Скачать или смотреть The Keller-Segel model on the sphere, with an improved simulation grid

This simulation of the Keller-Segel model on the sphere uses an improved simulation grid, reducing some artifacts coming from the numerical approximations. The original grid is obtained by projecting onto the sphere a regular grid on the faces of a cube. This grid is then improved by letting the grid points evolve as if they were interacting by repulsive forces. I also corrected a small error in one of the edge projections (a neighbor was uncorrectly identified), which may have been responsible for some of the artifacts seen in previous simulations.
Chemotaxis is the motion of life forms induced by a chemical, such as a nutrient. The Keller-Segel model involves two fields: the concentration of the life form, for instance slime molds, and the concentration of the nutrient. The organisms follow the gradient of concentration of the nutrient to reach higher concentrations, thereby depleting the nutrient, which regenerates at a given rate. If u and v denote the concentrations of slime molds and nutrient, the equations are reaction-diffusion equations of the form
d_t u = Delta(u) - div(k(u)*grad(v)) + u(1-u)
d_t v = D*Delta(v) + u-a*v,
where Delta denotes the Laplace operator, div is the divergence and grad is the gradient. D measures the diffusion of the nutrient, while a measures how fast the organisms deplete the nutrient. k(u) measure the influence of the organisms' concentration on how quickly they follow the nutrient gradient, and is given here by k(u) = c*u*(1+u²). The usual choice is k(u) = c*u/(1+u²), but I did not find parameter values leading to interesting dynamics with that k.
This video has two parts, showing the same simulation with two different representations.
3D view: 0:00
2D view: 1:06
The color hue and the z-coordinate depend on the concentration of the organisms. The peaks have been truncated at a given height for more visibility, actually they form much higher cusps. The observer rotates around the sphere on a circular orbit in a plane containing the center of the sphere. A line starting from above the north pole, and perpendicular to the polar axis, aims at making this motion more visible.
The simulation mesh is obtained by projecting a regular grid on the faces of a cube onto the sphere, and the instability occurs at the projection of a corner of the cube. Some limiters have been put on the fields to avoid blow-up. This may be what causes the system to reach a uniform organism distribution at the end of the simulation.

This simulation is inspired by the online simulator
https://visualpde.com/sim/?preset=Kel...
that allows you to explore the effect of the different parameters on the system.

Render time: 3D part - 15 minutes 23 seconds
2D part - 15 minutes 51 seconds
Color scheme: Cividis by Jamie R. Nuñez, Christopher R. Anderton, Ryan S. Renslow
https://journals.plos.org/plosone/art...

Music: "Dulce Sospecho" by Cumbia Deli‪@CumbiaDeli‬

See also https://images.math.cnrs.fr/Des-ondes... for more explanations (in French) on a few previous simulations of wave equations.

#chemotaxis #reaction_diffusion #Keller_Segel

The simulation solves a partial differential equation by discretization.
C code: https://github.com/nilsberglund-orlea...
https://www.idpoisson.fr/berglund/sof...