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 Incoming links: Critical length for a linearly growing morphogen

Put

a = f^{'} (0) > 0

C = 0

gives

C_{t} = D C_{xx} + a C .

Seek separated modes

C (x, t) = X (x) e^{λ t}

. Then

D X^{''} + a X = λ X, X (0) = 0, X^{'} (L) = 0.

X_{n} (x) = sin (k_{n} x), k_{n} = \frac{( n + 1/2 ) π}{L}, n = 0, 1, 2, \dots .

Indeed, the Dirichlet condition selects sine functions, while the Neumann condition requires

cos (k_{n} L) = 0

λ_{n} = a - D k_{n}^{2} = a - D \frac{( n + 1/2 ) ^{2} π ^{2}}{L ^{2}} .

The largest growth rate is the lowest mode,

λ_{0} = a - \frac{D π ^{2}}{4 L ^{2}} .

λ_{0} < 0

, after which all higher modes also decay. Therefore the critical length for a linearly growing morphogen is

L < \frac{π}{2} \frac{D}{f ^{'} ( 0 )} .

Equality gives a neutral lowest mode, while a larger domain is linearly unstable.