Variations on Lyapunov's stability criterion and periodic prey-predator systems
type: Item
instance of: scholarly article
title: Variations on Lyapunov's stability criterion and periodic prey-predator systems
zbMATH Open document ID: 1493.34153
author: Rafael Ortega
published in: Electronic Research Archive
publication date: 2021-12-03T00:00:00Z
review text: The author recovers a stability criterion applied to the linear equation \(\ddot{x} + \alpha(t) x =0\) where \(\alpha(t)\) is \(T\)-periodic and proved by Lyapunov. The author obtains related stability criteria for several equations. First for the same equation, then for the dissipative Hill's equation \[ \ddot{x} + c \dot{x} + \alpha(t) x =0, \] where \(c>0\) is a constant. Then, he considers the linear prey-predator system \[ \dot{x}_1 = -a_{11}(t) x_1 -a_{12}(t) x_2, \ \dot{x}_2 = a_{21}(t) x_1 -a_{22}(t) x_2, \] where all the coefficients \(a_{ij}(t)\) are \(T\)-periodic and positive. The author looks for applications of these results to Lotka--Volterra prey-predator systems of the type \[ \begin{array}{lll} \dot{u} & = & \displaystyle u\left( a(t)- b(t) u - c(t) v\right), \vspace{0.2cm} \\ \dot{v} & = & \displaystyle v \left(d(t)+e(t)u-f(t)v \right), \end{array} \] where \(u>0\), \(v>0\), all the coefficients are \(T\)-periodic and \(b(t)\), \(c(t)\), \(e(t)\) and \(f(t)\) are positive.
reviewed by: Maite Grau
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 37C60
zbMATH DE Number: 7438963
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: asymptotic stability
zbMATH Keywords: prey-predator
zbMATH Keywords: coexistence state
zbMATH Keywords: Turing instability
MaRDI profile type: Publication
full work available at URL: era.2021069
OpenAlex ID: W3197794688
Wikidata QID: Q113201418
cites work: A periodic prey-predator system
cites work: Periodic differential operators
cites work: Q5617840
cites work: Turing instabilities for systems of two equations with periodic coefficients
cites work: A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations
cites work: The periodic predator-prey Lotka-Volterra model
cites work: Dependence of solutions and eigenvalues of measure differential equations on measures
cites work: The First Interval of Stability of a Periodic Equation of Duffing Type
cites work: Periodic solutions of a Newtonian equation: Stability by the third approximation
cites work: Periodic Differential Equations in the Plane
cites work: Optimal bounds for bifurcation values of a superlinear periodic problem
cites work: Coexistence in seasonally varying predator-prey systems with Allee effect
cites work: On the asymptotic behavior of some population models. II
cites work: Differential- und Integral-Ungleichungen
cites work: A Lyapunov-type stability criterion using $L^\alpha $ norms
DOI: 10.3934/ERA.2021069
description: scientific article; zbMATH DE number 7438963