الملخص
Abstract: Fractional calculus is a branch of mathematical analysis, which studies the possibility of that the power of the differentiation operator is real or complex number has positive real part. In recent years, fractional calculus successfully applied to many areas such as mathematical biology, fluid mechanics, signal processing, etc.The differential equations. are generalized to the fractional differential equations. In this thesis, we introduced the basic concepts of fractional calculus and we studied some basic methods for solving fractional differential equations, The main aim of this thesis is to obtain numerical solutions of the system of fractional order (their order is positive real number) of a two-prey one-predator model without and with the help between the tow teams of prey against the one team of predator using Predictor-corrector method. This thesis consisted of four chapters, which are: In chapter 1, we introduced historical brief for fractional calculus and some special functions in fractional calculus as Gamma function, Beta function and Mittag-Leffler functions. Also, we gave some of their properties. We give definition and properties of fractional derivatives as Griinwald-Letnikov fractional derivative, Riemann- Liouville fractional integral and fractional derivative and Caputo-type fractional derivative and we gave some properties of these derivatives. Also, we reviewed the stability of solutions of fractional and differential equations systems by studying steady states and gave conditions of stability of fractional and differential equations. In chapter 2, we displayed exact solution of linear fractional differential equations with constant coefficients by using Laplace transform and we solved some examples by this transform and plot solutions. After that, we present the numerical algorithm for theapproximate solutions of fractional differential equations using Predictor-Corrector method. In chapter 3, we studied a system of fractional differential equations which represent the relation between two teams of prey against one team of predator without help between the two teams of prey against the one team of predator. Firstly, the existence and the uniqueness of the solutions of the model are proved. Secondly, the stability of the steady states is investigated. Also we gave an example of the steady state, which is locally asymptotically stable for its fractional order counterpart but is a center for the integer order system In chapter 4, we proposed a system of fractional differential equations which represent the relation between two teams of prey against one team of the predator with help between the two teams of preys. Firstly, the existence and the uniqueness of the solutions of the model are proved, Secondly, the stability of the steady states is investigated. Also, we gave an example of the steady state, which is locally asymptotically stable for its fractional order counterpart but is a center for the integer order system
