Solution landscape of reaction-diffusion systems reveals a nonlinear mechanism and spatial robustness of pattern formation

Spontaneous pattern formation in homogeneous systems is ubiquitous in nature. Although Turing demonstrated that spatial patterns can emerge in reaction-diffusion (RD) systems when the homogeneous state becomes linearly unstable, it remains unclear whether the Turing mechanism is the only route for pattern formation. Here, we develop an efficient algorithm to systematically map the solution landscape to find all steady-state solutions. By applying our method to generic RD models, we find that stable spatial patterns can emerge via saddle-node bifurcations before the onset of Turing instability. Furthermore, by using a generalized action in functional space based on large deviation theory, our method is extended to evaluate stability of spatial patterns against noise. Applying this general approach in a three-species RD model, we show that though formation of Turing patterns only requires two chemical species, the third species is critical for stabilizing patterns against strong intrinsic noise in small biochemical systems.