Lesson 9: Grids and diffusion

"""Turing patterns: the Gray-Scott reaction-diffusion rule on a 128 x 128 grid.
Run: python turing-patterns.py   (needs: pip install numpy matplotlib; takes a few seconds)
"""
import numpy as np
import matplotlib.pyplot as plt

plt.style.use("dark_background")

size, steps = 128, 6000
f, k = 0.0545, 0.062  # feed and kill rates: coral
DA, DB = 1.0, 0.5  # A spreads twice as fast as B

A, B = np.ones((size, size)), np.zeros((size, size))
rng = np.random.default_rng()
for y, x in rng.integers(10, size - 18, (6, 2)):
    B[y : y + 8, x : x + 8] = 1.0  # six drops of B


def laplacian(Z):
    """Each cell against its neighbours: 0.2 for the four sides, 0.05 for the corners (Karl Sims)"""
    sides = np.roll(Z, 1, 0) + np.roll(Z, -1, 0) + np.roll(Z, 1, 1) + np.roll(Z, -1, 1)
    corners = (np.roll(np.roll(Z, 1, 0), 1, 1) + np.roll(np.roll(Z, 1, 0), -1, 1)
               + np.roll(np.roll(Z, -1, 0), 1, 1) + np.roll(np.roll(Z, -1, 0), -1, 1))
    return 0.2 * sides + 0.05 * corners - Z


for _ in range(steps):
    reaction = A * B * B  # one A and two B make three B
    A += DA * laplacian(A) - reaction + f * (1 - A)
    B += DB * laplacian(B) + reaction - (f + k) * B

plt.figure(figsize=(6, 6))
plt.imshow(B, cmap="magma", vmin=0, vmax=0.5)
plt.axis("off")
plt.show()

# Try: f, k = 0.039, 0.058 (a cell net) or 0.034, 0.063 (spots) -> the same rule, another pattern

No Python on this computer? The notebook runs the same script, the exercises and the answers in your browser, on Google Colab.

Line by line

1. A, B = np.ones((size, size)), np.zeros((size, size))

   Two 128 × 128 grids: the food A everywhere, no B yet.

2. B[y : y + 8, x : x + 8] = 1.0

   A drop of B: an 8 × 8 square of the grid, set at once by slicing.

3. def laplacian(Z):

   How much a cell differs from its neighbours: positive where the neighbours have more, so diffusion fills it up.

4. sides = np.roll(Z, 1, 0) + np.roll(Z, -1, 0) + np.roll(Z, 1, 1) + np.roll(Z, -1, 1)

   np.roll shifts the whole grid by one cell up, down, left or right; the edges wrap around, so the dish has no border.

5. reaction = A * B * B

   The reaction, one A and two B make three B, for every cell at once.

6. A += DA * laplacian(A) - reaction + f * (1 - A)

   The Gray–Scott rule for A: it spreads, is eaten, and is fed back towards 1 at the rate f.

7. plt.imshow(B, cmap="magma", vmin=0, vmax=0.5)

   The grid of B as a picture, one pixel per cell.

Try it yourself

1. Try f, k = 0.034, 0.063.

   Hint

   One line near the top.

   Answer

   Spots: round patches of B, evenly spaced, instead of the coral’s winding lines.

2. Make B spread as fast as A: DB = 1.0. What happens?

   Hint

   DA, DB = 1.0, 1.0.

   Answer

   No pattern at all: the drops fade away to nothing. Without A spreading faster than B, nothing holds the patches apart. This is Turing’s point.

3. Use one drop in the middle instead of six.

   Hint

   Replace the loop with B[60:68, 60:68] = 1.0.

   Answer

   The coral grows outwards from the one drop with a fourfold symmetry that copies the square drop: one seed gives an ordered pattern, six random drops a natural-looking one.

Quiz