  In theoretical ecology, models describing the spatial dispersal and the
temporal evolution of species having non-overlapping generations are often
based on integrodifference equations. For various such applications the
environment has an aperiodic influence on the models leading to nonautonomous
integrodifference equations. In order to capture their long-term behaviour
comprehensively, both pullback and forward attractors, as well as forward limit
sets are constructed for general infinite-dimensional nonautonomous dynamical
systems in discrete time. While the theory of pullback attractors, but not
their application to integrodifference equations, is meanwhile
well-established, the present novel approach is needed in order to understand
their future behaviour.
