= Linear noise of two species with joint removal
{title2=$\eta=\frac1{2\mu(1-E^2)}\begin{pmatrix}2-E^2&-E\\-E&2-E^2\end{pmatrix}$}
For $E=C\mu/(\beta+C\mu)<1$ and mean lifetime $\tau=(\beta+C\mu)^{-1}$, the normalized restoring matrix is $M=\tau^{-1}\begin{pmatrix}1&E\\E&1\end{pmatrix}$ and the diffusion matrix is $D=(\tau\mu)^{-1}\begin{pmatrix}2&E\\E&2\end{pmatrix}$. 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 $E\to1$, giving strong negative correlations and invalidating small-fluctuation closure near exclusive joint removal.
// Target: mathematical-biology.bigb
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