Attractors of deterministic and random lattice difference equations

Abstract

Lattice difference equations are essentially difference equations on a Hilbert space of bi-infinite sequences. They are motivated by the discretization of the spatial variable in integrodifference equations arising in theoretical ecology. It is shown here that under similar assumptions to those used for such integrodifference equations they have a global attractor, to which the global attractors of finite-dimensional approximations converge upper-semi-continuously. Corresponding results are also shown for the lattice difference equations when only a finite number of interconnection weights are nonzero and when the interconnection weights themselves vary and converge in an appropriate manner.