Bet-hedging when Targets May Disappear: Optimal Mate-seeking or Prey-catching Trajectories and the Stability of Leks and Herds

When a female frog moves towards a calling male, the male may suddenly stop calling and the female have to switch to another male. Analogous situations where “hunters” move towards “targets” that can disappear unpredictably include predators stalking prey and plants growing towards gaps in the canopy. I use dynamic programming to show that when the hunter has a choice of such targets it is optimal to take a curved bet-hedging trajectory, initially heading between two targets so that if one target disappears the other is closer. Also hunters should prefer groups of targets, even if a solitary target is somewhat closer, because it is unlikely that all targets in a group will disappear. Assuming that hunters follow these optimal trajectories I then ask whether it will pay targets to form herds or leks. The extra attractiveness of groups in this model turns out not to be sufficient to outweigh the advantages of herding, but the net benefits of herding are considerably reduced.