Abstract
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.
Highlights
Integrodifference equations occur as temporal discretisations of integrodifferential equations or as time-1-maps of evolutionary differential equations, but are of interest in themselves
First and foremost, they are a popular tool in theoretical ecology to describe the dispersal of species having non-overlapping generations
While the theory of Urysohn or Hammerstein integral equations is rather classical [19], both numerically and analytically, our goal is here to study their iterates from a dynamical systems perspective
Summary
Integrodifference equations occur as temporal discretisations of integrodifferential equations or as time-1-maps of evolutionary differential equations, but are of interest in themselves. – This led to the development of forward attractors, which are compact and invariant nonautonomous sets [15] This dual concept depends on information from the future and given a fixed initial time, the actual time increases beyond all bounds — they are a moving target problem. The situation for forward attractors and limit sets is not as well-established as their pullback counterparts and deserves to be developed for the above reasons Their initial construction in [12,16] requires a locally compact state space, but recent continuous-time results in [6], which extend these to infinite-dimensional dynamical systems, will be transferred here. On a Banach space X , L(X ) denotes the space of bounded linear operators and ρ(L) is the spectral radius of a L ∈ L(X )
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