 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/14a/a/solution

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

For a positive homogeneous equilibrium,

v = 1 + u, αu = β v,

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}) .

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

 New to topics? Read the docs here!