Trade-off geometries and frequency-dependent selection

Life-history evolution is determined by the interplay between natural selection and adaptive constraints. The classical approach to studying constrained life-history evolution - Richard Levins's geometric comparison of fitness sets and adaptive functions is applicable when selection pressures are frequency independent. Here we extend this widely used tool to frequency-dependent selection. Such selection pressures vary with a population's phenotypic composition and are increasingly recognized as ubiquitous. Under frequency dependence, two independent properties have to be distinguished: evolutionary stability ( an evolutionarily stable strategy cannot be invaded once established) and convergence stability ( only a convergence stable strategy can be attained through small, selectively advantageous steps). Combination of both properties results in four classes of possible evolutionary outcomes. We introduce a geometric mode of analysis that enables predicting, for any bivariate selection problem, evolutionary outcomes induced by trade-offs of given shape, shapes of trade-offs required for given evolutionary outcomes, the set of all evolutionary outcomes trade-offs can induce, and effects of ecological parameters on evolutionary outcomes independent of trade-off shape.