Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 6A / c / Solution

{c}

When $N>N_c=(\mu+\nu)/\beta$, the positive equilibrium is
$$
X^*=N_c,
\qquad
Y^*=\frac{\mu}{\mu+\nu}(N-N_c),
\qquad
Z^*=\frac{\nu}{\mu+\nu}(N-N_c).
$$
On the invariant plane $X+Y+Z=N$, the $(X,Y)$ Jacobian at this equilibrium has negative trace $-(\mu+\beta Y^*)$ and positive determinant $\beta Y^*(\mu+\nu)$, so both eigenvalues have negative real part and the equilibrium is locally asymptotically stable.


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