  We report the results of an HCO+ (3-2) and N2D+ (3-2) molecular line survey
performed toward 91 dense cores in the Perseus molecular cloud using the James
Clerk Maxwell Telescope, to identify the fraction of starless and protostellar
cores with systematic radial motions. We quantify the HCO+ asymmetry using a
dimensionless asymmetry parameter $\delta_v$, and identify 20 cores with
significant blue or red line asymmetries in optically-thick emission indicative
of collapsing or expanding motions, respectively. We separately fit the HCO+
profiles with an analytic collapse model and determine contraction (expansion)
speeds toward 22 cores. Comparing the $\delta_v$ and collapse model results, we
find that $\delta_v$ is a good tracer of core contraction if the optically-thin
emission is aligned with the model-derived systemic velocity. The contraction
speeds range from subsonic (0.03 km/s) to supersonic (0.4 km/s), where the
supersonic contraction speeds may trace global rather than local core
contraction. Most cores have contraction speeds significantly less than their
free-fall speeds. Only 7 of 28 starless cores have spectra well-fit by the
collapse model, which more than doubles (15 of 28) for protostellar cores.
Starless cores with masses greater than the Jeans mass (M/M$_J$ > 1) are
somewhat more likely to show contraction motions. We find no trend of
optically-thin non-thermal line width with M/M$_J$, suggesting that any
undetected contraction motions are small and subsonic. Most starless cores in
Perseus are either not in a state of collapse or expansion, or are in a very
early stage of collapse.
