There are some empirical evidences that multiple clones of prey population play a major role of prey-predator cycles and the 1/2 period lag (1)

We have shown that a simple prey-predator model with prey evolution can explain prey-predator cycles (2), although we did not discussed the 1/2 period lag.

Prey evolution often destabilizes prey-predator systems. This is intuitively understandable because the prey availability (the number of prey that is not vigilant for the predator) oscilates 1/4 prior to the prredator. The larger rate of prey evolution, the prey-predator system destabilizes and has a stable limit cycle or chaos.

1) Yoshida T et al. (2003) Nature 424: 303-306.

2) Abrams PA, Matsuda H(1997) Evolution 51(6):1740-1748.