 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/6b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Along a solution, the chain rule and antisymmetry of

K

give

\frac{d H}{d t} = (\nabla H)^{T} \dot{x} = (\nabla H)^{T} K \nabla H = 0,

because

v^{T} K v = 0

for every real vector

v

. Therefore

H (x_{1} (t), x_{2} (t)) is constant .

The closed level curves around the strict minimum at the coexistence equilibrium are the periodic population cycles of the Lotka-Volterra predator-prey model.

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