Demographic feedbacks during evolutionary rescue can slow or speed adaptive evolution

Abstract

Populations declining toward extinction can persist via genetic adaptation in a process called evolutionary rescue. Predicting evolutionary rescue has applications ranging from conservation biology to medicine, but requires understanding and integrating the multiple effects of a stressful environmental change on population processes. Here we derive a simple expression for how generation time, a key determinant of the rate of evolution, varies with population size during evolutionary rescue. Change in generation time is quantitatively predicted by comparing how intraspecific competition and the source of maladaptation each affect the rates of births and deaths in the population. Depending on the difference between two parameters quantifying these effects, the model predicts that populations may experience substantial changes in their rate of adaptation in both positive and negative directions, or adapt consistently despite severe stress. These predictions were then tested by comparison to the results of individual-based simulations of evolutionary rescue, which validated that the tolerable rate of environmental change varied considerably as described by analytical results. We discuss how these results inform efforts to understand wildlife disease and adaptation to climate change, evolution in managed populations and treatment resistance in pathogens.

Keywords: adaptation; demography; evolutionary rescue; evolutionary theory.

Figure 1.

Predicted equilibrium generation times, $\hat{T}$, relative to $d_0^{-1},$vary according to the difference between ϕ and β. (a) Generation time given a phenotypic mismatch x = 5 (equation (2.3)). Roman numerals represent parameter combinations depicted in (b). (b) Responses of relative generation times to variation across a range of values of the degree of maladaptation x. (c) Equilibrium population size, as a ratio N/K. Note that equilibrium population size declines with x but is not perturbed by β or ϕ.

Figure 2.

Evolutionary rescue simulations across a range of ϕ and β. (a,b) Examples of mean (black) and optimal phenotype (z_opt; blue) and population sizes (grey; (c,d)) for a persisting population (a,c) and one that goes extinct (b,d); k indicates the rate of change per time unit. d₀ is 0.1, producing a base generation time of 10 units. Example logistic regression (e) of 200 replicates across a range of rates; k₅₀, the rate for which 50% of populations are estimated via logistic regression to survive via evolutionary rescue, is determined as the inflection point of the fitted curve. (f) k₅₀ values across a range of ϕ and β. A minimum of 125 observations of both extinctions and rescues were performed for each of 81 parameter combinations. Fitness is determined by f(x) = exp(x/2.28); genomic mutation rate U = 0.01.

Figure 3.

Predicted rates of evolution $V_A/({\sigma }_s\hat{T})$ (x-axis) compared to measured k₅₀ values from the 81 combinations of ϕ and β underlying figure 2f. Predictions are calculated at a high level of maladaptation (x = 5). Line indicates x = y. Simulation data and parameters are those in figure 2.

Figure 4.

Ensemble means of mean phenotype (a), change in mean phenotype per time unit (b), and additive genetic variance (c) for three values of ϕ and β. The rate of environmental change, depicted by the dotted line (a,b), is k = 0.02. Additive genetic variance is measured as the slope of midparent–offspring regression. Changes in phenotypes are smoothed with a simple uniform moving average of five units. At least 300 replicates are averaged for each condition.

Figure 5.

Capacity for demographic compensation determines capacity for evolutionary rescue. (a) The x-intercept denotes the parameter M = K(1 − d₀), which is used to scale the relationship between density-dependent population growth and d₀. This scaling ensures that the equilibrium population size is equal to K across values of d₀. Higher values of h(N) indicate less restriction by density dependence. (b) Estimated rate of environmental change at which 50% of populations survive (k₅₀) based on simulations for parameter combinations satisfying ϕ + β = 1. This equation is the diagonal from slowest to fastest generation times in figure 1 and captures the main axis of variation in k₅₀ in figure 2f. (c) The ratio of k₅₀ (ϕ = 0, β = 1) : k₅₀ (ϕ = 1, β = 0) as estimated from simulated data in (b), plotted against the reciprocal of d₀. Note that (b) is on a log–linear scale, making the ratio of the highest to lowest values an appropriate measure of the slope.