 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-4/14c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

When

D_{u} = D_{v} = D

, the linearized operator is

\partial_{t} (v u) = (M + D \nabla^{2} I) (v u) .

k

, its two growth rates are

λ_{j} (k) = λ_{j} (0) - D k^{2} .

Both homogeneous growth rates have negative real part, and subtracting the nonnegative scalar

D k^{2}

moves them farther into the left half-plane. Thus no spatial mode can become unstable. This is the impossibility of a two-species Turing instability at equal diffusivities.

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