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

Florian Munteanu

doi:10.1051/itmconf/20203403010

Abstract

In this paper we will consider a community of three mutually competing species modeled by the Lotka–Volterra system: $$ {\left\{ {\dot x} \right._i} = {x_i}\left( {{b_i} - \sum\limits_{i = 1}^3 {{a_{ij}}{x_j}} } \right),i = 1,2,3 $$ where xi(t) is the population size of the i-th species at time t, Ẋi denote $${{dxi} \over {dt}}$$ and aij, bi 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.

Highlights

The systems of differential equations that represent the competition between species, the so called Lotka-Volterra systems, were introduced by Lotka ([8]) and Volterra ([10], [11]) almost 90 years ago, their qualitative study, from the point of view of dynamical systems, is still of great interest for mathematical research and more
The competitive LotkaVolterra equations are a simple model of the population dynamics of species competing for some common resource
The form is similar to the LotkaVolterra equations for predation in that the equation for each species has one term for self-interaction and one term for the interaction with other species

Summary

Introduction

The systems of differential equations that represent the competition between species, the so called Lotka-Volterra systems, were introduced by Lotka ([8]) and Volterra ([10], [11]) almost 90 years ago, their qualitative study, from the point of view of dynamical systems, is still of great interest for mathematical research and more. For the three dimensional competitive system, the algebraic study discovers very important results about the dynamics of the competition between three species like in [3], [12] and [13] Under some conditions this 3-dimensional system of ordinary differential equations has eight equilibrium points with positive coordinates and represent a class of Kolmogorov systems. The competitive LotkaVolterra equations are a simple model of the population dynamics of species competing for some common resource. The definition of a competitive Lotka-Volterra system (5) assumes that all values in the interaction matrix ai j are striclty positive ([5]). Each species is assumed to be self-regulating (aii > 0), and in the absence of other species, to have a positive density independent growth rate constant bi

Local analysis of the 2D Lotka-Volterra ODE system

Local analysis of the 3D Lotka-Volterra ODE system

Conclusions