You can run - or you can hide: optimal strategies for cryptic prey against pursuit predators

We consider the optimal behavior of a cryptic prey individual as it is approached by a predator searching for prey. Although the predator has not yet discovered the prey, it has an increasing likelihood of doing so as it gets closer to the prey. Further, the closer the predator is to the prey when it discovers it, the more likely the predator will be to capture the prey. These arguments suggest that the prey should flee before the predator discovers it. However, the act of fleeing will alert the predator to the presence of the prey and trigger an attack that might not have occurred otherwise. We capture these conflicting outcomes in a mathematical model, which we then use to predict the optimal behavior of the prey and predator. We argue that the optimal strategy for the prey is either to run as soon as they detect a predator approaching or to only flee in response to having been detected by the predator. Running as soon as the predator is detected is associated with low predator search speeds, a low nonpredation cost to running, a large advantage to the prey in initiating chases rather than reacting, limited ability to spot the predator at distance, a high ability to spot prey by the predator, and a high probability that chases will be successful. The optimal strategy for the predator depends on whether its current trajectory is taking it closer to or further from the prey. In the latter case, the predator should attack immediately on discovering the prey; in the former case, it should delay its attack until it reaches the point on its current trajectory where distance to the prey is minimized.