Diffusion-Reaction-Models and their Application to Biological and Astrophysical Problems

In this thesis we study several problems from biophysics and astrophysics, which can be all be described by reaction-diffusion systems. The first part of this thesis is concerned with biophysical quasispecies models. These are deterministic models, describing the interaction of mutations and selection (i.e. fitness advantage by adaption). We investigate the influence of robustness against deleterious mutations on the stationary states of these models. Here, robustness means that a certain number of mutations in the individual's information string is tolerated before the fitness of the individual is diminished. The equations of state for the quasispecies models can be represented by a reaction-diffusion equation for a special type of reaction term. We give analytic results for the robustness effect on the mean fitness of a population. These results become exact in the limit of infinite information-sequence length. By exploiting a mapping to a Schr\"odinger-type equation, we find correction terms for finite sequence length, essential for applications. The provided solutions allow to answer the question under what circumstances robustness is preferable to fitness, a question often referred to as \emph{survival of the flattest}. Additionally, we investigate the occurence of the error threshold (a phase transition of the population's state) in a general class of epistatic fitness landscapes. We show that diminishing epistasis is necessary but not sufficient for the emergence of an error threshold. All analytic work is supported and verified by numerical studies. In the second part we investigate diffusion-mediated reactions on two-dimen-sional surfaces drawing on the example of hydrogen formation on interstellar dust grains. The surfaces of these dust grains play an important role in molecule production in the interstellar medium, by acting as catalysts. We are interested in the influence of (quenched) surface disorder on the production rate of molecules. As model system, we study the not yet completely understood reaction H+H -> H2 of hydrogen formation. We confirm the earlier proposed significant enhancement in the production rate of this process by disorder in the binding energies of the surface and moreover give analytic results for different distributions of binding energies. We identify the main mechanism leading to an enhanced production rate, enabling us to give temperature dependent mappings from systems with discrete and continuous binding energy distributions to effective systems with only a binary energy distribution. The analytical results on all models are confirmed by numerical solutions of the full rate equations as well as by kinetic Monte Carlo simulations