Stability Analysis of Periodic Solutions to the Nonstandard Discretized Model of the Lotka-volterra Predator-prey System

The standard classical discretization methods of differential equations often produce difference equations that do not share dynamics with their continuous counterparts; Recently,[4] has developed successful non-standard discretization schemes that produce dynamical consistency, which numerical analysis value highly. Many authors have adapted these methods to various biological models. We reviewed a non-standard discretized biological model of a Lotka-Volterrs Predator-Prey system in a general form and discussed the stability analysis of its periodic solutions. We also discussed a numerical example of this analysis using the non-standard discretized PredatorPrey model the name of executed program for drawing and calculation is “MATLAB 7.0”. Introduction A wide variety of numerical schemes are available to solve the dynamical systems that cannot be solved analytically. The standard classical discretization methods involved in these numerical schemes often produce systems of difference equations that do not inherit the dynamical properties of their continuous counterparts. When they exist, stability of fixed points and periodic solutions are the most important properties of continuous dynamical systems and discretized model. Thus, a discretization methods involved in numerical scheme is useful if the solution of that scheme is exact for at least a subclass of original system, if it preserves the dynamics, and if it conserves energy like its continuous analogue. Mickens developed non standard discretization methods that have proved to be very fruitful, producing numerical schemes that are highly desirable because they meet the criteria above. These methods are relatively easy to implement and have much greater computational efficiency than standard numerical methods. The relative importance of advection and biological and chemical reaction is directly incorporated into the corresponding numerical scheme, large time steps can be taken without affecting the accuracy of the numerical solutions. Generally, non-standard methods can be used in numerical schemes to construct highly accurate algorithms for solving a varity of stiff dynamical systems,[8]. Many researchers [Dohtani, 1992; Gopalsamy & Liu, 1999; Jian & Rogers, 1987], applyied these techniques to obtain numerical solutions to the various differential equations that rise in interesting problems in the natural and engineering sciences. [Al-Kahby al, 2000] and his Co-workers have used non-standard discretization methods with some biological models, they applied this approach to discretize the competitive and cooperative models of predator-prey. In that work, they consider the simple predator-prey model: