virtuell

Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in specific situations, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray-Scott equations constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear. In the derivation of a macroscopic model such as the deterministic Gray-Scott equations from basic physical principles, certain aspects of microscopic dynamics, e.g.~fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations. The randomness leads to a variate of new phenomena and may have a highly non-trivial impact on the behaviour of the solution. E.g. it has been shown by numerical modelling that the stochastic extension leads to a broader range of parameter with Turing patterns by a genetically engineered synthetic bacterial population in which the signalling molecules form a stochastic activator-inhibitor system. The stochastic extension may lead to multistability and noise-induced transitions between different states. In the talk, we will introduce the Gray Scott system, which is a special case of an activator-inhibitor system. Then, we introduce its numerical modelling and highlight the proof of convergence. References: Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray-Scott equations. J. Comput. Appl. Math. 364, Joint work with Jonas Toelle: A Schauder Tychonoff type Theorem and the stochastic Klausmeier system (Archive)

Hörer sollten sich bitte im Moodle-Raum https://moodle.tu-dortmund.de/enrol/index.php?id=23191 des Seminars anmelden. An alle Angemeldeten werden die Zugangsdaten zum meeting geschickt Weitere Vorträge in der Reihe sind unter http://www.mathematik.tu-dortmund.de/lsix/lehre/WiSe2021/GRK-Seminar/index.php einsehbar.

Erika Hausenblas

Montanuniversität Leoben