O SISTEMA LOTKA-VOLTERRA: MODELAGEM ECOLÓGICA DE PREDADORES E PRESAS E IMPLEMENTAÇÕES COM O SAGEMATH

Abstract

We will investigate a particular case of the Lotka-Volterra system that models a variety of problems in our world, including ecological dynamics involving two species, one of prey (with an abundant source of food) and the other of predators (which feeds on prey). The simplest mathematical model (classic model) is described using a two-dimensional system of coupled nonlinear ordinary differential equations. Initially, we will use the approach through linearization around critical points in order to understand how the system trajectories locally are. Still in this analysis, from the solution for the linear case, we will prove a series of results for the system solutions. Although the classical system is nonlinear, it is equivalent to a separable ODE, which allows us to obtain the solutions through an implicit equation. To fully understand the system, we use the open software SageMath, using several methods to obtain explicit, numerical solutions and generate associated graphics such as the vector field, phase plane trajectories, and how the solutions evolve in time, and we also compare the numerically obtained solutions with the explicit solutions. Furthermore, we will develop qualitative notions through the analysis of system stability. All implementations were made by the authors and are made available at the end of this work and online so that interested students and professors can use these resources already built to enrich their studies and research.