Fig. 1.

Optimal control of a finite population under Wright–Fisher evolution to maintain a polymorphism. The intrinsic selection coefficient is $\sigma =10$ and control shifts selection to $\sigma +u$. (A) Sample trajectories starting at $x_0=0.5$: without control ($u\equiv 0$, black line) the polymorphism is lost on a time scale of $1/\sigma$. With optimal control under perfect information (gray line, $\sigma +u=\sigma =10$ for $x<x_c\approx 0.64$, else $\sigma +u=\sigma -u_c=-20$), it can be maintained for an average of 8,000 N generations. With finite monitoring ($\mathrm{\Delta }=0.1$, measurements $x_i$ at circles), naive control $\left(u\equiv \overline{u}\left(x_i\right)\right)$, red line, is prone to overshooting, whereas preemptive control (blue line) tries to avoid this by switching to a neutral regime after a short time. (B) Loss of control under finite monitoring: as $\mathrm{\Delta }$ grows, so does the probability that the polymorphism is already lost at the next measurement. Shown is the mean survival time over 5,000 trajectories with $N={10}^4$ on a logarithmic scale. (C) Under preemptive control, some of the loss of control can be regained, especially for intermediate values of $\mathrm{\Delta }$.