Pattern Formation in Spatially Extended Systems: Interplay of Variability and Noise

Many spatially extended systems show a complex long-range dynamics and one can observe coherent pattern formation. For a fundamental understanding of the systems one has to know the pattern forming mechanisms. However the spatiotemporal dynamics of biological systems is even more complicated, because the omnipresent stochasticity may play an important role. Biological systems are exposed to large fluctuations and have to be adapted to noise. A second omnipresent source of stochasticity is the variability, which denotes static stochastic differences between otherwise equal elements of a system. It means that e.g. in a colony of bacteria the single cells are slightly different from each other. In this thesis the influence of variability and noise on spatially extended systems is investigated in detail. The model systems are oscillatory and excitable nets of FitzHugh-Nagumo, Hodgkin-Huxley and reduced Hodgkin-Huxley elements, respectively. It is shown that the interplay of variability and noise may be essential for pattern formation mechanisms, stochastic resonance effects and synchronisation. Multiplicative stochastic terms have in addition a systematic influence on the system dynamics, which may cause a wide range of interesting phenomena, like variability and/or noise induced transitions between different regimes.