 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/6a/a/solution

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

x_{n + 1} = g (x_{n}), g (x) = x (r + k e^{- λ x}) .

A positive equilibrium satisfies

1 = r + k e^{- λ x_{*}},

and hence

x_{*} = \frac{1}{λ} lo g (\frac{k}{1 - r}) .

It is strictly positive exactly when

k > 1 - r .

This is the nonzero equilibrium of the survival-augmented Ricker map.

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