  We study the qualitative properties of population cycles in a predator-prey
system where genetic variability allows contemporary rapid evolution of the
prey. Previous numerical studies have found that prey evolution in response to
changing predation risk can have major quantitative and qualitative effects on
predator-prey cycles, including: (i) large increases in cycle period, (ii)
changes in phase relations (so that predator and prey are cycling exactly out
of phase, rather than the classical quarter-period phase lag), and (iii)
"cryptic" cycles in which total prey density remains nearly constant while
predator density and prey traits cycle. Here we focus on a chemostat model
motivated by our experimental system [Fussmann et al. 2000,Yoshida et al. 2003]
with algae (prey) and rotifers (predators), in which the prey exhibit rapid
evolution in their level of defense against predation. We show that the effects
of rapid prey evolution are robust and general, and furthermore that they occur
in a specific but biologically relevant region of parameter space: when traits
that greatly reduce predation risk are relatively cheap (in terms of reductions
in other fitness components), when there is coexistence between the two prey
types and the predator, and when the interaction between predators and
undefended prey alone would produce cycles. Because defense has been shown to
be inexpensive, even cost-free, in a number of systems [Andersson and Levin
1999, Gagneux et al. 2006,Yoshida et al. 2004], our discoveries may well be
reproduced in other model systems, and in nature. Finally, some of our key
results are extended to a general model in which functional forms for the
predation rate and prey birth rate are not specified.
