Stochastic Turing Patterns

Pattern formation is a phenomenon based on the interaction of different components, possibly under the influence of their surroundings. Alan Turing, a cryptographer and a pioneer in computer science, developed algorithms to describe complex patterns using simple inputs and random fluctuation. In 1952, he proposed that the interaction between two biochemical substances with different diffusion rates have the capacity to generate biological patterns. In his mathematical framework, there is one activating protein (activator) that activates both itself and an inhibitory protein (inhibitor), which only inhibits the activator. He detected that a stable homogeneous pattern could become unstable if the inhibitor diffuses more rapidly than the activator. The interplay between the concentrations of these substances forms a pattern whose spatiotemporal evolution is governed by coupled reaction-diffusion systems (activator-inhibitor model). By his equation he could generate a pattern of regularly-spaced spots, fingerprints, or only simple the stripes of a zebra. This phenomenon is called diffusion-driven instability Turing instability. Thus, the most fundamental phenomenon in pattern-forming activator-inhibitor systems is that a slight deviation from spatial homogeneity has vital positive feedback leading to increase further. The presence of nonlinearities in the local dynamics, for example, due to the inhibitor concentration, saturates the Turing instability into a stable and spatially inhomogeneous pattern. Usually, one models these equations in a deterministic framework. The deterministic model, i.e., the macroscopic system of equations, is derived from the microscopic behavior studying the limit behavior. From the microscopic perspective, one interprets the movements of the molecules as a result of microscopic irregular movement. Taking the limit and passing from the microscopic to the macroscopic equation, one neglects the fluctuations around the mean value. In addition, biological systems are frequently subject to noisy environments, inputs, and signalling. These stochastic perturbations are crucial when considering the ability of such models to reproduce results consistently. In our project we investigate the impact of the randomness to systems generating Turing patterns

How do stripes on zebras, spots on animal coats, patterns in chemical reactions, or spatial structures in biological systems arise? Such patterns can be created through the interplay of diffusion and chemical reactions. Alan Turing already showed that two chemical substances, which react with each other and spread at different rates can generate spatial patterns from an initially uniform state. This phenomenon is now known as Turing instability. In reality, however, biological, chemical, and ecological systems are never completely deterministic. They are constantly exposed to random influences: molecules move irregularly, environmental conditions fluctuate, and external disturbances cannot be fully controlled. Therefore, this project investigates Turing patterns under the influence of randomness. Mathematically, this leads to stochastic partial differential equations, that is, equations which describe both spatial and temporal evolution, as well as random perturbations. The aim of the project is to understand how random effects change the formation, stability, and long-time development of patterns. In particular, we study classical activator-inhibitor models such as the Grey-Scott model and the Gierer-Meinhardt model. These models play an important role in mathematical biology, chemistry, ecology, and physics. The project pursues several goals. First, we investigate the existence, uniqueness, and regularity of mathematical solutions. Then, we develop numerical methods which allow us to simulate the stochastic models reliably. Particular attention is paid to numerical schemes which preserve the essential structural properties of the models, also over long time intervals. Furthermore, we analyse the long-time behaviour of the systems, for example, whether typical statistical states arise, or whether random perturbations can generate new patterns, transitions between different states, or metastable behaviour. The project combines methods from stochastic analysis, partial differential equations, numerical mathematics, and dynamical systems. The results are expected to contribute to a better mathematical understanding of random pattern formation and to the development of more realistic models for applications, for example in biology, chemistry, and ecology.