Implementación de métodos cualitativos y numéricos para el estudio de un modelo de dinámica poblacional

Abstract

This work presents the study and analysis of the solutions of the Lotka-Volterra model with the use of qualitative and numerical analysis with Runge Kutta RK4 Method. In order to carry out the study and analysis of this model, concepts and results of qualitative analysis of systems of differential equations were used to examine: the behavior of the system trajectories, the stability and classification of the equilibrium points of the system. For this purpose, a brief exposition of the qualitative analysis is included from the most relevant referenced bibliographic information. MATLAB software was used to implement the Runge Kutta RK4 Method and thus find approximate solutions to the Lotka-Volterra model, which allowed us to: program the method and examine the behavior of the solutions with their respective phase plane in an illustrative, specific and accurate way. These analyses are applied to a given problem describing the interaction between prey and predators using the Lotka-Volterra system to explain the different behaviors of the solutions presented by the problem in a certain period of time, to support the analysis performed in an application. It is concluded that to analyze the behavior of the Lotka-Volterra model it is not necessary to obtain its explicit solution. Thanks to the qualitative analysis and through a numerical method it is possible to arrive at similar responses of the system without knowing its explicit solution. It is recommended to consider the phase plane to determine the type of trajectory that has the solution of the system, and to continue the study using these methods to study other types of models, in particular the Lotka-Volterra model for two competing species.