The DeAngelis-Beddington functional response and the evolution of timidity of the prey

We study the evolution of "timidity" of the prey (i.e., its readiness to seek refuge) in a predator-prey model with the DeAngelis-Beddington functional response. Using the theory of adaptive dynamics, we find that a predator-prey population at equilibrium always favours less timidity. Low levels of timidity, however, may destabilise the population and lead to cycles. Large-amplitude cycles favour a positive level of timidity, but if such cycles do not occur, timidity will evolve all the way to zero, where the prey no longer responds to the predator by seeking refuge, and in which case the DeAngelis-Beddington functional response has become identical to the Holling type-II functional response.