Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 355 / 2 / a

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- Solution a

Solution

For homogeneous fields the gradient terms vanish, leaving

\overset{n}{˙} = γ - δ n, \overset{c}{˙} = α n - β c .

Thus

n_{*} = \frac{γ}{δ}, c_{*} = \frac{α γ}{β δ} .

The homogeneous Jacobian is triangular,

(- δ α 0 - β),

with strictly negative eigenvalues

- δ

and

- β

. The fixed point is therefore linearly stable to homogeneous perturbations.

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