Linear coupling of patterning systems can have nonlinear effects.

Isolated patterning systems have been repeatedly investigated. However, biological systems rarely work on their own. This paper presents a theoretical and quantitative analysis of a two-domain interconnected geometry, or bilayer, coupling two two-species reaction-diffusion systems mimicking interlayer communication, such as in mammary organoids. Each layer has identical kinetics and parameters, but differing diffusion coefficients. Critically, we show that despite a linear coupling between the layers, the model demonstrates nonlinear behavior; the coupling can lead to pattern suppression or pattern enhancement. Using the Routh-Hurwitz stability criterion multiple times, we investigate the pattern-forming capabilities of the uncoupled system, the weakly coupled system, and the strongly coupled system, using numerical simulations to back up the analysis. We show that although the dispersion relation of the entire system is a nontrivial octic polynomial, the patterning wave modes in the strongly coupled case can be approximated by a quartic polynomial, whose features are easier to understand.