Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 6A / b / Solution

Let the conserved population be $S>0$. At equilibrium $Y^{\prime }=-\beta YN=0$, so $Y$ and $N$ cannot both be positive. The other equations then force $U=0$. The only equilibria are

Near $E_Y$, using $(N,U)$ as coordinates gives the linearizationIts determinant is $-\beta S\zeta <0$, so $E_Y$ is a saddle and is unstable. Near $E_N$, coordinates $(Y,U)$ give the triangular linearization with eigenvalues $-\beta S$ and $-\zeta$. Hence $E_N$ is asymptotically stable.