  The Taylor-Frank method for making kin selection models when fitness is a
nonlinear function of a continuous phenotype requires this function to be
differentiable. This assumption sometimes fails for biologically important
fitness functions, for instance in microbial data and the theory of repeated
n-person games, even when fitness functions are smooth and continuous. In these
cases, the Taylor-Frank methodology cannot be used, and a more general form of
direct fitness must replace the standard one to account for kin selection, even
under weak selection.
