  We present the results of a detailed set of one-zone models that account for
the coupling between pulsation and convection following the original
prescriptions of Stellingwerf (1986). Motivated by the arbitrary nature of the
input parameters adopted in this theoretical framework, we computed several
sequences of models that cover a substantial fraction of the parameter space
and a longer integration time. We also included a turbulent pressure term and
found that this physical mechanism plays a crucial role in the pulsation
characteristics of the models by removing the sharp discontinuities along the
light and the velocity curves showed by models that do not account for
turbulent pressure. Finally, we investigated the vibrational and the
pulsational stability of completely convective models. We consider as the most
important finding of the present work the identification of a well-defined
region in the parameter space where they approach limit-cycle stability.
Several numerical experiments performed by adopting different values of the
adiabatic exponent and of the shell thickness indicate that the coupling
between pulsation and convection is the key driving mechanism for LPVs, a
finding supported by recent theoretical predictions.
