equilibria of the system have been discussed. It is examined that, two types of bifurcation occur in the system. The two types of bifurcations have been analyzed, and it has been found by analyzing the saddle-node bifurcation that, there is a maximum sustainable yield. It is ob- served that if harvesting rate is greater than the maximum sustainable yield, the prey population abolish from the system and then extinction of the preda- tor population happen. But if harvesting rate is lesser than the maximum sustainable yield, the extinction of the prey population can not be possible. By analyzing the Hopf bifurcation, it is obtained that, there exists an unstable limit cycle around the interior equilibrium point. Several numerical simulations are performed to check the results.

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“…In recent years, time-delay differential equation and difference equation models have been studied by many authors (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]) as they are useful tools for modeling a wide variety of systems in areas including traditional areas such as physics and engineering and newer areas such as disease transmission, medical research, optimal drug treatment, bioeconomics, agriculture, finance, insurance, and environmental protection. In many of these time-delay models, bifurcations occur as values of parameters are changed.…”

Section: Introductionmentioning

confidence: 99%

Andronov–Hopf and Neimark–Sacker bifurcations in time-delay differential equations and difference equations with applications to models for diseases and animal populations

2020

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In many areas, researchers might think that a differential equation model is required, but one might be forced to use an approximate difference equation model if data is only available at discrete points in time. In this paper, a detailed comparison is given of the behavior of continuous and discrete models for two representative time-delay models, namely a model for HIV and an extended logistic growth model. For each model, there are seven different time-delay versions because there are seven different positions to include time delays. For the seven different time-delay versions of each model, proofs are given of necessary and sufficient conditions for the existence and stability of equilibrium points and for the existence of Andronov-Hopf bifurcations in the differential equations and Neimark-Sacker bifurcations in the difference equations. We show that only five of the seven time-delay versions have bifurcations and that all bifurcation versions have supercritical limit cycles with one having a repelling cycle and four having attracting cycles. Numerical simulations are used to illustrate the analytical results and to show that critical times for Neimark-Sacker bifurcations are less than critical times for Andronov-Hopf bifurcations but converge to them as the time step of the discretization tends to zero.

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Chaotic behavior of predator-prey model with group defense and non-linear harvesting in prey

Cited by 48 publications

References 25 publications

Dynamics for a fractional-order predator-prey model with group defense

Dynamics for a fractional-order predator-prey model with group defense

Mathematical study on a dynamical predator-prey model with constant prey harvesting and proportional harvesting in predator

Andronov–Hopf and Neimark–Sacker bifurcations in time-delay differential equations and difference equations with applications to models for diseases and animal populations

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