Numerical Dynamics of Integro Differenc...

Over the last decades, integrodifference equations (IDEs for short) proved to be valuable models for dispersal processes being discrete in time, but continuous in space. The aim of the two projects at hand is to investigate the behavior of such infinite-dimensional discrete dynamical systems under spatial approximation and to relate their behaviour to the actual long-term dynamics. This refers to both convergence and persistence properties. We thus not only provide a first contribution to the numerical dynamics of IDEs under a general class of discretizations and their specific convergence theory, but rather also enrich the field of nonautonmous dynamics:

• Full discretizations are based on collectively compact operators, which require innovative methods replacing the commonly suitable uniform convergence in numerical dynamics.
• We contribute to the still rudimentary perturbation theory for the dichotomy spectrum.
• The recent concept of forward attractors is extended to infinite-dimensional systems.
• Methods preserving dynamical features (dissipativity, center manifolds, monotonicity) are to be developed.

The mentioned field is sufficiently wide to provide questions for (at least) two PhD projects. This flexibility to choose between different tasks consequently allows to develop individual profiles of the candidates, but also carries potential for synergetic effects.

Last, but not least, the obtained results are fundamental to validate frequently applied numerical simulations, and suggest more appropriate methods (a priori).