Limit cycles for the competitive three dimensional Lotka-Volterra system
type: Item
instance of: scholarly article
title: Limit cycles for the competitive three dimensional Lotka-Volterra system
zbMATH Open document ID: 0960.34022
author: Dongmei Xiao
author: Wenxia Li
published in: Journal of Differential Equations
publication date: 2000-08-27T00:00:00Z
review text: Here, the following competitive three-dimensional Lotka-Volterra system is investigated \[ \dot x= X(b- Ax), \] where \(x= \text{col}(x_1, x_2, x_3)\) is a three-dimensional state vector \(X= \text{diag}(x_1, x_2, x_3)\) is a \(3\times 3\) diagonal matrix, \(b= \text{col}(b_1, b_2, b_3)\) is a positive real vector and \(A= (a_{ij})_{3\times 3}\) is a positive matrix. First, it is proved that the number of limit cycles of the system in \(\mathbb{R}^3_+\) is finite if the system has not any heteroclinic polycycles in \(\mathbb{R}^3_+\). Second, a particular 3-dimensional competitive Lotka-Volterra system with two small parameters is discussed. It is proved that there exists one parameter range in which the system is persistent and has at least two limit cycles, and there exists other parameter ranges in which the system is not persistent and has at least one limit cycle. Hence, some open questions are answered partly in this paper.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34C60
zbMATH DE Number: 1496032
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: competitive three-dimensional Lotka-Volterra system
zbMATH Keywords: limit cycles
reviewed by: Lansun Chen
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3729
OpenAlex ID: W2043036629
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DOI: 10.1006/JDEQ.1999.3729
description: scientific article; zbMATH DE number 1496032