A note on dissipativity and permanence of delay difference equations

• Main Author: Ábel Garab
• Format: Article
• Language: English
• Published: University of Szeged 2018-06-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: delay difference equation; higher order difference equation; absorbing set; global pullback attractor; permanence; positive invariance; population dynamics
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=6907

We give sufficient conditions on the uniform boundedness and permanence of non-autonomous multiple delay difference equations of the form \[x_{k+1}=x_k f_k(x_{k-d},\dots,x_{k-1},x_k),\] where $f_k\colon D \subseteq (0,\infty)^{d+1}\to (0,\infty)$. Moreover, we construct a positively invariant absorbing set of the phase space, which implies also the existence of the global (pullback) attractor if the right-hand side is continuous. The results are applicable for a wide range of single species discrete time population dynamical models, such as (non-autonomous) models by Ricker, Pielou or Clark.