Analysis & Applied Math

Modern biomorphology models such as Murray-Oster and Scianna-Bell-Preziosi involve pattern formation in systems with mechanical/hydrodynamical effects taking the form of convection-reaction-diffusion models with conservation laws. Here, extending previous work of Matthews-Cox and H\"acker-Schneider-Zimmerman in pattern formation with conservation laws, and of Eckhaus, Mielke,and Schneider on stability of Turing patterns in reacti on diffusion models, we investi gate diffusive stability of Turing patterns for convection-reaction-diffusion models with conservation laws. Formal multi scale expansion yields a singular system of amplitude equations coupling Complex Ginzburg Landau with a singular convection-diffusion system, similar to partially coupled systems found by H\"acker-Schneider-Zimmerman in the context of thin film flow, but with the singular convection part now fully engaged in long term stability and behavior rather than transient as in the (triangular) partially coupled case. The resulting complicated two-parameter matrix perturbation problem governing spectral stability can nonetheless be solved, yielding (m+1) simple stability criteria analogous to the Eckhaus and Benjamin-Feier-Newell criteria of the classical (no conservation law) case, where m is the number of conservation laws. It is to be hoped that these can play the same important role in the study of biopattern formation as the classical ones in myriad other applications.

Indiana University and Fields Institute