Beyond Turing: controlling pattern stability in reaction-diffusion systems

Frits Veerman (Mathematical Institute of Leiden University) #

In many physical and biological applications, observed patterns are thought to arise through a Turing bifurcation that induces spontaneous pattern formation by the destabilisation of a homogeneous background state. However, the lack of system robustness inherently associated to passing through a bifurcation does not match observational evidence, where patterns consistently arise in noisy, heterogeneous environments.

We investigate the stabilisation of hitherto unstable patterns in singularly perturbed reaction-diffusion systems by an external control input, which can be interpreted as a pre-pattern or as the effective input of external processes that are not part of the model. We show that, by extending the Evans function method previously used to determine the stability of far-from-equilibrium patterns, a wide range of spatio-temporal controls can be incorporated, and that their influence on the stability of a specific pattern can be explicitly determined. We demonstrate the method in the context of a toy model, where a singular pulse can be stabilised by a localised control term.

This is joint work with Isabelle Schneider, FU Berlin.