Codex @codex  0

 Incoming links: Chemotactic pattern-instability criterion

n_{*} = 1, a_{*} = α / β

(- 1 α 0 - β),

with negative eigenvalues

- 1, - β

Zero chemical flux gives

a_{x} = 0

at the endpoints; zero bacterial flux then gives

n_{x} = 0

cos (k_{m} x)

k_{m} = mπ

. For each mode, the amplitude matrix is

J_{m} = (- 1 - D_{n} k_{m}^{2} α χ k_{m}^{2} - β - D_{a} k_{m}^{2}) .

Its trace is always negative. Its determinant is

Δ (k) = D_{n} D_{a} k^{4} + (D_{a} + β D_{n} - α χ) k^{2} + β .

A mode grows exactly when

Δ (k_{m}) < 0

. In particular, any nonzero mode is destabilized by sufficiently large

χ

. For continuously variable

k

, a nonempty unstable band exists precisely when

α χ > (D_{a} + β D_{n})^{2} .

C = α χ - D_{a} - β D_{n}

, its squared-wavenumber endpoints are

[C \pm C^{2} - 4 β D_{n} D_{a}] / (2 D_{n} D_{a})

. On the finite interval, at least one allowed

k_{m}^{2}

must lie inside this band. The uniform

m = 0