Dynamical analysis for a fractional-order prey–predator model with Holling III type functional response and discontinuous harvest

Xie, Yingkang; Wang, Zhen; Meng, Bo; Huang, Xia

doi:10.1016/j.aml.2020.106342

Cited by 28 publications

(12 citation statements)

References 11 publications

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“…The fractional-order models are also well-liked due to their capability in providing an exact description of different nonlinear phenomena [32]. In recent years, the development of fractional-order models grows rapidly and becomes popular in studying the dynamical behavior of predator-prey interaction [17,[33][34][35][36][37][38]. It has been shown that the order of the fractional derivate significantly affects the dynamical behavior of the models, which is in contrast to the first-order derivative models that depend only on the values of parameters.…”

Section: Variables and Parameters Description X(t)mentioning

confidence: 99%

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Panigoro

Suryanto

Kusumawinahyu

et al. 2020

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The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator.

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Section: Variables and Parameters Description X(t)mentioning

confidence: 99%

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Panigoro

Suryanto

Kusumawinahyu

et al. 2020

Axioms

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“…Recently, a series of mathematical models has been developed to better understand the dynamics of the predator-prey system with different kinds of functional responses [6,[16][17][18][19][20][21][22][23][24]. Dalziel et al [16] studied a modified Holling type-II predator-prey model, based on the premise that the search rate of predators is dependent on the prey density, rather than constant.…”

Section: Introductionmentioning

confidence: 99%

“…Khajanchi [28] studied a predator-prey model with age-structured incorporating Beddington-DeAngelis-type response function, in which the author performed global stability analysis. Xie et al [20] studied a predator-prey system by using a system of fractional order differential equations with Holling type-III response function and incorporate the effect of discontinuous harvesting. Predatorprey system has been investigated by introducing discrete time delay with disease in the predator by Huang et al [21].…”

Section: Introductionmentioning

confidence: 99%

Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

et al. 2020

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In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

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“…For example, in [39], a new SIS epidemic model considering the infection rates of multiple edges interfere with each other on complex networks was proposed. In [40], a class of fractional-order prey-predator models with Holling III type functional response and discontinuous harvest is investigated.…”

Section: Introduction Echinococcosis Is Mainly a Zoonotic Parasitic mentioning

confidence: 99%

“…In this paper, motivated by the results given in [2,35,34,38,39,40], we use the compartment modelling method (see [12,25]) to establish a discrete-time dynamical model for the transmission of echinococcosis. The technical contribution in this paper is that we use Euler backward difference method to transform the partial differential equations into discrete equations, thus simplifying the solution process.…”

Section: Introduction Echinococcosis Is Mainly a Zoonotic Parasitic mentioning

confidence: 99%

The threshold dynamics of a discrete-time echinococcosis transmission model

Cui-cui¹,

Zhou²,

Teng³

et al. 2020

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Dynamical analysis for a fractional-order prey–predator model with Holling III type functional response and discontinuous harvest

Cited by 28 publications

References 11 publications

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

A Rosenzweig–MacArthur Model with Continuous Threshold Harvesting in Predator Involving Fractional Derivatives with Power Law and Mittag–Leffler Kernel

Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

The threshold dynamics of a discrete-time echinococcosis transmission model

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