Determination of the Fixed Point of Lotka-Volterra Function

Abstract

The paper focuses on the Lotka-Volterra function in its discrete form. The purpose of the study was to determine the fixed points of the function. The study employs the Banach Fixed Point Theorem and Contraction Mapping in Metric Space on the function to demonstrate the uniqueness of the fixed points and its continuous stability after several iterations, using the fixed points as the initial conditions. The study has shown that $(0,0),\left(0,\frac{\alpha-1}{\beta}\right),\left(\frac{1+\gamma}{\delta},0\right)$ and $\left(\frac{1+\gamma}{\delta},\frac{\alpha-1}{\beta}\right) $are the fixed points of the function, with the initial pair serving as a trivial one and the other three solely depending on the parameter values for the behavior of the function. The outcome of the limit points of the function as the fixed points after several iterations forms a fixed orbit structure of the function, irrespective of the value of the parameter. The study also showed the uniqueness of the fixed points, demonstrating the stability and continuity of the function in its steady state.