Dynamical analysis of a delayed ratio-dependent Holling–Tanner predator–prey model

Saha, Tapan Kumar; Chakrabarti, C. G.

doi:10.1016/j.jmaa.2009.03.072

Cited by 43 publications

(14 citation statements)

References 22 publications

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“…Delay differential equation models are capable of generating rich, more effective and accurate dynamics compared to ordinary differential equation models when it is necessary to capture oscillatory dynamics [6,16,18]. Recently, many theoreticians and experimentalists have discussed dynamical behavior of prey-predator system with Holling-Tanner functional response, it reveals that time delay may cause the loss of stability and other complicated dynamical behavior such as the periodic structure and bifurcation phenomenon [29,31,32,35,41,44,47]. By considering time delay for both prey and predator population, system (3) is extended in [47], which is as follows:…”

Section: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ẋ (T) = X(t )(1 − X(t )) − X(t )Y(t) X(t ) + αYmentioning

confidence: 99%

Modeling and analysis in a prey–predator system with commercial harvesting and double time delays

Liu¹,

Lü²,

Zhang³

et al. 2016

Applied Mathematics and Computation

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Section: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ẋ (T) = X(t )(1 − X(t )) − X(t )Y(t) X(t ) + αYmentioning

confidence: 99%

Modeling and analysis in a prey–predator system with commercial harvesting and double time delays

Liu¹,

Lü²,

Zhang³

et al. 2016

Applied Mathematics and Computation

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“…In these models, x(t) and (t) denote the density of the prey and predator populations, respectively. Saha and Chakrabarti 55 proposed the following ratio-dependent Holling-Tanner delayed model where the time delay is due to negative feedback of the predator given by…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Dynamics of a delayed predator‐prey model with Allee effect and Holling type II functional response

Anacleto

Vidal

2020

Math Methods in App Sciences

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In this paper, a delayed with Holling type II functional response (Beddington-DeAngelis) and Allee effect predator-prey model is considered.The growth of the prey is affected by the parameter M, which defines the Allee effect. In addition, the delay also influences the logistic growth of the prey, which can be interpreted as the maturity time or the gestation period. In the study of the characteristic equation, we observe that the delay also depends on the parameter M, which affects the dynamics in the prey population. Considering the delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. On the other hand, we find that the system can also suffer a Hopf bifurcation in the positive equilibrium when the delay passes through a sequence of critical values. In particular, we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, an explicit algorithm is provided applying the normal form theory and center manifold reduction for the functional differential equations. Finally, numerical simulations that support the theoretical analysis are included. KEYWORDSAllee effect, delay predator-prey model, equilibrium point, Holling type II functional response, Hopf bifurcation, stability/unstablity MSC CLASSIFICATION 34K17; 34K18; 34K20Math Meth Appl Sci. 2020;43:5708-5728. wileyonlinelibrary.com/journal/mma

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“…There is extensive literature related to these topics for ordinary differential equation models (see [1]- [6] and the references cited therein). We know that more realistic prey-predator models were introduced by Holling suggesting three kinds of functional responses for different species to model the phenomena of predation [3].…”

Section: Introductionmentioning

confidence: 99%

Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay

Du¹,

Duan²,

Liao³

2014

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This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with Beddington-DeAngelis functional response and a discrete time delay. By introducing a partial quasi-monotonicity condition and constructing a pair of upper-lower solutions, we establish the existence of traveling wave solutions. Moreover, a numerical simulation is carried out to illustrate the theoretical results.

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Dynamical analysis of a delayed ratio-dependent Holling–Tanner predator–prey model

Cited by 43 publications

References 22 publications

Modeling and analysis in a prey–predator system with commercial harvesting and double time delays

Modeling and analysis in a prey–predator system with commercial harvesting and double time delays

Dynamics of a delayed predator‐prey model with Allee effect and Holling type II functional response

Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay

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