Below are several videos highlighting different aspects of pattern formation or spatiotemporal behaviour in growing Reaction-Diffusion Systems.
Above are 2D and 3D simulations of spot formation in a Gierer-Meinhardt system with growth along one axis. This growth is uniform within two different regions of the domain. In the Lagrangian (original) frame, 90% of the domain grows at some rate, and the other 10% grows at a rate 50 times larger. Where the change in growth rate occurs is the line (in 2D) or plane (in 3D) demarcated by the small black dots. In the 3D simulations, only regions of large activator are shown, and only in the lower half of the domain to see the internal concentrations of the really spherical regions of high activation.
Three examples of a rectangle deforming to a square, and then to a rectangle of inverse aspect ratio. Such a transformation preserves volume (and hences introduces no dilution in the concentrations). The simulations are arranged in order of increasing rate of deformation, so that the leftmost has very slow domain evolution (compared to the timescales of reaction and diffusion), whereas the right has very rapid domain evolution. For further details and discussion, see Figure 12 of this paper.
These are simulations using surface finite-elements of the FitzHugh-Nagumo reaction-diffusion model within a Turing parameter regime (so that a homogeneous initial condition leads to spatial patterning) on a growing torus. As the torus "grows" this patterning should become finer. However, in the left video only 3,976 triangular finite-elements are used, and this leads to a breakdown in the patterning, apparently leading to emergent spatiotemporal patterns. The video on the right uses 136,470 triangular finite-elements, and so we observe the predicted behaviour of finer spatial patterning during growth. More details about the system and its analysis can be found here.