Selection gradient ($g(x)={\partial }_{y}f(y,x){|}_{y=x}$)

For a differentiable rare-mutant invasion fitness $f(y,x)$ against resident trait $x$, the selection gradient gives the first-order growth advantage of a nearby mutant. A zero gradient is a candidate interior evolutionary singularity. Strict local uninvadability follows from negative mutant-fitness curvature; evolutionary approach additionally requires the gradient to point toward the candidate. Boundary traits require one-sided tests. A gradient alone does not prove global evolutionary stability.