AAU

Christian Pötzsche (Jun 2022)

Numerical dynamics of integrodifference equations: Periodic solutions and invariant manifolds in C𝜶(𝜴)

submitted (26p.)

Integrodifference equations are versatile models in theoretical ecology to describe the spatial dispersal of species involving in non-overlapping generations. The dynamics of these infinite-dimensional discrete dynamical systems is often illustrated using computer simulations.

This paper studies the effect of numerical discretization of Nyström-, collocation and degenerate kernel type to the local dynamics of periodic integrodifference equations having the (Hölder) continuous functions over a compact domain as state space. We provide a satisfactory understanding for the behavior of hyperbolic periodic solutions and their associated stable and unstable invariant manifolds. Both persistence and convergence results respecting the order of the particular method are established and illustrated using numerical examples.

Numerical dynamics of integrodifference equations:

Forward dynamics and pullback attractors

(with Huy Huynh and P.E. Kloeden) submitted (42p.)

In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior needs to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pullback attractors, as well as forward limit sets persist and that the latter two notions even converge under perturbation. As concrete application, we study integrodifference equations under spatial discretization of collocation type.