Phenotypic lag and population extinction in the moving-optimum model: insights from a small-jumps limit

Continuous environmental change-such as slowly rising temperatures-may create permanent maladaptation of natural populations: Even if a population adapts evolutionarily, its mean phenotype will usually lag behind the phenotype favored in the current environment, and if the resulting phenotypic lag becomes too large, the population risks extinction. We analyze this scenario using a moving-optimum model, in which one or more quantitative traits are under stabilizing selection towards an optimal value that increases at a constant rate. We have recently shown that, in the limit of infinitely small mutations and high mutation rate, the evolution of the phenotypic lag converges to an Ornstein-Uhlenbeck process around a long-term equilibrium value. Both the mean and the variance of this equilibrium lag have simple analytical formulas. Here, we study the properties of this limit and compare it to simulations of an evolving population with finite mutational effects. We find that the "small-jumps limit" provides a reasonable approximation, provided the mean lag is so large that the optimum cannot be reached by a single mutation. This is the case for fast environmental change and/or weak selection. Our analysis also provides insights into population extinction: Even if the mean lag is small enough to allow a positive growth rate, stochastic fluctuations of the lag will eventually cause extinction. We show that the time until this event follows an exponential distribution, whose mean depends strongly on a composite parameter that relates the speed of environmental change to the adaptive potential of the population.