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In the two-class globally coupled underdominant metapopulation, the rectangle $3/4\le x\le 1$, $0\le y\le 1/4$ 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.

Use a globally coupled underdominant metapopulation with equal habitat sizes:Symmetry preserves a common frequency $x$ in the initially $AA$ habitats and $y$ in the initially $aa$ habitats. With $q=m/N$, the equations reduce exactly toThe maximum positive selection rate is $f(u_m)=s/(6\sqrt{3})$ at $u_m=(3+\sqrt{3})/6$. Comparing it with the worst outward migration flux gives a sufficient barrier condition $\sigma \max (q,1-q)<f(u_m)/u_m$. A slightly stronger optimized condition follows from maximizing the per-capita restoring term $f(u)/u=s(1-u)(2u-1)$ over $u>1/2$: its maximum is $s/8$ at $u=3/4$.

Indeed, on $x=3/4$, with $0\le y\le 1/4$, $\dot{x}\ge 3s/32-(3/4)\sigma (1-q)>0$. On $y=1/4$, with $3/4\le x\le 1$, $\dot{y}\le -3s/32+(3/4)\sigma q<0$. The other two edges point inward as well. The invariant-rectangle criterion for underdominant coexistence therefore gives the explicit robust conditionsStarting from the specified pure habitats, the populations then remain in $x\ge 3/4$, $y\le 1/4$. Their global frequency lies between $3q/4$ and $q+(1-q)/4$, so neither allele fixes. This is a sufficient condition, not a claim that the bound is sharp. If $m=0$ or $m=N$, the population is already fixed; if $\sigma =0$, any mixture of the two pure habitat types persists.

The exactly balanced case illustrates why robustness and the prescribed trajectory differ. When $q=1/2$, symmetry forces $y=1-x$ andFor $\sigma <s/2$ its heterogeneous limiting state isHere $f^{\prime }(x_{\ast })=f^{\prime }(y_{\ast })=-s+3\sigma$, so the full Jacobian matrix has diagonal entries $f^{\prime }(x_{\ast })-\sigma /2$ and off-diagonal entries $\sigma /2$. Its two full-system eigenvalues are $-s+3\sigma$ and $-s+2\sigma$: the symmetric underdominant two-cluster equilibrium is robustly attracting only for $\sigma <s/3$. For larger migration the exactly symmetric initial condition can still avoid fixation, eventually approaching the spatially uniform frequency $1/2$ when $\sigma \ge s/2$, but the symmetry-broken mean-frequency mode is unstable. A finite population with genetic drift need not maintain deterministic coexistence indefinitely.