If this does not hold the strategy is evolutionarily unstable and, provided that it is also convergence stable, evolutionary branching will eventually occur. For a singular strategy formula_17 to be convergence stable monomorphic populations with slightly lower or slightly higher trait values must be invadable by mutants with trait values closer to formula_17. That this can happen the selection gradient formula_4 in a neighbourhood of formula_17 must be positive for formula_28 and negative for formula_29. This means that the slope of formula_4 as a function of formula_9