 Solution

Let $m_i=⟨x_i⟩$. The mean equations give $m_1={\lambda }_1/{\beta }_1$ and $m_2={\lambda }_2{m}_1/{\beta }_2$. For relative deviations, the drift Jacobian and normalized reaction-noise covariance areThe normalized stationary fluctuation-dissipation relation for a reaction network is $A\eta +\eta A^T+D=0$ with this negative-drift convention. Its entries successively giveThe birth and death channels act on one species at a time, so the diffusion matrix is diagonal even though the stationary covariance is not.