Modelo de Lotka-Volterra para a Dinâmica Predador-Presa

Abstract

The present Undergraduate Thesis has as main theme of study the equations of Lotka-Volterra. Among its diverse applications, this work is restrict to those who model the populational dynamic of two species, the predator and its prey. These equations describe many predator-prey relationships present in nature. The interaction is described by nonlinear ordinary differential equations, which consequently form a nonlinear system. The main objective of this work is qualitatively (geometrically) analyze the Lotka-Volterra system solutions. For that, a technique of solving linear systems and linearization around the equilibrium points of the system and obtaining the trajectories that determine its solutions were used. Initially, concepts considered essential for the understanding of nonlinear systems were presented, such as, concepts about linear systems with constant coefficients and obtaining their solutions; autonomous systems and determination of their trajectories; locally linear systems, among others. Then the Lotka-Volterra system was studied. By examining it, one can see that the equations don’t describe all the existing population dynamics of predators and prey, since the qualitative analysis of the solutions allowed to conclude that the behavior of the species tends to a cyclical oscillation. However, this behavior is not common for all dynamics, it is possible that one of the species is prone to extinction, or that both evolve to coexistence. Finally, in view of the diversity of interactions, the present work presents changes that can be made in the equations to make them more realistic, allowing the description of more complex systems.