Singular Amplitude Equations and Eckhaus Type Stability Criteria for Convective Turing Bifurcation with Conservation Laws

In modern biomorphology models like Murray-Oster, the reaction diffusion scenario of Turing is augmented by mechanical forces, leading to reaction convection diffusion equations with conservation laws. This leads to formal Eckhauss-type amplitude equations that are *singular* with respect to the small bifurcation parameter epsilon, possessing a fast-slow time scale structure. We show that nonetheless one may reduce the question of stability to constant coefficient (singular) and, by a detailed 2-parameter matrix perturbation analysis, obtain Eckhaus type stability criteria analogous to those of the classical Turing case that are necessary and sufficient for spectral, linearized and nonlinear stability of associated bifurcating Turing patterns