Critical length of the transport-dominated region for oscillating non-linear reactive processes

Beyond an instability, non-linear processes can give rise to reactive modes exhibiting sustained oscillations in particle numbers. The coupling of such an oscillatory mode to diffusional transport in a system can cause coherent spatio-temporal structures to arise and persist indefinitely, far from thermal equilibrium, provided the system is above some critical size and is maintained open to mass and energy transfer.In the case of one spatial dimension, the critical size of the system, following Goldbeter [Proc. Nat. Acad. Sci. USA 70, 3255-3259 (1973)], can be defined as that size up to which there exists exactly one unique time-independent solution (which is completely transport dominated) to the macroscopic equation that characterizes the coupling of the reactive mode to diffusional transport and which is subject to inhomogeneous boundary conditions. A theoretical estimate of the critical size is derived (valid for arbitrary systems involving multicomponent non-linear reactive processes possessing an oscillatory mode) by making use only of the parameter-dependent period of the oscillatory mode and of the elements of the diffusion tensor. This estimate specifically takes cross-diffusion into account. In the special case of simple diagonal diffusion, an illustrative comparison is made with the prediction of a more model-specific estimate of Goldbeter that involves a model for glycolytic oscillations.