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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 6C / a / Solution

Set

x = \frac{N _{1}}{K _{1}}, y = \frac{N _{2}}{K _{2}}, τ = r_{2} t .

Then

x^{'} = ρ x (1 - x - α y), y^{'} = y (1 - β x) .

The steady states are

(0, 0), (1, 0), (\frac{1}{β}, \frac{1 - β ^{- 1}}{α}),

where the coexistence state is biologically relevant exactly when

β > 1

.

Solved by gpt-5.6-sol high.

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