DYNAMICAL BEHAVIORS OF FRACTIONAL ORDER PREY PREDATOR INTERACTIONS .

A. George maria selvam 1 , R. Dhineshbabu 2 and S. Britto jacob 1 . 1. Sacred Heart College, Tirupattur 635601, S. India . 2. DMI College of Engineering, Chennai 600123, S. India. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

Recently, population models have received increasing attention by scientists due to their importance in ecology. Indeed, there are different approaches to study population models, e.g. ordinary differential equations, difference equations, partial differential equations and fractional order differential equations. Fractionalorder differential equations (FOD) are used since they are naturally related to systems with memory which exists in most biological systems [2,4]. Many phenomena in population dynamics can be described successfully by the models using fractional order differential equations.
The fractional derivatives have several definitions. One of the most common definitions is the Caputo definition of fractional derivatives, which is often used in real applications.
Function. The operator D  is called the  -order Caputo differential operator.
In this paper, we study the dynamical behaviors of fractional-order LotkaVolterra predator prey system. It is shown that the discretized fractional-order system produces a much richer set of patterns than those observed in the systems counterpart.

Fractional Order Prey -Predator Model and its Discretization:-
The Lotka Volterra equations, also known as the predator prey equations, are a pair of first-order, nonlinear, differential equations frequently used to describe the dynamics in which two species interact, one as a predator and the other as prey. The populations change through time according to the following pair of equations: ; ' x ax bxy y cy dxy x is the number of prey (for example rabbits), y is the number of some predators (for example foxes) and , , , a b c d are positive real parameters describing the interaction of two species: a -Growth rate of prey, b -attack rate, c -Predator mortality rate, d -Growth rate of predator.
A more general model of predator prey interactions is the following system of differential equations, 22 ' ; ' x ax bxy ex y cy dxy fy        Here the term ex reflects the internal competition of the prey x for their limited external resources and the term fy reflects the competition among the predators for the limited number of prey. We shall consider the fractional order Lotka Volterra predator prey system which is given as follows where 0 t  and  is the fractional order satisfying (0,1]   . Now, applying the discretization process for a fractional-order system described in [3,8], we obtain the discrete fractional order predator prey system as follows:

Fixed Points:-
The fixed points of (1) are the points of intersections at which

Trivial and Semi-trivial fixed points and Stability Analysis;-
The Jacobian matrix of the linearized system of model (1)

Interior fixed point and its Stability
Here we investigate the linear stability of (1) at the interior fixed point 3 E . Therefore, the corresponding Jacobian

Dynamical behaviors of the Discretized Fractional Order Model:-
Let us investigate the dynamics of the discretized fractionalorder LotkaVolterra predator prey model (2). The dynamical behaviors of model (2) is determined by five parameters , , , , , , a b c d e f s and  .
Here we discuss the stability of fixed points of (2). The Jacobian matrix for (2) evaluated at any fixed point where Tr is the trace and Det is the determinant of the Jacobian matrix ** ( , ) J x y and they are given as x d e y f b a c x ad ce y bc af dex bfy efx y ac In order to study stability analysis of the fixed points of the model (2), we give the following theorems that can be easily proved by using the relation between roots and coefficients of the characteristic equation (3).