 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-3/13b/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

At the threshold, the determinant quadratic has a double zero. Its vertex lies at

s = (χ α n_{0} - β D_{n} - γ n_{0} D_{c}) / (2 D_{n} D_{c})

. Substituting the equality case of part (iii) gives

k_{c}^{2} = \frac{β γ n _{0}}{D _{n} D _{c}}, k_{c} = (\frac{β γ n _{0}}{D _{n} D _{c}})^{1/4} .

The mode is neutral exactly at onset and grows when the chemotactic coefficient increases beyond that threshold.

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