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Chemotactic telegraph equation Created 2026-09-28 Updated 2026-11-05

A chemotactic telegraph equation describes persistent motion with finite turning rate and a turning bias set by a chemical gradient. Its large-turning-rate limit is a drift--diffusion equation with chemotactic drift.

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Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 356 / 3 / b / Solution 2026-09-28

The two transport equations are

\partial_{t} p_{+} = - s \partial_{x} p_{+} - λ_{+} p_{+} + λ_{-} p_{-},

\partial_{t} p_{-} = s \partial_{x} p_{-} + λ_{+} p_{+} - λ_{-} p_{-} .

For

p = p_{+} + p_{-}

and

j = s (p_{+} - p_{-})

they become

p_{t} = - j_{x}, j_{t} = - s^{2} p_{x} + 2 s b c^{'} (x) p - 2 λ_{0} j .

Eliminating

j

gives the closed chemotactic telegraph equation

p_{tt} + 2 λ_{0} p_{t} = s^{2} p_{xx} - 2 s b \partial_{x} [c^{'} (x) p] .

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