Bifurcation characteristics and spatial patterns in an integro-differential equation

Spatial patterns are studied in reaction-diffusion systems where chemical reactions are coupled with diffusion. An integro-differential equation is obtained when the inhibitor has a much larger reaction rate than the activator. The periodic spatial pattern near the bifurcation point β_c is described with a time-dependent Ginzburg-Landau equation derived using a perturbative method, which can be applied to the situation where the amplitude of the patterned solution develops gradually and also where it appears discontinuously at β_c. Furthermore, a new type of potential is proposed to explain comprehensively the mechanisms of the bifurcation and the localized spatial pattern appearing very far from equilibrium. Since the potential is constructed from two internal variables, the bifurcation can be understood from the analogy with the first-order phase transition in equilibrium systems, and at the same time the appearance of patterns is shown to be analogous with the phase separation. The theoretical results are compared with numerical calculations.