Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 6A (Mathematical Biology)

The $\beta YN$ interaction converts supporters into opponents. The $\alpha YN$ interaction converts opponents into undecided people, while $\zeta U$ converts undecided people into opponents. Summing the three equations gives $(Y+N+U{)}^{\prime }=0$, as required for a fixed population.

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.

No positive parameter choices can make everyone eventually favour the proposition for every initial condition. The all-opposed state $E_N$ is itself an equilibrium, and part (b) shows that an entire neighborhood is attracted to it. Thus initial conditions in that neighborhood cannot converge to $E_Y$.