Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 6C (Mathematical Biology)

SetThenThe steady states arewhere the coexistence state is biologically relevant exactly when $\beta >1$.

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.

There is no stable positive coexistence state: the only one is a saddle for every $\rho ,\alpha >0$ when $\beta >1$. Moreover, the Bendixson-Dulac criterion with Dulac function $B(x,y)=1/(xy)$ applies in the positive quadrant, sinceThus no periodic orbit can support long-term coexistence. Except on the saddle's stable manifold, one species is excluded: species 2 becomes extinct for initial data in the basin of $(1,0)$, ecologically when species 1 is sufficiently abundant and species 2 sufficiently scarce. On the other side, species 1 becomes extinct and the unregulated second species grows.