One of the scenarios actively studied during the 2003 trace tritium experimental campaign performed on JET includes a gas puff of small amounts of tritium into sawtooth-unstable H-mode plasmas heated by deuterium beams. The sawtooth crashes observed in these discharges, with the electron cyclotron emission diagnostics, are frequently accompanied by the oscillations of the 14 MeV neutron emission, produced mainly due to the reaction between the deuterium beam and the thermal tritium. Such oscillations are clearly seen in low density plasmas while they are weak or absent at high density. The goal of this work is to explain the different behaviour of the oscillations of neutron emission with density. For this purpose, the dynamics of the trace tritium and beam deuterium during the sawtooth crash in discharges with different plasma density is studied numerically using the TRANSP code. In addition, a possibility of using the neutron measurements in sawtoothing low density plasmas for the test of the full and partial reconnection models is discussed and the sensitivity of the simulations to the key parameters of these models is presented.