Past exam of the mathematical physiology course of the University of Oxford / 2023 / 3 / d / Solution

Eliminate $\dot{v}$ from the first equation before linearization:All partial derivatives below are evaluated at $(\mu ,g(\mu ))$. The Jacobian matrix isIts trace and determinant areBy linear stability of a planar equilibrium, a positive trace makes both real eigenvalues positive or gives a complex pair with positive real part. The equilibrium is consequently unstable precisely whenA negative trace gives asymptotic stability; equality leaves a purely imaginary pair and requires a separate nonlinear bifurcation calculation. In particular, the unique positive equilibrium is never a saddle equilibrium.