Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge

To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management.

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“…Lemma (Li et al) Let us consider a continuous function f ( t ) on [ t 0 , ∞ ), which satisfy

D_{t}^{α} f false(t false) \leq - λ f false(t false) + μ

f false(t_{0} false) = f_{t_{0}}

, where 0 < α < 1, ( λ , μ ) ∈ ℜ 2 , λ ≠ 0 and t 0 ≥ 0 is the initial time. Then its solution has the form

f (t) \leq (f_{t_{0}} - \frac{μ}{λ}) E_{α} [- λ {(t - t_{0})}^{α}] + \frac{μ}{λ} .

Applying this lemma and letting ϵ →0, we get

N (t) \leq (N (0) - \frac{l}{ν}) E_{α} [- ν t^{α}] + \frac{l}{ν},

\leq N (0) E_{α} [- ν t^{α}] + \frac{l}{ν} (1 - E_{α} [- ν t^{α}]) .

Therefore, both the solutions x ( t ) and y ( t ) are uniformly bounded in the region

normalΩ = false{false(x, y false) \in ℜ_{+}^{2} false| N false(t false) \leq \frac{l}{ν} + ϕ,

for any ϕ > 0}.…”

Section: Well Posednessmentioning

confidence: 99%

Dynamical study of fractional order differential equations of predator‐pest models

Mandal

Sha

Chattopadhyay

2019

Math Methods in App Sciences

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“…For brevity, we here mention only some review papers and books [6,7,8,9,10]. Fractional order models have also been used to understand the dynamics of interacting populations [11,12,13,14,15,16,17,18]. In recent past, Aziz-Alaoui [19] studied the following three-dimension coupled nonlinear autonomous system of integer order differential equations to understand the underlying dynamics of food chain model:…”

Section: Introductionmentioning

confidence: 99%

Global stability of a Leslie-Gower-type fractional order tritrophic food chain model

Mondal¹,

Bairagi²,

'Guerekata³

2019

Fractional Differential Calculus

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Recently, the dynamical behaviors of a fractional order three species food chain model was studied by Alidousti and Ghahfarokhi (Nonlinear Dynamics, doi: org/10.1007/s11071-018-4663-6, 2018). They proved both the local and global asymptotic stability of all equilibrium points except the interior one. This work extends their work and gives proof of both the local and global stability analysis of the interior equilibrium point. Numerical examples are also provided to substantiate the analytical findings.Research of NB is supported by JU-RUSA 2.0.

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“…Recently, Yang et al proposed new and interesting fractional derivatives without singular kernel [11,12], and analytic and computational methods for solving nonlinear fractional-order partial differential equations [13,14], which can be effectively used in the modeling of the fractional-order heat flow [15,16]. In the last decade, the dynamics of fractional-order prey-predator systems, such as stability, bifurcations, chaos, have been researched by many investigators [17][18][19][20][21][22][23][24][25][26]. In particular, Li et al [19] investigated the global asymptotic stability of predator-extinction equilibrium point and coexistence equilibrium point for a fractional-order predator-prey model incorporating a prey refuge.…”

Section: Introductionmentioning

confidence: 99%

“…In the last decade, the dynamics of fractional-order prey-predator systems, such as stability, bifurcations, chaos, have been researched by many investigators [17][18][19][20][21][22][23][24][25][26]. In particular, Li et al [19] investigated the global asymptotic stability of predator-extinction equilibrium point and coexistence equilibrium point for a fractional-order predator-prey model incorporating a prey refuge. Panja [17] studied the stability and dynamics of a three-species predator-prey model with prey, middle predator and top predator.…”

Section: Introductionmentioning

confidence: 99%

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Periodic pulse control of Hopf bifurcation in a fractional-order delay predator–prey model incorporating a prey refuge

Liu

Fang

2019

Adv Differ Equ

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This paper is concerned with periodic pulse control of Hopf bifurcation for a fractional-order delay predator-prey model incorporating a prey refuge. The existence and uniqueness of a solution for such system is studied. Taking the time delay as the bifurcation parameter, critical values of the time delay for the emergence of Hopf bifurcation are determined. A novel periodic pulse delay feedback controller is introduced into the first equation of an uncontrolled system to successfully control the delay-deduced Hopf bifurcation of such a system. Since the stability theory is not well-developed for nonlinear fractional-order non-autonomous systems with delays, we investigate the periodic pulse control problem of the original system by a semi-analytical and semi-numerical method. Specifically, the stability of the linearized averaging system of the controlled system is first investigated, and then it is shown by numerical simulations that the controlled system has the same stability characteristics as its linearized averaging system. The proposed periodic pulse delay feedback controller has more flexibility than a classical linear delay feedback controller guaranteeing the control effect, due to the fact that the pulse width in each control period can be flexibly selected.

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Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge

Cited by 287 publications

References 28 publications

Dynamical study of fractional order differential equations of predator‐pest models

Dynamical study of fractional order differential equations of predator‐pest models

Global stability of a Leslie-Gower-type fractional order tritrophic food chain model

Periodic pulse control of Hopf bifurcation in a fractional-order delay predator–prey model incorporating a prey refuge

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