 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/13a/c/solution

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

λ

⟨ n ⟩ = λ, ⟨ n (n - 1)⟩ = λ^{2} .

Substitution into part (b) gives the closed Riccati equation

\dot{λ} = α + β λ - γ λ^{2} .

Its positive equilibrium is stable on the nonnegative half-line and equals

λ_{*} = \frac{β + β ^{2} + 4 α γ}{2 γ} .

Thus under the Poisson approximation,

t \to \infty lim ⟨ n (t)⟩ = \frac{β + β ^{2} + 4 α γ}{2 γ} .

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