The stability of harmonic solutions of Lotka-Volterra systems
type: Item
instance of: scholarly article
title: The stability of harmonic solutions of Lotka-Volterra systems
zbMATH Open document ID: 0844.34053
author: Bin Liu
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-08-29T00:00:00Z
review text: Consider the periodic Lotka-Volterra system (1) \(\dot p = h_1 (t,E) p - h_3 (t,E) pq\), \(\dot q = - h_2 (t,E) q + h_4 (t,E) pq\), where \(E\) is a parameter in \(J \subset \mathbb{R}\), and the coefficient functions \(h_i (t,E) > 0(i = 1, \dots, 4)\) are smooth in \((t,E)\), and \(2 \pi\)-periodic in \(t \in \mathbb{R}\). In recent years, existence theorems for \(2 \pi\)-periodic solutions of (1) are considered in literature under various conditions for the above coefficient functions. In 1993, the reviewer and F. Zanolin proved the existence of at least one \(T\)-periodic solution of (1) with continuous positive coefficient \(T\)-periodic functions, as well as at least \(2N_k\) subharmonic solutions of any order \(k \geq k_0\) (for some integer \(k_0 > 1)\) with \(N_k \to \infty\) as \(k \to \infty\). The purpose of this paper is to study the stability of the \(2 \pi\)-periodic solution of (1), denoted by \(z(t,E) = (p(t,E), q(t,E))\). Its main result is stated as follows: There is a subset \(J_1\) of \(J\), such that for each \(E \in J_1\), the harmonic solution \(z(t,E)\) is stable in the Lyapunov sense. Moreover, there is at least one subharmonic solution of (1) with minimal period \(2k \pi\) for every sufficiently large \(k\) and \(E \in J_1\). Furthermore, \(\text{mes} [J \backslash J_1] = 0\).
reviewed by: Ding Tongren
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 859231
zbMATH Keywords: periodic solutions
zbMATH Keywords: periodic Lotka-Volterra system
zbMATH Keywords: stability
zbMATH Keywords: subharmonic solution
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1326
OpenAlex ID: W1989385057
DOI: 10.1006/JMAA.1995.1326
description: scientific article; zbMATH DE number 859231