Pattern formation by boundary forcing in convectively unstable, oscillatory media with and without differential transport

Abstract

Motivated by recent experiments and models of biological segmentation, we analyze the excitation of pattern-forming instabilities in convectively unstable reaction-diffusion-advection systems, occuring by constant or periodic forcing at the upstream boundary. Such boundary-controlled pattern selection is a generalization of the flow-distributed-oscillation (FDO) mechanism that can be modified to include differential diffusion (Turing) and differential flow (DIFI) modes. Our goal is to clarify the relationships among these mechanisms in the general case where there is differential flow as well as differential diffusion. We do so by analyzing the dispersion relation for linear perturbations and showing how its solutions are affected by differential transport. We find a close relationship between DIFI and FDO modes, while the Turing mechanism gives rise to a distinct set of unstable modes. Finally, we illustrate the relevance of the dispersion relations using nonlinear simulations and we discuss experimental implications of our results.

Flow and diffusion distributed structures with noise at the inlet

“…Similar approach in a context of FDO/FDS patterns was developed by McGraw and Menzinger [8] who studied small perturbations to the steady state and considered linear modes having real frequencies. This type of modes corresponds to a stationary forcing at the inlet.…”

“…The essential difference of our study is the consideration of small perturbations to a fully developed FDS pattern. The linearized equations (8) in this case have periodic coefficients and we should apply the Floquet theorem to analyze its stability properties.…”

“…In particular, these papers explore the relationship between FDS and other known types of instabilities in reaction-diffusion-advection systems: Turing patterns, DIFI and Hopf instabilities. As shown by McGraw and Menzinger in [8], the FDS (they use the term FDO, see the remark above) is closely related to the Hopf instability and DIFI, while the Turing pattern has a different nature.…”

“…Extensive theoretical studies of new structures are provided in [9,6,7,8]. In particular, these papers explore the relationship between FDS and other known types of instabilities in reaction-diffusion-advection systems: Turing patterns, DIFI and Hopf instabilities.…”

“…In this paper we adhere to the last version and refer to the structures under consideration as FDS. (See also the discussion of the terminology in [8]). …”