Figure – Subcritical Turing bifurcation: formation of a hexagonal pattern from noisy initial conditions in the two-component reaction-diffusion system of Fitzhugh-Nagumo type. From left to rigth:a) Noisy initial conditions at t = 0. b) State of the system at t = 10. c) Almost converged state at t = 100. (source link)
Figure – Other patterns found in the above two-component reaction-diffusion system of Fitzhugh-Nagumo type. From left to rigth: a) Rotating spiral.b) Target pattern. c) Stationary localized pulse (dissipative soliton). (source link)
When an activator-inhibitor system undergoes a change of parameters, one may pass from conditions under which a homogeneous ground state is stable to conditions under which it is linearly unstable. The corresponding bifurcation may be either a Hopf bifurcation to a globally oscillating homogeneous state with a dominant wave number k=0 or a Turing bifurcation to a globally patterned state with a dominant finite wave number. The latter in two spatial dimensions typically leads to stripe or hexagonal patterns. [p.s. – a related lovely beach bay!.. state-phase diagram; Reaction-Diffusion by the Gray-Scott model: Pearson’s Parameterization (link) ]
Video – “BIG BANG BIG BOOM”: an unscientific point of view on the beginning and evolution of life … and how it could probably end. Direction and animation by BLU blublu.org / production and distribution by ARTSH.it / sountrack by Andrea Martignoni.
[…] It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically, unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading. […], A. M. Turing, “The Chemical Basis of Morphogenesis“, in Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, Vol. 237, No. 641. (Aug. 14, 1952), pp. 37-72. (link)