Quantifying feedback among traits in coevolutionary models

Phenotypic traits rarely evolve in isolation. Instead, multiple traits typically interact, resulting in complex coevolutionary dynamics. Such dynamics can be predicted using mathematical frameworks such as adaptive dynamics and quantitative genetics. Selection gradients play a crucial role in these frameworks, describing the direction and strength of selection and predicting evolutionary trajectories and potential endpoints. Current theory focuses mainly on analysing how traits change in response to selection. However, selection gradients also change over time as traits evolve. For a complete understanding of coevolutionary dynamics, it is consequently essential to examine how trait changes feed back to influence the selection environment. Here, we develop a general framework for investigating coevolutionary feedback between traits and selection gradients. Rather than relying on verbal interpretation of causal relationships, our approach explicitly quantifies the various pathways by which traits and selection gradients influence one another. Our framework can be applied both to adaptive-dynamic models and to quantitative-genetic models under the weak selection limit. We illustrate our approach with three examples that showcase its potential to deepen our understanding of established models.