= Clamped-quartic solvability in shallow bioconvection
{title2=$W_1=-R_0\alpha^2z^2(1-z)^2/24$}
For the constant leading cell-concentration mode normalized to one, the leading vertical <velocity> mode in shallow long-wave <bioconvection> solves $W_1^{(4)}=-R_0\alpha^2$ and $W_1=W_1'=0$ at both $z=0,1$. The displayed quartic satisfies all four conditions and is unique because a homogeneous cubic with those conditions is zero. Its integral is $-R_0\alpha^2/720$. The next cell-conservation solvability condition is $\alpha^2+\sigma_2=-\int W_1dz$, hence $\sigma_2=\alpha^2(R_0/720-1)$. The leading threshold is independent of the distinguished leading gyrotactic coefficient; a horizontally uniform conservation mode does not select it.
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