  The study of prestellar cores is critical as they set the initial conditions
in star formation and determine the final mass of the stellar object. To date,
several hypotheses are describing their gravitational collapse. We perform
detailed line analysis and modelling of H$_{2}$D$^{+}$ 110 -111 and
N$_{2}$H$^{+}$ 4-3 emission at 372 GHz, using 2'x2' maps (JCMT). Our goal is to
test the most prominent dynamical models by comparing the modelled gas
kinematics and spatial distribution (H$_{2}$D$^{+}$ and N$_{2}$H$^{+}$) with
observations towards four prestellar (L1544, L183, L694-2, L1517B) and one
protostellar core (L1521f). We perform a detailed non-LTE radiative transfer
modelling using RATRAN, where we compare the predicted spatial distribution and
line profiles of H$_{2}$D$^{+}$ and N$_{2}$H$^{+}$ with observations towards
all cores. To do so, we adopt the physical structure for each core predicted by
three different dynamical models taken from literature: Quasi-Equilibrium
Bonnor-Ebert Sphere (QE-BES), Singular Isothermal Sphere (SIS), and
Larson-Penston (LP) flow. Our analysis provides an updated picture of the
physical structure of prestellar cores. We find that the SIS model can be
clearly excluded in explaining the gas emission towards the cores, but a larger
sample is required to differentiate clearly between the LP flow, the QE-BES and
the static models. All models of collapse underestimate the intensity of the
gas emission by up to several factors towards the only protostellar core in our
sample, indicating that different dynamics take place in different evolutionary
core stages. If the LP model is confirmed towards a larger sample of prestellar
cores, it would indicate that they may form by compression or accretion of gas
from larger scales. If the QE-BES model is confirmed, it means that quasi
hydrostatic cores can exist within turbulent ISM.
