Mixed-mode chemotactic instability threshold (${\beta }_{crit}=5{\pi }^2+2+\frac{1}{5{\pi }^2}$)

When $\alpha =\kappa =\eta =1$, the instability threshold of a spatial mode with squared 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 $s\ge 5{\pi }^2$. 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 instability begins above this value.