Evolutionary-branching lines and areas in bivariate trait spaces

Aims: Evolutionary branching is a process of evolutionary diversification induced by frequency-dependent ecological interaction. Here we show how to predict the occurrence of evolutionary branching in bivariate traits when populations are evolving directionally. Methods: Following adaptive dynamics theory, we assume low mutation rates and small mutational step sizes. On this basis, we generalize conditions for evolutionary-branching points to conditions for evolutionary-branching lines and areas, which delineate regions of trait space in which evolutionary branching can be expected despite populations still evolving directionally along these lines and within these areas. To assess the quality of predictions provided by our new conditions for evolutionary-branching lines and areas, we analyse three eco-evolutionary models with bivariate trait spaces, comparing the predicted evolutionary-branching lines and areas with actual occurrences of evolutionary branching in numerically calculated evolutionary dynamics. In the three examples, a phenotype’s fitness is affected by frequency-dependent resource competition and/or predator–prey interaction. Conclusions: In the limit of infinitesimal mutational step sizes, evolutionary branching in bivariate trait spaces can occur only at evolutionary-branching points, i.e. where the evolving population experiences disruptive selection in the absence of any directional selection. In contrast, when mutational step sizes are finite, evolutionary branching can occur also along evolutionary-branching lines, i.e. where disruptive selection orthogonal to these lines is sufficiently strong relative to directional selection along them. Moreover, such evolutionarybranching lines are embedded in evolutionary-branching areas, which delineate all bivariate trait combinations for which evolutionary branching can occur when mutation rates are low, while mutational step sizes are finite. Our analyses show that evolutionary-branching lines and areas are good indicators of evolutionary branching in directionally evolving populations. We also demonstrate that not all evolutionary-branching lines and areas contain evolutionarybranching points, so evolutionary branching is possible even in trait spaces that contain no evolutionary-branching point at all.