Comparative Analysis of Taylor Series and Runge-Kutta Fehlberg Methods in Solving the Lotka-Volterra Competitive Model

Abstract

This paper primarily scrutinizes the comparative efficacy of numerical methods, namely the Taylor Series and the Runge-Kutta Fehlberg methods, against the exact solution in solving mathematical models. These methods exhibit the capacity to handle the non-linearity of the Lotka-Volterra competitive model with a high degree of accuracy and reliability. The comparison of results obtained from both methods and the exact solution shows that the Runge-Kutta Fehlberg method provides a more precise approximation for the model than the Taylor Series method. This conclusion is supported by computations carried out using the Mathematica 13.2 software. The research data involves two species, Paramecium Caudatum and Stylonychia Pustulata, derived from Gause's experiment. Both species demonstrate an intraspecific interaction, with their populations rising steadily until reaching a constant level. For Paramecium Caudatum, the population peaks at 202 cells on the 16^th day, while for Stylonychia Pustulata, it reaches a maximum of 41 cells on the 8^th day. Equilibrium and stability analysis offer vital insights into the long-term behavior of the system and its reaction to perturbations. In mixed populations, when the carrying capacity of both species is less than the carrying capacity of another species divided by the competition coefficient, the species coexist in a stable equilibrium. Conversely, if the carrying capacity of one species is less than the carrying capacity of the other divided by the competition coefficient, one species may outcompete the other, leading to an unstable equilibrium or even extinction of the weaker species. The research thus provides valuable insights into the dynamics of competition and survival, with profound implications for the fields of ecology, conservation, and environmental management.