Homotopy perturbation method applied to the solution of fractional Lotka-Volterra equations with variable coefficients

Cui, Zhoujin; Yang, Zuodong

doi:10.20454/jmmnm.2014.314

Cited by 13 publications

(19 citation statements)

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“…(i) If D(F) is positive then all the roots of (12) are real and distinct. If not, let us assume that F( ) = 0 has one real root 1 and another two complex conjugate roots 2 and 3 . In terms of the roots, the discriminant of F( ) can be written as 13 D…”

Section: Dynamical Behaviormentioning

confidence: 99%

“…In recent past, fractional order differential equations have been used in several biological systems to explore the underlying dynamics. [1][2][3] Here, we assume an ecological system where a prey population grows logistically and a predator population feeds on this prey population. Now, assume that the prey population is infected by some microparasites.…”

Section: Introductionmentioning

confidence: 99%

“…Note that it is an integer order system of differential equations and its dynamics was studied by Chattopadhyay and Bairagi. 5 Considering the fractional derivatives in the sense of Caputo derivative, and assuming 0 < ≤ 1, we have the following fractional order eco-epidemiological model corresponding to the model (1):…”

Section: Introductionmentioning

confidence: 99%

“…where c 0 D t is the Caputo fractional derivative. The main advantage of Caputo approach is that the initial conditions for the fractional differential equations with Caputo derivatives takes the similar form as for integer-order differential equations, [1,6] i.e., it has advantage of defining integer order initial conditions for fractional order differential equations. We analyze system (2) with the initial conditions…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a fractional order eco‐epidemiological model with prey infection and type 2 functional response

Mondal

Lahiri

Bairagi

2017

Math Methods in App Sciences

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In this paper, we introduce fractional order into an ecoepidemiological model, where predator consumes disproportionately large number of infected preys following type 2 response function. We prove different mathematical results like existence, uniqueness, nonnegativity, and boundedness of the solutions of fractional order system. We also prove the local and global stability of different equilibrium points of the system. The results are illustrated with several examples. KEYWORDS ecological model, epidemic model, fractional order, global stability, local stability 6776

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Section: Dynamical Behaviormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analysis of a fractional order eco‐epidemiological model with prey infection and type 2 functional response

Mondal

Lahiri

Bairagi

2017

Math Methods in App Sciences

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“…where c 0 D m t is the Caputo fractional derivative with fractional-order m (0 < m ≤ 1). The main advantage of Caputo's approach is that the initial conditions for the fractional differential equations with Caputo derivatives takes the similar form as for integer-order differential equations [27,28], and thus takes the advantage of defining integer order initial conditions for fractional-order differential equations. We analyze system (2) with the initial conditions x(0) > 0, y(0) > 0.…”

Section: Introductionmentioning

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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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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A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods

Kumar

Agarwal

et al. 2020

Math Methods in App Sciences

279

123

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The Lotka-Volterra (LV) system is an interesting mathematical model because of its significant and wide applications in biological sciences and ecology. A fractional LV model in the Caputo sense is investigated in this paper. Namely, we provide a comparative study of the considered model using Haar wavelet and Adams-Bashforth-Moulton methods. For the first method, the Haar wavelet operational matrix of the fractional order integration is derived and used to solve the fractional LV model. The main characteristic of the operational method is to convert the considered model into an algebraic equation which is easy to solve.To demonstrate the efficiency and accuracy of the proposed methods, some numerical tests are provided. KEYWORDS Adams-Bashforth-Moulton method, fractional LV model, Haar wavelet method, operational matrix MSC CLASSIFICATION 26A33; 34A08; 34A34 Math Meth Appl Sci. 2020;43:5564-5578. wileyonlinelibrary.com/journal/mma

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Homotopy perturbation method applied to the solution of fractional Lotka-Volterra equations with variable coefficients

Cited by 13 publications

References 0 publications

Analysis of a fractional order eco‐epidemiological model with prey infection and type 2 functional response

Analysis of a fractional order eco‐epidemiological model with prey infection and type 2 functional response

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

A study of fractional Lotka‐Volterra population model using Haar wavelet and Adams‐Bashforth‐Moulton methods

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