= Mixed-mode chemotactic instability threshold
{title2=$\beta_{crit}=5\pi^2+2+\frac1{5\pi^2}$}
When $\alpha=\kappa=\eta=1$, the <Lyapunov stability>[instability] threshold of a spatial mode with squared <wavenumber>[wave number] $s$ is $\beta=s+2+1/s$. On a rectangle $L_1=1$, $L_2=1/2$, modes varying in both coordinates have $s\ge5\pi^2$. Since the threshold increases for $s>1$, the first such mode is $(n,m)=(1,1)$, at $\beta_{crit}=5\pi^2+2+(5\pi^2)^{-1}$. At equality it is neutral; strict <Lyapunov stability>[instability] begins above this value.
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