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 evolutionary 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 us to predict, for any bivariate selection problem, (i) evolutionary outcomes induced by trade-offs of given shape, (ii) shapes of trade-offs required for given evolutionary outcomes, (iii) the set of all evolutionary outcomes trade-offs can induce, (iv) effects of ecological parameters on evolutionary outcomes independent of trade-off shape.