Robust uniform persistence in discrete and continuous nonautonomous systems
type: Item
instance of: scholarly article
title: Robust uniform persistence in discrete and continuous nonautonomous systems
zbMATH Open document ID: 1267.34086
author: Paul L. Salceanu
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-12-20T00:00:00Z
review text: The paper extends earlier results of the author on nonautonomous equations. In detail, it addresses difference equations \[ x_{n+1}=f_n(x_n,y_n),\quad y_{n+1}=A(x_n,y_n)y_n, \] as well as ordinary differential equations \[ \dot x=f(t,x,y),\quad \dot y=A(t,x,y)y \] in the positive cone of \({\mathbb R}^p\times{\mathbb R}^q\) which exhibit a positively invariant boundary hyperplane \(X\). It is shown that robust uniform persistence for the dynamics in \(({\mathbb R}^p\times{\mathbb R}^q)\setminus X\) is obtained, provided a compact subset of \(X\) attracting all solutions in \(X\) is a robust uniform weak repellent. Here, the nonautonomous nature of the equations requires some additional uniformity assumptions. The time-periodic case is treated in detail. Finally, the results are illustrated using a periodic discrete-time epidemic model and a continuous-time SIRS model with \(n\) infection strains. It has to be pointed out that convergence or persistence statements are always understood in the forward rather than the pullback sense.
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 37C60
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 92D30
zbMATH DE Number: 6117534
zbMATH Keywords: robust uniform persistence
zbMATH Keywords: nonautonomous systems
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: disease persistence
zbMATH Keywords: epidemic models
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.09.005
OpenAlex ID: W2045212120
DOI: 10.1016/J.JMAA.2012.09.005
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description: scientific article; zbMATH DE number 6117534