Mathematical and numerical simulations of the Lotka-Volterra Equations

Abstract

In this study, we perform several Mathematical and numerical simulations of the classic Lotka-Volterra equations and their extensions. We first study the classic Lotka Volterra equations in the context of a two species Predator-Prey model. We then extend the classic two species model into a three-species Lotka Volterra Food chain with a top predator species, a mid-level Predator/Prey species and a lower-level prey. In our two species classic model, we find the critical points of our system, model the phase plot of the two species to understand their behavior around their critical point and then plot the variation of the two species with respect to time. As for our three species Lotka Volterra food chain, we take extreme cases of our system by assuming each of the species to be absent discretely and model the behavior of the two remaining species. We also study the behavior of our three species system around its critical points and vary the constants to understand the fate of each of our species in different contexts. Lastly, we plot, the variation of our three species through time, given varying assumptions. Our two species Lotka-Volterra System shows oscillatory behavior through time, where the predator and prey both persist, and their population oscillates over time. As for our Three species Lotka-Volterra food chain, the destiny of each of our species is dictated by the values of 4 constants in our system.