Optimal investment to enable evolutionary rescue

Abstract

“Evolutionary rescue” is the potential for evolution to enable population persistence in a changing environment. Even with eventual rescue, evolutionary time lags can cause the population size to temporarily fall below a threshold susceptible to extinction. To reduce extinction risk given human-driven global change, conservation management can enhance populations through actions such as captive breeding. To quantify the optimal timing of, and indicators for engaging in, investment in temporary enhancement to enable evolutionary rescue, we construct a model of coupled demographic-genetic dynamics given a moving optimum. We assume “decelerating change”, as might be relevant to climate change, where the rate of environmental change initially exceeds a rate where evolutionary rescue is possible, but eventually slows. We analyze the optimal control path of an intervention to avoid the population size falling below a threshold susceptible to extinction, minimizing costs. We find that the optimal path of intervention initially increases as the population declines, then declines and ceases when the population growth rate becomes positive, which lags the stabilization in environmental change. In other words, the optimal strategy involves increasing investment even in the face of a declining population, and positive population growth could serve as a signal to end the intervention. In addition, a greater carrying capacity relative to the initial population size decreases the optimal intervention. Therefore, a one-time action to increase carrying capacity, such as habitat restoration, can reduce the amount and duration of longer term investment in population enhancement, even if the population is initially lower than and declining away from the new carrying capacity.

Appendices

Appendix A: Initialization and convergence of optimal paths

We initialized all optimization runs from the optimal path found with zero discount rate with uniform random initial conditions, and run for 2,500 iterations (Fig. 4). To assess convergence, we re-ran each parameter combination for 1000, 2000, and 2500 iterations for different random seeds. The longer runs showed consistent paths (Figs. 5 and 6), which is a criterion for convergence recommended for global optimization algorithms like the augmented Lagrangian method we employed (Johnson 2016).

Fig. 5

Optimal paths for increasing run times (lighter grays) to show convergence of control paths for different budgets (columns) and discount rates (subpanel rows) at three carrying capacities corresponding to aK = 10,000, bK = 15,000. There are three runs shown in each panel: 1000, 2000, and 2500 iterations. In most cases, the two longest runs resulted in the same path

Fig. 6

Optimal paths for increasing run times (lighter grays) to show convergence of population trajectories for different budgets (columns) and discount rates (subpanel rows) at three carrying capacities corresponding to aK = 10,000, bK = 15,000. There are three runs shown in each panel: 1000, 2000, and 2500 iterations. In most cases, the two longest runs resulted in the same path.

Appendix B: Code and graphics

We performed all numerical analyses in R using the nloptr package to perform optimization and dplyr to manage numeric outputs; we provide R code and metadata for optimal paths in Online Resource 1; the optimal path used for initial conditions is provided in Online Resource 2 and the optimal paths for all parameter combinations are provided in Online Resource 3. We produced all graphics using R packages ggplot2 and cowplot.