= Invariant-rectangle criterion for underdominant coexistence
{title2=$\sigma\max(q,1-q)<s/8$}
In the two-class <globally coupled underdominant metapopulation>, the rectangle $3/4\le x\le1$, $0\le y\le1/4$ is forward invariant under the displayed bound. On its inner edges, selection points inward with magnitude $3s/32$ while outward migration is at most $(3/4)\sigma(1-q)$ or $(3/4)\sigma q$; the outer edges also point inward. Thus prescribed initial state $(1,0)$ cannot approach global <allele fixation> when $0<q<1$. Maximizing $f(u)/u=s(1-u)(2u-1)$ gives $s/8$ at $u=3/4$, making the bound optimal within this particular symmetric worst-case rectangle argument, not a universal sharp migration threshold.
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