A Discretization of the Lotka-Volterra System of Equations

Abstract

The Lotka-Volterra system of differential equations is commonly used to model predator-prey relationships. This research focuses on the discretization of the nonlinear differential equations using the framework of time scale calculus to predict behavior of this commonly studied system of differential equations. A brief overview of the mathematics concepts required (including time scale calculus, phase portraits, the delta derivative, etc.) will be given. Initial analysis of this equation led to a phase-portrait diagram revealing a predictable pattern of behavior for the system in the forward direction. Sixteen regions of behavior were discovered, and much time has been devoted to proving the behavior seen within these regions as the equation jumps in the forward direction. Further exploration of the backwards direction will be discussed in attempt to identify regions in a similar manner to prove that the system of equations has predictable behavior in both the forward and backward directions.