Numerical solution of time-fractional three-species food chain model arising in the realm of mathematical ecology

2020 ◽  
Vol 13 (02) ◽  
pp. 2050011 ◽  
Author(s):  
Ved Prakash Dubey ◽  
Rajnesh Kumar ◽  
Devendra Kumar
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Time Derivative ◽  
Chain Model ◽  
Homotopy Analysis ◽  
Food Chain Model ◽  
Derivatives Of

This research paper implements the fractional homotopy analysis transform technique to compute the approximate analytical solution of the nonlinear three-species food chain model with time-fractional derivatives. The offered technique is a fantastic blend of homotopy analysis method (HAM) and Laplace transform (LT) operator and has been used fruitfully in the numerical computation of various fractional differential equations (FDEs). This paper involves the fractional derivatives of Caputo style. The numerical solutions of this selected fractional-order food chain model are evaluated by making use of the associated initial conditions. It is revealed by the adopting procedure that the more desirable estimation of the solution can be easily acquired through the calculation of some number of iteration terms only — a fact which authenticates the easiness and soundness of the suggested hybrid scheme. The variations of fractional order of time derivative on the solutions for different specific cases have been depicted through graphical presentations. The outcomes demonstrated through the graphs expound that the adopted scheme is very fantastic and accurate.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

scholarly journals ON NONLINEAR FRACTIONAL-ORDER MATHEMATICAL MODEL OF FOOD-CHAIN

Fractals ◽  
2021 ◽  
Author(s):  
KOTTAKKARAN SOOPPY NISAR ◽  
MATI UR RAHMAN ◽  
GHAYLEN LAOUINI ◽  
MESHAL SHUTAYWI ◽  
MUHAMMAD ARFAN
Keyword(s):  
Mathematical Model ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Order ◽  
Food Chain ◽  
Existence Results ◽  
Chain Model ◽  
Top Predator ◽  
The Fixed Point Theorem ◽  
Food Chain Model

This paper investigates the dynamical semi-analysis of the delayed food chain model under the considered fractional order. The food chain model is composed of three compartments, namely, population of the prey, intermediate predator and a top predator. By using the fixed point theorem approach, we exploit some conditions for existence results and stability for the considered system via Atangana–Baleanu–Caputo derivative with fractional order. Also, using the well-known Adam–Bashforth technique for numerics, we simulate the concerning results for the interference between the prey and intermediate predator. Graphical results are discussed for different fractional-order values for the considered model.

Stability and Hopf Bifurcation of a Fractional-Order Food Chain Model With Disease and Two Delays

2020 ◽  
Vol 15 (3) ◽  
Author(s):  
Xinhe Wang ◽  
Zhen Wang ◽  
Xiao Shen
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Positive Equilibrium ◽  
Chain Model ◽  
Two Delays ◽  
Existence Conditions ◽  
Positive Equilibrium Point ◽  
The Stability ◽  
Food Chain Model ◽  
Bifurcation And Stability

Abstract In this study, a fractional-order food chain model with disease and two delays is proposed. The existence conditions for a positive equilibrium point are given, and the stability conditions without the effects of delays are established. The effects of a single time delay and two time delays are discussed, the bifurcation and stability criteria are obtained, and the bifurcation points are calculated. To support the theoretical analysis, numerical simulations are presented.

scholarly journals Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 161
Author(s):  
Sameh Askar ◽  
Abdulrahman Al-khedhairi ◽  
Amr Elsonbaty ◽  
Abdelalim Elsadany
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
High Sensitivity ◽  
Phase Portraits ◽  
Chain Model ◽  
Encryption Scheme ◽  
Dynamical Behaviors ◽  
Food Chain Model

Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It is demonstrated that the model can exhibit a variety of dynamical behaviors including stable steady states, periodic and quasiperiodic dynamics. Then, a hybrid encryption scheme based on chaotic behavior of the model along with elliptic curve key exchange scheme is proposed for colored plain images. The hybrid scheme combines the characteristics of noise-like chaotic dynamics of the map, including high sensitivity to values of parameters, with the advantages of reliable elliptic curves-based encryption systems. Security analysis assures the efficiency of the proposed algorithm and validates its robustness and efficiency against possible types of attacks.

scholarly journals Dynamical analysis of fractional-order Holling type-II food chain model

2021 ◽  
Vol 18 (5) ◽  
pp. 5221-5235
Author(s):  
Cuimin Liu ◽  
◽  
Zhen Wang ◽  
Bo Meng
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Dynamical Analysis ◽  
Chain Model ◽  
Type Ii ◽  
Food Chain Model

Approximate analytical solution of fractional order biochemical reaction model and its stability analysis

2019 ◽  
Vol 12 (05) ◽  
pp. 1950059 ◽  
Author(s):  
Ved Prakash Dubey ◽  
Rajnesh Kumar ◽  
Devendra Kumar
Keyword(s):  
Stability Analysis ◽  
Analytical Solution ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Approximate Analytical Solution ◽  
Reaction Model ◽  
Biochemical Reaction ◽  
Classical Motion ◽  
Homotopy Analysis

Approximate analytical solution of the system of coupled nonlinear Ordinary Differential Equations (ODEs) of a biochemical reaction model is much relevant due to its practical significance to biochemists. In this paper, an effective and powerful mathematical technique, viz. fractional homotopy analysis transform method (FHATM), is employed to get the numerical solutions of biochemical reaction model with time fractional derivatives. The adopted scheme is the beautiful copulation of homotopy analysis technique and Laplace transform algorithm. This paper shows that the adopted scheme is quite easy as well as computationally attractive in the context of a solution procedure. The Caputo-type fractional derivatives are considered in the present paper. Approximate results of the probability density functions of the time fractional biochemical reaction model are computed for miscellaneous fractional Brownian motions as well as for classical motion and are presented graphically. The time fractional biochemical reaction model with respect to stability analysis for various values of fractional order [Formula: see text] is also analyzed. In the context of stability discussion, we have used the fractional Routh–Hurwitz stability criterion to establish the local stability of the biochemical reaction model of fractional order.

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