Linear noise of two species with joint removal ($\eta =\frac{1}{2\mu (1-E^2)}\left(\begin{array}{cc}2-E^2& -E\\ -E& 2-E^2\end{array}\right)$)

For $E=C\mu /(\beta +C\mu )<1$ and mean lifetime $\tau =(\beta +C\mu {)}^{-1}$, the normalized restoring matrix is $M={\tau }^{-1}\left(\begin{array}{cc}1& E\\ E& 1\end{array}\right)$ and the diffusion matrix is $D=(\tau \mu {)}^{-1}\left(\begin{array}{cc}2& E\\ E& 2\end{array}\right)$. Solving the normalized stationary fluctuation-dissipation relation for a reaction network gives the displayed covariance. The sum mode remains damped, but the difference mode's restoring rate is $\beta =(1-E)/\tau$. Its noise grows without bound relative to that restoring rate as $E\to 1$, giving strong negative correlations and invalidating small-fluctuation closure near exclusive joint removal.

// Target: mathematical-biology.bigb