Global stability and bifurcation of time delayed prey–predator system incorporating prey refuge

Jana, Soovoojeet; Chakraborty, Milon; Chakraborty, Kunal; Kar, Tapan Kumar

doi:10.1016/j.matcom.2012.10.003

Cited by 57 publications

(30 citation statements)

References 29 publications

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“…In this section, it is investigated the dynamics of discretized fractional-order prey-predator model (12). The Jacobian matrix J * of the system (12) at any equilibrium point (x * , y * ) is given by…”

Section: Dynamical Behavior Of Discretized Fractional Ordermentioning

confidence: 99%

“…In view of the biological meaning, we only consider the stability for all nonnegative equilibria of model (12). Then the following theorem as follows:…”

Section: Dynamical Behavior Of Discretized Fractional Ordermentioning

confidence: 99%

“…[7][8][9][10] The delay due to maturation time of prey, gestation time of predator, and hunting can induce periodic solutions as well as chaos in prey-predator model. [11][12][13][14][15] Hence, the exploration of prey-predator model with several natural factors is getting more popularity because of interesting biological phenomenon.…”

Section: Introductionmentioning

confidence: 99%

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Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

Singh

Elsadany

Elsonbaty

2019

Math Methods in App Sciences

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A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.

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Section: Dynamical Behavior Of Discretized Fractional Ordermentioning

confidence: 99%

“…In view of the biological meaning, we only consider the stability for all nonnegative equilibria of model (12). Then the following theorem as follows:…”

Section: Dynamical Behavior Of Discretized Fractional Ordermentioning

confidence: 99%

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Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

Singh

Elsadany

Elsonbaty

2019

Math Methods in App Sciences

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“…The predator population, y(t) is a generalist predator and is modeled as modified version of the Leslie-Gower formulation with Holling type II functional response. Predation process follows Type II ratio-dependent functional response [24,25,36,41] because a ratio-dependent prey-predator model does not show the so called paradox of enrichment [41][42][43] and biological control paradox [44]. The model system is written as…”

Section: The Mathematical Modelmentioning

confidence: 99%

“…This model predicted a minor effect of the refuge on the prey density at equilibrium. Jana et al [36] described a time delayed prey-predator system incorporating prey refuge with Holling type II functional response. Ko and Ryu [37] investigated the asymptotic behavior of spatially inhomogeneous solution and local existence of periodic solution under the homogeneous Neumann boundary condition in a model with Holling type II functional response incorporating a prey refuge.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of generalist predator in a stochastic environment: Effect of delayed growth and prey refuge

Jana

Agrawal

Upadhyay

2015

Applied Mathematics and Computation

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A delayed prey–predator model with Crowley–Martin‐type functional response including prey refuge

Maiti

Dubey²,

Tushar³

2017

Math Methods in App Sciences

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In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin‐type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.

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Global stability and bifurcation of time delayed prey–predator system incorporating prey refuge

Cited by 57 publications

References 29 publications

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

Dynamics of generalist predator in a stochastic environment: Effect of delayed growth and prey refuge

A delayed prey–predator model with Crowley–Martin‐type functional response including prey refuge

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