  In this work we present a reduction result for discrete time systems with two
time scales. In order to be valid, previous results in the field require some
strong hypotheses that are difficult to check in practical applications.
Roughly speaking, the iterates of a map as well as their differentials must
converge uniformly on compact sets. Here, we eliminate the hypothesis of
uniform convergence of the differentials at no significant cost in the
conclusions of the result. This new result is then used to extend to nonlinear
cases the reduction of some population discrete models involving processes
acting at different time scales. In practical cases, some processes that occur
at a fast time scale are often only measured at slow time intervals, notably
mortality. For a general class of linear models that include such kind of
processes, it has been shown that a more realistic approach requires the
re-scaling of those processes to be considered at the fast time scale. We
develop the same type of re-scaling in some nonlinear models and prove the
corresponding reduction results. We also provide an application to a particular
model of a structured population in a two-patch environment.
