Quantifying feedback among traits in coevolutionary models

Phenotypic traits rarely evolve in isolation. Instead, multiple traits typically interact to influence fitness, 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 thus predicting evolutionary trajectories and potential endpoints. Current theory focuses mainly on analysing how traits change in response to selection, which changes over time as traits evolve. However, the extent to which changes in each trait contribute to changes in the selection environment remains unquantified, leaving much of our understanding of trait coevolution reliant on verbal reasoning. To advance a more comprehensive and quantitative understanding of coevolutionary dynamics, we develop a general framework that examines how trait changes feedback to influence the selection environment. This framework enables a fine-grained and systematic investigation of coevolutionary feedback between traits and selection gradients by quantifying the pathways through which they 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.