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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 6A / a / Solution

The population map is

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.

Solved by gpt-5.6-sol high.

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