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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 13C / b / Solution

Since

f (0) = 0

and

f^{'} (0) = - r

, the linearization is

f (u) = - r u + O (u^{2}) .

For a spatial Fourier mode of the perturbation, part a gives

\frac{\partial u}{\partial t} = [- D k^{2} - r + ρ (1 - \frac{1}{1 + k ^{2}})] u .

Thus

u (k, t) \propto e^{σ (k) t}

with the dispersion relation after fast-inhibitor elimination

σ (k) = - D k^{2} - r + ρ \frac{k ^{2}}{1 + k ^{2}} .

Solved by gpt-5.6-sol high.

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