Fractional Predator-prey Model Research Articles

Three-dimensional pattern dynamics of a fractional predator-prey model with cross-diffusion and herd behavior

In this paper, we study the pattern dynamics in a spatial fractional predator-prey model with cross fractional diffusion, herd behavior and prey refuge. In this model, herd behavior exists in the population of predators and the prey. The spatial dynamics of the system are obtained through appropriate threshold parameters, and a series of three-dimensional patterns are observed, such as tubes, planar lamellae and spherical droplets. Specifically, linear stability analysis is applied to obtain the conditions of Hopf bifurcation and Turing instability. Then, by utilizing the central manifold reduction theory analysis, the amplitude equation near the critical point of Turing bifurcation is deduced to study the selection and stability of pattern formation. The theoretical results are verified by numerical simulation.

Existence and Uniqueness of Caputo Fractional Predator-Prey Model of Holling-Type II with Numerical Simulations

We suggested a new mathematical model for three prey-predator species, predator is considered to be divided into two compartments, infected and susceptible predators, as well as the prey and susceptible population based on Holling-type II with harvesting. We considered the model in Caputo fractional order derivative to have significant consequences in real life since the population of prey create memory and learn from their experience of escaping and resisting any threat. The existence, uniqueness, and boundedness of the solution and the equilibrium points for the considered model are studied. Numerical simulations using Euler’s method are discussed to interpret the applicability of the considered model.

A numerical study of fractional order population dynamics model

In this paper, the population dynamics model including the predator-prey problem and the logistic equation are generalized by using fractional operator in term of Caputo-Fabrizio derivative (CF-derivative). The models under study include of fractional Lotka-Volterra model (FLVM), fractional predator-prey model (FPPM) and fractional logistic model of population growth (FLM-PG) with variable coefficients. After that a numerical scheme is presented to obtain numerical solutions of these fractional models. These solutions are made using three-step Adams-Bashforth scheme. To show the efficiency and the accuracy of the present scheme, a few examples are evaluated. The numerical simulations of the results are depicted the accuracy of the present scheme.

On a problem of linearized stability for fractional difference equations

This paper discusses the problem of linearized stability for nonlinear fractional difference equations. Computational methods based on appropriate linearization theorem are standardly applied in bifurcation analysis of dynamical systems. However, in the case of fractional discrete systems, a theoretical background justifying its use is still missing. Therefore, the main goal of this paper is to fill in the gap. We consider a general autonomous system of fractional difference equations involving the backward Caputo fractional difference operator and prove that any equilibrium of this system is asymptotically stable if the zero solution of the corresponding linearized system is asymptotically stable. Moreover, these asymptotic stability conditions for equilibria of the system are described via location of all the characteristic roots in a specific area of the complex plane. In the planar case, these conditions are given even explicitly in terms of trace and determinant of the appropriate Jacobi matrix. The results are applied to a fractional predator-prey model and the fractional Lorenz model. Related experiments are supported by a numerical code that is appended as well

Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

Stability and Bifurcation Control in a Fractional Predator–Prey Model via Extended Delay Feedback

This paper explores the bifurcation control of a fractional predator–prey system with an active extended delayed feedback controller. Delay-induced bifurcations criteria for such an uncontrolled system are firstly derived by selecting time delay as a bifurcation parameter. Then, an extended delayed feedback controller is cleverly devised to control Hopf bifurcation for the proposed system. It means that the bifurcation dynamics can be efficaciously controlled for a given system with the adjustment of the fractional order, feedback gain and extended feedback delay provided that the remnant parameters are fixed. The obtained results significantly extend the preceding studies concerning bifurcation control of delayed fractional-order systems. To verify the correctness of the established theory, some numerical results are presented.

Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel.

One of the major applications of the nonlinear system of differential equations in biomathematics is to describe the predator-prey problem. In this framework, the fractional predator-prey model with Beddington-DeAngelis is examined. This model is formed of three nonlinear ordinary differential equations to describe the interplay among populations of three species including prey, immature predator, and mature predator. The fractional operator used in this model is the Atangana-Baleanu fractional derivative in Caputo sense. We show first that the fractional predator-prey model has a unique solution, then propose an efficient numerical scheme based on the product integration rule. The numerical simulations indicate that the obtained approximate solutions are in excellent agreement with the expected theoretical results. The numerical method used in this paper can be utilized to solve other similar models.

A Fractional Predator-Prey Model and its Solution

A Fractional Predator-Prey Model and its Solution