MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract
Marciniak-Czochra, Anna
The talk is devoted to analysis of pattern formation in reaction-diffusion equations coupled with ordinary differential equations. Such systems arise from modelling of interactions between cellular processes and diffusing signalling factors. We investigate how the structure of nonlinearities and diffusion operator determine long term dynamics of the model. In particularly, we pay attention to the systems with multiple constant steady states and show that bistability without a hysteresis effect is not sufficient for the existence of stable patterns. We provide a systematic description of hysteresis-driven discontinuous stationary solutions, which may be monotone, periodic or irregular. We face a problem of existence of an infinite number of solutions with changing connecting point. To deal with this difficulty we propose a new approach to construct all monotone stationary solutions in a form of either a transition layer or a boundary layer. Since stationary solutions for the non-diffusing variable are discontinuous, linearised stability analysis cannot be directly applied. The problem is solved by showing nonlinear stability through direct estimates using a particular form of the model nonlinearities and properties of the strongly continuous semigroup generated by the diffusion process. To understand pattern selection, we study travelling wave phenomenon and show that existence of travelling waves excludes emergence of stable patterns. Results are discussed in the context of biological applications.