Pattern selection in reaction-diffusion-ODE and mechanochemical systemsOberseminar Analysis

by Dr Finn Münnich (University Heidelberg)

Friday Nov 13, 2026, 2:15 PM → 4:15 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

 Spontaneous pattern formation is classically explained by Turing's diffusion-driven instability (DDI), typically realised by a slowly diffusing activator and a fast-diffusing inhibitor. In this talk, I discuss two settings beyond this classical picture and ask, in each, which pattern is actually selected.
First, I consider reaction-diffusion systems in which some components do not diffuse, so-called reaction-diffusion-ODE systems. Their stationary solutions can be less regular than in classical reaction-diffusion systems, exhibiting for example jump discontinuities, and the linearised operator has nontrivial essential spectrum. I derive sufficient conditions for DDI in systems with three spatial scales (non-diffusing, slow- and fast-diffusing components), and show that the destabilisation mechanism influences the stability of the emerging patterns. I also construct patterns with jump discontinuities, which exist largely independently of the diffusion coefficients. Deciding between regular and jump-discontinuous patterns requires a nonlinear stability theory for bounded, possibly discontinuous, steady states. I show that, in the space of bounded functions, a negative spectral bound of the linearisation implies nonlinear stability and a positive one implies instability, without requiring a spectral gap.
Second, I present a mechanochemical model for head formation in the freshwater polyp Hydra, in which tissue mechanics takes the role of the fast-diffusing inhibitor. A one-dimensional reduction leads to a nonlocal reaction-diffusion equation, for which only single-peak patterns can be stable, consistent with the single body axis of Hydra. A bifurcation analysis reveals subcritical branches and bistability between the constant state and nonconstant stationary solutions, so that in this regime patterns emerge only from sufficiently large perturbations of the homogeneous state.
Based on joint work with T. André, S. Cygan, C. Kowall, A. Marciniak-Czochra and D. Oelz.