Adaptive dynamics of quantitative traits in a steadily changing environment.

Understanding the adaptation of quantitative traits to changing environments is a central challenge in evolutionary biology. However, the precise roles of the speed of environmental change and trait architecture in adaptive dynamics remain unclear. Here, we investigate their roles through an adaptive walk model and individual-based polygenic simulations within a moving-optimum framework. We introduce a dimensionless parameter, the effective selection strength, to characterize quantitative traits in both models. We show that adapting populations attain a stationary state below a critical speed of environmental change. Properties of the stationary state are compared between the two approaches, and it is shown that the adaptive walk model provides a good approximation for long-term polygenic adaptation. Additionally, in the adaptive walk model, we derive an explicit formula for the logarithmic divergence of the stationary phenotypic gap upon approaching the critical speed. To elucidate population extinction in biologically realistic terms, we define a fitness-threshold-based critical speed and reveal its power-law dependence on the effective selection strength in both models. Finally, results of our polygenic simulations show how the genetic architecture in the stationary regime is shaped by a dynamic equilibrium between the influx of de novo mutations and their loss through fixation and extinction constrained by the rate of phenotypic change enforced by the optimum speed. Taken together, our study advances the understanding of polygenic adaptation and provides insights to simplify analytical work and computational simulations for future studies.