Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 14A / a

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- Solution a

Solution

For a positive homogeneous equilibrium,

v = 1 + u, αu = β v,

so

u_{*} = \frac{β}{α - β}, v_{*} = \frac{α}{α - β},

requiring

α > β

. The reaction Jacobian there is

J = \frac{1}{α - β} (β α β - β - β^{2}) .

Its determinant is

β^{2} / (α - β) > 0

, while its trace is

β (1 - β) / (α - β)

. Stability therefore requires

β > 1, α > β .

This is the region above the line

α = β

and to the right of

β = 1

in the positive quadrant.

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