Levins metapopulation model ($\dot{p}=cp(1-p)-mp$)

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.