scholarly journals Computational dynamics of a Lotka-Volterra Model with additive Allee effect based on Atangana-Baleanu fractional derivative

2021 ◽  
Vol 2 (2) ◽  
pp. 96-103
Author(s):  
Hasan S. Panigoro ◽  
Emli Rahmi
Keyword(s):  
Hopf Bifurcation ◽  
Convergence Rate ◽  
Fractional Derivative ◽  
Limit Cycle ◽  
Fractional Order ◽  
Allee Effect ◽  
Volterra Model ◽  
Fractional Order Derivative ◽  
Computational Dynamics ◽  
Dynamical Behaviors

This paper studies an interaction between one prey and one predator following Lotka-Volterra model with additive Allee effect in predator. The Atangana-Baleanu fractional-order derivative is used for the operator. Since the theoretical ways to investigate the model using this operator are limited, the dynamical behaviors are identified numerically. By simulations, the influence of the order of the derivative on the dynamical behaviors is given. The numerical results show that the order of the derivative may impact the convergence rate, the occurrence of Hopf bifurcation, and the evolution of the diameter of the limit-cycle.

scholarly journals Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg–de Vries equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Choonkil Park ◽  
R. I. Nuruddeen ◽  
Khalid K. Ali ◽  
Lawal Muhammad ◽  
M. S. Osman ◽  
...  
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Mathematical Physics ◽  
Order Derivative ◽  
Conformable Fractional Derivative ◽  
Fractional Order Derivative ◽  
De Vries ◽  
Fifth Order ◽  
2D And 3D

Abstract This paper aims to investigate the class of fifth-order Korteweg–de Vries equations by devising suitable novel hyperbolic and exponential ansatze. The class under consideration is endowed with a time-fractional order derivative defined in the conformable fractional derivative sense. We realize various solitons and solutions of these equations. The fractional behavior of the solutions is studied comprehensively by using 2D and 3D graphs. The results demonstrate that the methods mentioned here are more effective in solving problems in mathematical physics and other branches of science.

A Local Fractional Derivative

2003 ◽  
Author(s):  
Xiaorang Li ◽  
Christopher Essex ◽  
Matt Davison
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Local Property ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Local Fractional Derivative ◽  
Basic Properties ◽  
Differentiable Functions ◽  
Definition Of

A new definition of fractional order derivative is given and its basic properties are investigated. This definition is based on the Weyl derivative and is a local property of functions. It can be applied to non-differentiable functions and may be useful for studying fractal curves.

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

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 122
Author(s):  
Hasan S. Panigoro ◽  
Agus Suryanto ◽  
Wuryansari Muharini Kusumawinahyu ◽  
Isnani Darti
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Power Law ◽  
Existence And Uniqueness ◽  
Dynamical Analysis ◽  
Prey Interaction ◽  
Uniqueness Of The Solution ◽  
Over Exploitation ◽  
Caputo Fractional Order Derivative

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.

Bifurcation analysis of a fractional-order SIQR model with double time delays

2020 ◽  
Vol 13 (07) ◽  
pp. 2050067
Author(s):  
Shouzong Liu ◽  
Ling Yu ◽  
Mingzhan Huang
Keyword(s):  
Hopf Bifurcation ◽  
Convergence Rate ◽  
Fractional Order ◽  
Bifurcation Analysis ◽  
Time Delays ◽  
Oscillatory System ◽  
Endemic Equilibrium ◽  
Nonlinear Incidence Rate ◽  
Double Time ◽  
The Stability

In this paper, a fractional-order delayed SIQR model with nonlinear incidence rate is investigated. Two time delays are incorporated in the model to describe the incubation period and the time caused by the healing cycle. By analyzing the associated characteristic equations, the stability of the endemic equilibrium and the existence of Hopf bifurcation are obtained in three different cases. Besides, the critical values of time delays at which a Hopf bifurcation occurs are obtained, and the influence of the fractional order on the dynamics behavior of the system is also investigated. Numerically, it has been shown that when the endemic equilibrium is locally stable, the convergence rate of the system becomes slower with the increase of the fractional order. Besides, our studies also imply that the decline of the fractional order may convert a oscillatory system into a stable one. Furthermore, we find in all these three cases, the bifurcation values are very sensitive to the change of the fractional order, and they decrease with the increase of the order, which means the Hopf bifurcation gradually occurs in advance.

scholarly journals Complex Dynamical Behaviors of a Fractional-Order System Based on a Locally Active Memristor

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Yajuan Yu ◽  
Han Bao ◽  
Min Shi ◽  
Bocheng Bao ◽  
Yangquan Chen ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Hysteresis Loop ◽  
Fractional Order ◽  
Period Doubling ◽  
Order System ◽  
Periodic Signal ◽  
Dynamical Behaviors ◽  
Memristive System ◽  
Attraction Basins ◽  
The Stability

A fractional-order locally active memristor is proposed in this paper. When driven by a bipolar periodic signal, the generated hysteresis loop with two intersections is pinched at the origin. The area of the hysteresis loop changes with the fractional order. Based on the fractional-order locally active memristor, a fractional-order memristive system is constructed. The stability analysis is carried out and the stability conditions for three equilibria are listed. The expression of the fractional order related to Hopf bifurcation is given. The complex dynamical behaviors of Hopf bifurcation, period-doubling bifurcation, bistability and chaos are shown numerically. Furthermore, the bistability behaviors of the different fractional order are validated by the attraction basins in the initial value plane. As an alternative to validating our results, the fractional-order memristive system is implemented by utilizing Simulink of MATLAB. The research results clarify that the complex dynamical behaviors are attributed to two facts: one is the fractional order that affects the stability of the equilibria, and the other is the local activeness of the fractional-order memristor.

scholarly journals Further Study on Dynamics for a Fractional-Order Competitor-Competitor-Mutualist Lotka–Volterra System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Bingnan Tang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Natural World ◽  
Volterra Model ◽  
Sufficient Criterion ◽  
Volterra System ◽  
Stability Behavior ◽  
Predator Model ◽  
Set Up ◽  
The Stability

On the basis of the previous publications, a new fractional-order prey-predator model is set up. Firstly, we discuss the existence, uniqueness, and nonnegativity for the involved fractional-order prey-predator model. Secondly, by analyzing the characteristic equation of the considered fractional-order Lotka–Volterra model and regarding the delay as bifurcation variable, we set up a new sufficient criterion to guarantee the stability behavior and the appearance of Hopf bifurcation for the addressed fractional-order Lotka–Volterra system. Thirdly, we perform the computer simulations with Matlab software to substantiate the rationalisation of the analysis conclusions. The obtained results play an important role in maintaining the balance of population in natural world.

A Numerical Method for Calculating Fractional Derivative

2015 ◽  
Vol 775 ◽  
pp. 426-430
Author(s):  
Jun Huang ◽  
Yong Jun Li ◽  
Fan Huang
Keyword(s):  
Fractional Derivative ◽  
Numerical Method ◽  
Fractional Order ◽  
High Precision ◽  
Good Stability ◽  
Order Derivative ◽  
Singularity Problem ◽  
Numerical Examples ◽  
Fractional Order Derivative

A numerical method is proposed for calculating the fractional order derivative and successfully resolving the integrand singularity problem based on Zhang-Shimizu algorithm. And then a method is developed to calculate the twice nonlinear fractional derivative, numerical examples demonstrate the numerical method with high precision and good stability.

Turing-Hopf bifurcation in a diffusive mussel-algae model with time-fractional-order derivative

2020 ◽  
Vol 138 ◽  
pp. 109954 ◽  
Author(s):  
Salih Djilali ◽  
Behzad Ghanbari ◽  
Soufiane Bentout ◽  
Abdelheq Mezouaghi
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Order Derivative ◽  
Fractional Order Derivative

scholarly journals On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Erdal Karapinar ◽  
Ho Duy Binh ◽  
Nguyen Hoang Luc ◽  
Nguyen Huu Can
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Initial Value Problem ◽  
Caputo Fractional Derivative ◽  
Parabolic Systems ◽  
Initial Value ◽  
Fractional Order Derivative ◽  
Nonlinear Parabolic ◽  
Need To Evaluate ◽  
Derivative Order

AbstractIn this work, we study an initial value problem for a system of nonlinear parabolic pseudo equations with Caputo fractional derivative. Here, we discuss the continuity which is related to a fractional order derivative. To overcome some of the difficulties of this problem, we need to evaluate the relevant quantities of the Mittag-Leffler function by constants independent of the derivative order. Moreover, we present an example to illustrate the theory.

scholarly journals Dynamic Study of a Predator-Prey Model with Weak Allee Effect and Delay

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Yong Ye ◽  
Hua Liu ◽  
Yu-mei Wei ◽  
Ming Ma ◽  
Kai Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Theoretical Analysis ◽  
Bifurcation Analysis ◽  
Allee Effect ◽  
Stability Switches ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Predator Model

In this paper, a prey-predator model and weak Allee effect in prey growth and its dynamical behaviors are studied in detail. The existence, boundedness, and stability of the equilibria of the model are qualitatively discussed. Bifurcation analysis is also taken into account. After incorporating the searching delay and digestion delay, we establish a delayed predator-prey system with Allee effect. The results show that there exist stability switches and Hopf bifurcation occurs while the delay crosses a set of critical values. Finally, we present some numerical simulations to illustrate our theoretical analysis.

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