Spatial-temporal patterns in active medium caused by diffusion instability
The results of investigation of reaction-diffusion type models demonstrating diffusion instability are presented. In particular, in general case the condition for both Turing and wave instabilities are obtained for three equations of this type with the diagonal diffusion matrix. Qualitative properties of the system, in which bifurcations of each of the two types can take place, are clarified. Investigation of a set of amplitude equations, describing interaction of several modes which became unstable due to the wave bifurcation, is carried out. It is shown that as a result of competition between modes depending on the value of the parameter defining the strength of interaction only two regimes are possible: either quasi one-dimensional travelling waves (there exists only one nonzero mode) or standing waves (all the modes are nonzero). A possible mechanism for the transition from standing waves to traveling waves with a half wavelength, observed in the Belousov-Zhabotinsky reaction dispersed in a water-in-oil aerosol microemulsion, is considered.