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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 6A / c / Solution

Let

N_{t} = a_{t} + b_{t}

. Adding the two recurrences gives

N_{t + 1} - N_{t} = (3 μ - k) b_{t} .

Since

b_{t} \geq 0

, total population can increase only if

3 μ > k

. This is exactly

R_{0} = \frac{4 μ}{μ + k} > 1,

so the population-growth and offspring criteria agree.

Solved by gpt-5.6-sol high.

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