A note on dissipativity and permanence of delay difference equations

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) ,..., x(k-1), x(k)), where f(k) : D subset of (0, infinity)(d+1) -> (0, infinity). 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.