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

The Jacobian isFor $\beta >1$, the origin has eigenvalues $\rho ,1$ and is an unstable node. At $(1,0)$ the eigenvalues are $-\rho$ and $1-\beta$, so it is a stable node. At the coexistence state,so it is a saddle.

The nullclines are $x=0$, $x+\alpha y=1$, $y=0$, and $x=1/\beta$. The stable manifold of the coexistence saddle separates trajectories tending to $(1,0)$ from trajectories on which $x\to 0$ and $y$ grows without bound. The coordinate axes are invariant.