Pattern Formation in the Sel'kov Model

Pattern Formation in the Sel'kov Model

The video describes the formation of Turing-like patterns in a reaction-diffusion system where one species, which does not diffuse, is inhomogeneously distributed. It has been speculated that these instabilities are responsible for biological pattern formation, such as leopard spots. A chemical "spot", "stripe" or other inhomogeneous state with a characteristic wavelength evolves from the destabilization of a homogeneous state with no spatial structure.

Turing instabilities are driven by differences in the diffusion coefficients of chemical species. It is now believed that these difference are achieved by selective complexing with immobilized species. This is what is modeled in the simulations. The immobilized complexing agent is distributed in different geometries, ranging from strips with different widths and separations, to regular arrays of disks, to random arrays of disks.

The lengths and widths of the inhomogeneously distributed complexing agent are played against the intrinsic Turing wavelength to obtain the "zoo" of patterns in the video. In the absence of complexing agent the chemical system oscillates and these are the chemical waves one sees in the "holes" between the regions with complexing agents.

J. Voroney, A. Lawniczak and R. Kapral, "Turing Pattern Formation in Heterogeneous Media", Physica D, 99, 303 (1996).

[movie icon] Periodic stripes of disks with stripe width greater than the turing length (128KB mpeg-1 movie).

[movie icon] Same configuration; long time dynamics (134KB mpeg-1 movie).

[movie icon] Same configuration; even longer time dynamics (125KB mpeg-1 movie).

[movie icon] Hexagonal array of disks with a separation of one half of turing length (290KB mpeg-1 movie).

[movie icon] Square array of disks in a random distribution; concentration near the Turing-Hopf point (132KB mpeg-1 movie).