Clamped-quartic solvability in shallow bioconvection ($W_1=-R_0{\alpha }^2{z}^2(1-z{)}^2/24$)

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 $W_1=W_1^{\prime }=0$ 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 ${\alpha }^2+{\sigma }_2=-\int W_{1}dz$, 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.