Numerical Dynamics of Integrodifference Equations

Over the last decades, integrodifference equations proved to be valuable models for dispersal processes being discrete in time, but continuous in space. In applied sciences, their behavior is often demonstrated using numerical simulations. 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. We refers to both convergence and persistence properties. This verifies the results of numerical simulations. We thus not only provide a first contribution to the numerical dynamics of integrodifference equations under a general class of discretizations, but rather also enrich the field of nonautonmous dynamics by various aspects: Full discretizations are based on collectively compact operators, which we introduce to numerical dynamics. Explicitly time-dependent (nonautonomous) problems are considered, requiring innovative approaches to stability and attractor theory. Finally, we develop methods preserving dynamical properties under numerical discretisation (dissipativity, center manifolds, monotonicity). In conclusion, the obtained results are fundamental to validate frequently applied numerical simulations, and suggest more appropriate methods specifically designed for particular problems (a priori).

Integrodifference equations serve as succesful and popular models in theoretical ecology to describe the growth and dispersal of species having non-overlapping generations. Further applications allow to model the spread of diseases and infections in epidemiology. However, the structure of integrodifference equations requires that their long-term behavior is typically illustrated using computational simulations based on numerical discretizations. For this project located in the intersection of Dynamical Systems Theory and Numerical Analysis, we provided a rigorous mathematical foundation that such simulations reflect the actual dynamical behavior. This led to guidelines for non-mathematicians and simulating scientists, not only recommending the type of discretization method, but also specifying their convergence behavior and finally justifying the observed simulation results.