From bistability to temporal oscillations and spatial pattern formation

Abstract

The idea ‘‘from bistability to oscillations,’’ originally proposed by De Kepper and Boissonade for studying and designing chemical oscillations, is extended to ‘‘from bistability to spatial pattern formation.’’ To carry out such extension, the idea from bistability to oscillations is discussed again by intuitive arguments based on the analysis of the intersection of the nullclines of kinetic equations and the linear stability analysis. By analogy between the kinetic equations of homogeneous reactions and the stationary equations of reaction–diffusion processes, similar intuitive arguments and linear stability analysis are applied to the reaction–diffusion equations, which leads to the conclusion that the homogeneous steady state being on the middle branch of the nullcline of the bistable subsystem is a necessary condition for spatial pattern formation. This condition becomes sufficient if the diffusion coefficients of the species involved in the feedback subsystem are much larger than those of the species which constitute the bistable subsystem. To demonstrate the validity of these conclusions, an example of heterogeneous catalysis system is studied analytically and numerically, and various temporal-spatial patterns, including the coexistence of a homogeneous steady state and an oscillating state or a spatial pattern, are revealed. The potential applications of the idea from bistability to spatial pattern formation to the experimental study and design of spatial patterns are discussed.