A fractional calculus approach to Rosenzweig-MacArthur predator-prey model and its solution

Mondal, Shuvojit; Bairagi, Nandadulal; Lahiri, Abhijit

doi:10.20454/jmmnm.2017.1097

Cited by 6 publications

(6 citation statements)

References 11 publications

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“…It also has applications in other fields of science and engineering [7,8,9,10,11]. Some recent studies discuss about the approximate solution of nonlinear fractional-order differential population models [1,12] and some others study the qualitative behavior of nonlinear interactions of biological systems [13,14,15,16,17]. However, existence and proof of Hopf bifurcation, that causes oscillations in population densities due to fractional-order, is rare in the fractionalorder population model.…”

Section: Introductionmentioning

confidence: 99%

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Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Mondal¹,

Biswas

Bairagi

2020

J. Appl. Math. Comput.

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This paper is focused on local and global stability of a fractional-order predator-prey model with habitat complexity constructed in the Caputo sense and corresponding discrete fractional-order system. Mathematical results like positivity and boundedness of the solutions in fractional-order model is presented. Conditions for local and global stability of different equilibrium points are proved. It is shown that there may exist fractional-order-dependent instability through Hopf bifurcation for both fractional-order and corresponding discrete systems. Dynamics of the discrete fractional-order model is more complex and depends on both step length and fractional-order. It shows Hopf bifurcation, flip bifurcation and more complex dynamics with respect to the step size. Several examples are presented to substantiate the analytical results.This model says that the prey population x grows logistically with intrinsic growth rate r to its carrying capacity K. Predator y captures the prey at a maximum rate α in absence of any habitat complexity (c = 0). In presence of complexity, predation rate decreases to α(1 − c), where the dimensionless parameter c is called the degree or strength of complexity. The value of c ranges from 0 to 1. In particular, c = 0.4 implies that predation rate decreases by 40% due to habitat complexity. If c = 0, i.e. if there is no habitat complexity then the system (1) reduces to well known Rosenzweig-MacArthur model [25]. However, if c = 1 then y → 0 as t → ∞ and the prey population grows logistically to its maximum value K. The parameter θ (0 < θ < 1) is the conversion efficiency, measuring the number of newly born predators for each captured prey and d is the death rate of predator. All parameters are assumed to be positive. For construction and more explanation of the model, readers are referred to [26].Considering the fractional derivatives in the sense of Caputo, we have the following fractional-order model corresponding to the integer order model (1): Existence and uniquenessHere we study the existence and uniqueness of the solution of our system (2). We have the following Lemma due to Li et al [30]. Lemma 2.3 Consider the system . If f (t, x) satisfies the locally Lipschitz condition with respect to x then there exists a unique solution of the above system on [t 0 , ∞) × Ω.We study the existence and uniqueness of the solution of system (2) in the region Ω × [0, T ], where Ω = {(x, y) ∈ ℜ 2 | max{|x|, |y|} ≤ M}, T < ∞ and M is large. Denote X = (x, y),X = (x,ȳ). Consider a mapping H : Ω → ℜ 2 such that H(X) = (H 1 (X), H 2 (X)), where

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Section: Introductionmentioning

confidence: 99%

“…both roots of(12) are negative real or complex conjugate with negative real parts. Hence | arg(ξ 1,2 ) |> mπ 2 , ∀m ∈ (0, 1].…”

mentioning

confidence: 99%

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Mondal¹,

Biswas

Bairagi

2020

J. Appl. Math. Comput.

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“…Secondly and more importantly, fractional order derivatives not only depend on the local conditions but also on the history of the function [14] and, therefore, fractional derivatives have become an efficient tool for those systems, where the consideration of memory or hereditary properties of the function is essential to represent the system, e.g., in the case of biological systems. In the last two decades, fractional order calculus has found its many applications in biological sciences [15,16,17,18,19,20,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

Study of a discretized fractional-order eco-epidemiological model with prey infection

Mondal¹,

Cao²,

Bairagi³

2020

Fractional Differential Calculus

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In this paper, an attempt is made to understand the dynamics of a three-dimensional discrete fractional-order eco-epidemiological model with Holling type II functional response. We first discretize a fractional-order predator-prey-parasite system with piecewise constant arguments and then explore the system dynamics. Analytical conditions for the local stability of different fixed points have been determined using the Jury criterion. Several examples are given to substantiate the analytical results. Our analysis shows that stability of the discrete fractional order system strongly depends on the step-size and the fractional order. More specifically, the critical value of the step-size, where the switching of stability occurs, decreases as the order of the fractional derivative decreases. Simulation results explore that the discrete fractional-order system may also exhibit complex dynamics, like chaos, for higher step-size.

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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%

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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A fractional calculus approach to Rosenzweig-MacArthur predator-prey model and its solution

Cited by 6 publications

References 11 publications

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Local and global dynamics of a fractional-order predator–prey system with habitat complexity and the corresponding discretized fractional-order system

Study of a discretized fractional-order eco-epidemiological model with prey infection

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

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