Adaptive dynamics is a mathematical framework for studying evolution. It extends evolutionary game theory to account for more realistic ecological dynamics and it can incorporate both frequency- and density-dependent selection. This is a practical guide to adaptive dynamics that aims to illustrate how the methodology can be applied to the study of specific systems. The theory is presented in detail for a single, monomorphic, asexually reproducing population. We explain the necessary terminology to understand the basic arguments in models based on adaptive dynamics, including invasion fitness, the selection gradient, pairwise invasibility plots (PIP), evolutionarily singular strategies, and the canonical equation. The presentation is supported with a worked-out example of evolution of arrival times in migratory birds. We show how the adaptive dynamics methodology can be extended to study evolution in polymorphic populations using trait evolution plots (TEPs). We give an overview of literature that generalises adaptive dynamics techniques to other scenarios, such as sexual, diploid populations, and spatially-structured populations. We conclude by discussing how adaptive dynamics relates to evolutionary game theory and how adaptive-dynamics techniques can be used in speciation research.

Theoretical models suggest that sympatric speciation along environmental gradients might be common in nature. Here we present the first data-based model of evolutionary diversification along a continuous environmental gradient. On the basis of genetic analyses, it has been suggested that a pair of coregonid fishes (Coregonus spp.) in a postglacial German lake originated by sympatric speciation. Within this lake, the two species segregate vertically and show metabolic adaptations to, as well as behavioral preferences for, correspondingly different temperatures. We test the plausibility of the hypothesis that this diversifying process has been driven by adaptations to different thermal microhabitats along the lake's temperature-depth gradient. Using an adaptive-dynamics model that is calibrated with empirical data and allows the gradual evolution of a quantitative trait describing optimal foraging temperature, we show that under the specific environmental conditions in the lake, evolutionary branching of a hypothetical ancestral population into two distinct phenotypes may have occurred. We also show that the resultant evolutionary diversification yields two stably coexisting populations with trait values and depth distributions that are in agreement with those currently observed in the lake. We conclude that divergent thermal adaptations along the temperature-depth gradient might have brought about the two species observed today