  We examine the dynamics of small bodies in orbits similar to that of comet
29P/Schwassmann-Wachmann 1, i.e. near-circular orbits between Jupiter and
Saturn. As of late 2019, there are 14 other known bodies in this region that
lie in similar orbits. Previous research has shown that this region of the
solar system, which in this work we call the "near Centaur region" (NCR), is
not stable, suggesting that any bodies found in it would have very short
lifetimes. We performed 20 Myr high-precision numerical simulations of the
evolution of massless particles, initially located in the Kuiper belt but close
to Neptune, with perihelia slightly below 33 au ("Neptune crossers"). Some of
these particles quickly migrate inward, passing through the NCR before becoming
Jupiter Family Comets. We find that objects in the NCR do indeed generally
travel through it very quickly. However, our simulations reveal that resonant
behavior in this region is somewhat common and can trap objects for up to 100
Kyr. We summarize the dynamics of 29P and other observed bodies in the
region---two of which seem to be clearly exhibiting resonant behavior---and use
our simulations to put limits on the reservoir size of the Neptune crossers,
which at its time determines the reservoir size of the Kuiper belt. Finally, we
put some constraints on the current population of Centaurs and particularly of
the NCR population, based on the injection rates required to keep the observed
population of Jupiter family comets in steady-state.
