A study of a three-dimensional competitive Lotka–Volterra system

University of Craiova, Department of Applied Mathematics Al I Cuza 13, Craiova 200585, Dolj, Romania

^* e-mail: munteanufm@gmail.com

Abstract

In this paper we will consider a community of three mutually competing species modeled by the Lotka–Volterra system:

$${\{\dot{x}}_i=x_i\left(b_i-{\displaystyle \sum _{i=1}^3{a}_{ij}{x}_j}\right),i=1,2,3$$

where x_i(t) is the population size of the i-th species at time t, Ẋ_i denote $\frac{dxi}{dt}$ and a_ij, b_i are all strictly positive real numbers.

This system of ordinary differential equations represent a class of Kolmogorov systems. This kind of systems is widely used in the mathematical models for the dynamics of population, like predator-prey models or different models for the spread of diseases.

A qualitative analysis of this Lotka-Volterra system based on dynamical systems theory will be performed, by studying the local behavior in equilibrium points and obtaining local dynamics properties.