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This paper investigates the dynamical behavior of a Fractional order Prey -Predator interaction model. A discretization process is applied to obtain its discrete version. The fixed points are obtained and the stability properties are discussed. …………………………………………………………………………………………………….... Introduction:-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. Fractional -order 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. 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:

In ecology, by refuge an organism attains protection from predation by hiding in an area where it is unreachable or cannot simply be found. In population dynamics, once refuges are available, both prey-predator populations are expressively greater and meaningfully extra species can be sustained in the region. This examine the stability of a discrete predator prey model incorporating with constant prey refuge. Existence results and the stability conditions of the system are analyzed by obtaining fixed points and Jacobian matrix. The chaotic behavior of the system is discussed with bifurcation diagrams. Numerical experiments are simulated for the better understanding of the qualitative behavior of the considered model. Mathematics Subject Classification. [2010] : 37C25, 39A28, 39A30, 92D25.

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