Pullback permanence in a non-autonomous competitive Lotka--Volterra model
type: Item
instance of: scholarly article
title: Pullback permanence in a non-autonomous competitive Lotka--Volterra model
zbMATH Open document ID: 1027.35017
DOI: 10.1016/S0022-0396(02)00173-0
author: James C. Robinson
author: J. A. Langa
author: Antonio Suárez
published in: Journal of Differential Equations
publication date: 2003-05-25T00:00:00Z
review text: The goal of the paper is to study the long-time behaviour of the non-autonomous competitive system \[ \begin{aligned} u_t-\Delta u=u(\lambda -a(t)u-bv) &\quad\text{in } Q\times (s,+\infty),\\ v_t-\Delta v=v(\mu -cv-du) &\quad\text{in } Q\times (s,+\infty), \end{aligned} \] where \(Q\subset \mathbb{R}^N\) is a smooth bounded domain, \(b, c, d\) and \(a(t)\) are positive. Both the forwards and pullback behaviour of system are analysed. The notion of ''pullback permanence'' is introduced. The existence of a non-autonomous attractor and conditions for pullback permanence are estblished. The main tools in the paper are the theory of attractors for non-autonomous differential equations, the sub-supersolution method and the spectral theory for linear elliptic equations.
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1915659
zbMATH Keywords: attractors for non-autonomous differential equations
zbMATH Keywords: competitive diffusion system
zbMATH Keywords: pullback attractor
zbMATH Keywords: sub-supersolution method
reviewed by: E. A. Kalita
MaRDI profile type: Publication
cites work: Order-preserving random dynamical systems: equilibria, attractors, applications
cites work: Should a Park Be an Island?
cites work: Practical persistence in ecological models via comparison methods
cites work: Permanence in ecological systems with spatial heterogeneity
cites work: Ecological models, permanence and spatial heterogeneity
cites work: Q4285039
cites work: Attractors of non-autonomous dynamical systems and their dimension
cites work: Attractors for equations of mathematical physics
cites work: Order-preserving skew-product flows and nonautonomous parabolic systems
cites work: Random attractors
cites work: Persistence in Infinite-Dimensional Systems
cites work: Geometric theory of semilinear parabolic equations
cites work: Q3998030
cites work: On an abstract competition model and applications
cites work: Permanence and the dynamics of biological systems
cites work: Nonautonomous attractors of skew-product flows with digitized driving systems
cites work: Nonautonomous systems, cocycle attractors and variable time-step discretization
cites work: Asymptotic behaviour of nonautonomous difference inclusions
cites work: Stability, instability, and bifurcation phenomena in non-autonomous differential equations
cites work: Forwards and pullback behaviour of a non-autonomous Lotka–Volterra system
cites work: Q3149644
cites work: The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems
cites work: On the structure of the permanence region for competing species models with general diffusivities and transport effects
cites work: Coexistence States and Global Attractivity for Some Convective Diffusive Competing Species Models
cites work: Q3240942
cites work: Asymptotically Autonomous Semiflows: Chain Recurrence and Lyapunov Functions
cites work: Semilinear Parabolic Problems Define Semiflows on C k Spaces
cites work: Q5584121
cites work: Q4762866
cites work: Nonlinear elliptic boundary-value problems in unbounded domains
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Uniform persistence and permanence for non-autonomous semiflows in population biology
description: scientific article; zbMATH DE number 1915659