Metapopulation

A metapopulation describes occupied habitat patches with local colonization and extinction. The patch occupancy variable differs from population abundance within a patch.

This model couples deterministic underdominant allele-frequency dynamics in equal-sized habitats by global migration. Starting with $m$ pure habitats of one allele and $N-m$ of the other preserves two within-class frequencies $x,y$, reducing the system to $\dot{x}=f(x)-\sigma (1-q)(x-y)$, $\dot{y}=f(y)+\sigma q(x-y)$ with $q=m/N$. Migration is conservative in the mean, but selection need not conserve global allele frequency.

With equally many initially pure habitats of each allele, symmetry preserves $y=1-x$. The heterogeneous equilibrium exists for $\sigma <s/2$. Its full-system eigenvalues are $-s+3\sigma$ in the mean-frequency direction and $-s+2\sigma$ in the antisymmetric direction, so it attracts general nearby perturbations only for $\sigma <s/3$. Exact symmetry can preserve both alleles beyond this robust range; at stronger migration it approaches the unstable uniform midpoint.

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.

Colonization of empty habitat at rate $c$ and extinction of occupied habitat at rate $m$ give the displayed patch-occupancy equation. The positive equilibrium is $1-m/c$ when $c>m$. Competition can be incorporated through replacement fluxes between patches occupied by different species.

A resident trait $x$ occupies fraction $1-m/c(x)$ when its colonization rate exceeds fire extinction. With net competitive replacement given by ability difference $B(y)-B(x)$, a rare mutant has the displayed per-capita growth. The selection gradient is $g(x)=m{c}^{\prime }(x)/c(x)+[1-m/c(x)]B^{\prime }(x)$. A singular trait with negative mutant-fitness curvature is a strict local evolutionarily stable strategy. Global strict uninvadability requires that $mc(y)+[c(x)-m]B(y)$ be smaller than its resident value for every other trait. Monotonicity alone does not specify the shape or existence of an optimum.