  Adaptive dynamics is a widely used framework for modeling long-term evolution
of continuous phenotypes. It is based on invasion fitness functions, which
determine selection gradients and the canonical equation of adaptive dynamics.
Even though the derivation of the adaptive dynamics from a given invasion
fitness function is general and model-independent, the derivation of the
invasion fitness function itself requires specification of an underlying
ecological model. Therefore, evolutionary insights gained from adaptive
dynamics models are generally model-dependent. Logistic models for symmetric,
frequency-dependent competition are widely used in this context. Such models
have the property that the selection gradients derived from them are gradients
of scalar functions, which reflects a certain gradient property of the
corresponding invasion fitness function. We show that any adaptive dynamics
model that is based on an invasion fitness functions with this gradient
property can be transformed into a generalized symmetric competition model.
This provides a precise delineation of the generality of results derived from
competition models. Roughly speaking, to understand the adaptive dynamics of
the class of models satisfying a certain gradient condition, one only needs a
complete understanding of the adaptive dynamics of symmetric,
frequency-dependent competition. We show how this result can be applied to
number of basic issues in evolutionary theory.
