Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2009 / B / 6 / d / ii

Since the steady state is strictly positive, the positive-part function is differentiable in a neighbourhood of it. Write $p=p_{\ast }+q$. Retaining first-order terms givesSubstitute the exponential mode $q=e^{\lambda T}$ to obtainHere the last equality uses the exact steady-state equation. For small $\mu$, $\beta =\alpha \mu +O(\alpha {\mu }^2)$ and $\gamma =\alpha (1+\mu )+O(\alpha {\mu }^2)$. Linearizing at the exact root avoids carrying a spurious constant residual from the printed approximate equilibrium.