Dynamical Analysis of Fractional-Order Hastings-Powell Food Chain Model with Alternative Food

In this paper, a fractional-order Hastings-Powell food chain model is discussed. It is assumed that the top-predator population is supported by alternative food. Existence and local stability of equilibrium points of fractional-order system are investigated. Numerical simulations are conducted to illustrate analysis results. The analysis results show that alternative food can give a positive impact for top-predator population.


INTRODUCTION 
Nowadays, fractional calculus becomes the main focus for the researchers. Many problems of science and engineering can be modeled by using fractional derivatives. The process of developing a differential system of integer order into fractional-order becomes popular in dynamic systems [1]. Basically, a biological mathematical model in predicting the future, not only depends on the current but also the memory or the previous condition. In fractional derivatives, at some certain conditions contain information of previous condition, therefore fractional derivatives can be used to explain more realistic natural phenomena.
Interactions between populations can be described in a food chain model. One of the interactions in the food chain is predation process. Mathematical model used to describe interactions between predator and prey is called the predator-prey model. Furthermore, many interesting fenomena in ecology can be described by mathematical model through predator-prey models such as harvesting in predator population [1], supplying alternative food in a predator population [2], refuging prey population [3], spreading disease in ecosystem [4], and the effect of the present an omnivore [5]. In predator prey model [2], it is assumed that prey populations do not always exist, they also experience migration to find new habitats due to climate change factors and low food reserves. Therefore, predators need additional food or alternative prey to survive.
In this paper, a food chain model of threespecies fractional-order with alternative food is introduced.
Examples of three-species ecosystems in this model: vegetation-hare-lynx, mouse-snake-owl and worm-robin-falcon. Moreover, predator-prey food chains have been studied in structured populations in [5,6]. In this paper, Model is a modification of model [7]. Then the conditions of existence and stability of the equilibrium points of the fractional-order are examined in the result and discussion. Numerical simulations are illustrated by the Grunwald-Letnikov approximation [8].

MATERIALS AND METHODS Model Formulation
In this research, the predatory-prey model [7] is the main object of the study. The model construction is done by modifying the model of Sahoo and Poria [2] by changing the integer order into the fractional-order.

Determination of the Equilibrium Point
In dynamical analysis, the first step is to determine the equilibrium points. The equilibrium point is obtained when the population rate of the system is unchanged or zero. From this condition, the existence properties of each equilibrium points is also obtained.

Stability of the Equilibrium Point
In this paper, the local stability of equilibrium points is analyzed. The discussion of local stability is begun by linearizing the model by using Taylor series. The linearization around its equilibrium point is done to change the nonlinear model into linear form. Approximation of linear system using Taylors series will be in the form of Jacobian matrix. From the Jacobian matrix, it is determined the roots of the characteristic equation or eigenvalue. The determination of local stability can be obtained from the absolute of its eigen value argument.

Numerical Simulation
The behavior of the system [3] is described by numerical simulation. The numerical simulation approach uses the Grunwald-Letnikov scheme. The numerical simulations are conducted by using MATLAB software. An important step in this stage is to determine parameters that match the condition of existence and the stability of the equilibrium points. The behavior of local stability is visualized by graphic based on kinds of parameter values pointed. The last step at this stage is do the interpretation results of numerical simulation.

RESULT AND DISCUSSION Model Formulation
In this model, definition of Caputo fractional derivative is used. In [9] Caputo's definition of fractional derivatives can be written as follows with − 1 < < and Γ is a Gamma function and = .
Hastings and Powell [10] has discussed the food chain model of three species. The three species are prey population ( ), intermediatepredator ( ) and top-predator ( ). The prey population is hunted by the intermediatepredator population and the top-predator hunts intermediate-predators. Both predation processes use the Holling Type II response function. 0 and 0 express the rate of growth and carrying capacity of the prey population. 1 and 2 are the interaction rates between prey populations, intermediate-predators with predators, toppredators. respectively. Mataouk et al. [10] modified the Hastings-Powell model [8] by changing the integer order into the fractional-order as follows: with 0 < < 1.
Model (1) explains that top-predator food sources only depend on intermediate-predators. However, alternative prey (supplementary feeding) for top-predators can reduce predation rates in intermediate-predators [2], then to give this effect, model (1) can be modified to where is a time independent constant to get the alternative resource (0 < < 1). To make easier the model analysis, variables and some parameter are selected to be

Stability of Equilibrium Points
To determine the equilibrium points of differential equation (3) , let ).
The Jacobian matrix of system (3)  Proof. The Jacobian matrix at 1 is given by .
Eigenvalues of matrix ( 1 ) are obtained by solving the characteristic equation The eigenvalues corresponding to the equilibrium 1 are 1 = 1 > 0, 2 = − 1 , and it follows from convergence of Mittag-Leffler function [9] that the equilibrium 1 is always a saddle point. Proof. The Jacobian matrix of 2 is given by It follows from convergence of Mittag-Leffler function [9] that the equilibrium 2 of system (3) is locally asymptotically stable. Furthermore, the equilibrium points

Numerical Method and Simulations
Numerical method which is introduced by Grunwald and Letnikov [8] is used to solve nonlinear fractional differential equation [3]. As described in [8,12], by using the Grunwald-Letnikov approximation method, it is obtained the following nonstandard explicit scheme for system [3].

CONCLUSION
In this work, the Hastings-Powell food chain model with alternative foods has been modified into a system of fractional-order. The local stability of all the equilibrium points of the fractionalorder system is investigated. Numerical simulation results agree with the analytical result. It is also found that the fractional parameter has effects on the stability of solution behavior. Furthermore, our analysis predicts that providing a suitable amount of alternative food has a positive impact for top-predator population.