Comparative analysis of Taylor Series and Runge-kutta Fehlberg Methods in solving the Lotka-volterra Competitive Model

This research mainly focuses on the comparisons between numerical methods, such as Taylor Series and Runge­Kutta Fehlberg (RKF) methods with the exact solution. The advantage of these methods is that they can reliably and precisely solve the non-linearity of the model. Based on the results, both methods offer an accurate way to solve the model by comparing the results. RKF proves to be the most accurate approximation method for solving the Lotka-Volterra competitive model when compared to the exact solution than Taylor Series method by using software Mathematica 13.2 programming. This research will use data of two species, Paramecium Caudatum and Stylonychia Pustulata from Gause's experiment. Both species undergo intraspecific interaction and their population is increasing day by day until a level where they remain constant. For P. Caudatum, the population sizes are increasing day by day until day 16th, which is at 202 cells. Meanwhile for S. Pustulata the population sizes are increasing day by day until day 8th, which is at 41 cells. The equilibrium and stability analysis are performed to give critical insights into the long-term behaviour of the system and indicate how the system responds to perturbations, respectively. For mixed population, when the value of both species' carrying capacities is lower than the value of other species carrying capacity divided by the competition coefficient, the two species are able to coexist in this stable equilibrium. Meanwhile, when the value of both species' carrying capacities is lower than the value of other species carrying capacity divided by the competition coefficient, one of the species will undergo competitive exclusion which eventually outcompete the other species that leads to an unstable equilibrium or the extinction of the weaker species. By understanding the behavior of competition and survival, the research's outcomes shed light on the population behavior of these species and have far-reaching implications in the fields of ecology, conservation, and environmental management.