  We develop a generating functional description of the dynamics of
non-Markovian individual-based systems, in which delay reactions can be
terminated before completion. This generalises previous work in which a
path-integral approach was applied to dynamics in which delay reactions
complete with certainty. We construct a more widely applicable theory, and from
it we derive Gaussian approximations of the dynamics, valid in the limit of
large, but finite population sizes. As an application of our theory we study
predator-prey models with delay dynamics due to gestation or lag periods to
reach the reproductive age. In particular we focus on the effects of delay on
noise-induced cycles.
