Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 6B / Solution

The terms $u$ and $\alpha v$ represent intrinsic exponential population growth. The losses $-uv$ and $-\alpha uv$ describe competition between the species; $-{ϵ}_1{u}^2$ and $-\alpha {ϵ}_2{v}^2$ describe crowding within each species. The positive constant $\alpha$ sets the relative time scale of the second population.

The four equilibrium points in the nonnegative quadrant areThe last follows by solving ${ϵ}_{1}u+v=1$ and $u+{ϵ}_{2}v=1$ and is strictly positive under the assumptions. The Jacobian matrix isAt the origin its eigenvalues are $1,\alpha >0$, so it is unstable. At the first single-species equilibrium they are $-1$ and $\alpha (1-1/{ϵ}_1)<0$; at the second they are $1-1/{ϵ}_2<0$ and $-\alpha$. Both are asymptotically stable. At coexistence,Thus coexistence is a saddle; either species-only equilibrium is stable, and extinction is unstable. The saddle's stable manifold separates the competing basins of attraction.