A mathematical mechanism for instabilities in stripe formation on growing domains

A cascade process involving stripe splitting in reactiondiffusion systems with isotropically growing one-dimensional domains is studied. Such cascades, propagating from a smaller domain to a larger domain, have been proposed as an answer to the criticism that the Turing mechanism lacks robustness because many stable patterns can coexist on a large domain and, therefore, the final state is very sensitive to the initial conditions. In contrast, if the system starts with a small domain, very few stable patterns are possible, which limits the sensitivity to the initial conditions. In order to show the validity and limitations of this scenario, we clarify the underlying mathematical mechanism that drives various types of stripe-splitting via a reduction from partial differential equations to ordinary differential equations, as well as investigating global arrangements of the set of n-mode stripe branches with ^Dn-symmetry of the stripe locations. The mathematical simplification above allows us to reveal how each n-mode stripe branch is destabilized as the domain grows and to characterize the associated eigenprofiles that actually determine the manner of splitting at the infinitesimal level. We find that all the ^Dn-symmetry-breaking instabilities typically occur simultaneously up to leading order before the saddle-node point of the n-mode stripe branch is reached. The instability with the largest real part is of the alternate type: every other peak splits at the infinitesimal level. A symmetry-preserving instability appears at the saddle-node point, which drives the simultaneous type of splitting, i.e., mode-doubling. Due to competition between these two types of instabilities, the problem depends subtly on the growth speed. Alternate splitting typically arises for slow growth and simultaneous splitting for fast growth. For intermediate growth rates, the manner of splitting becomes mixed and sensitive to fluctuations.