  It was recently proposed that the electric field oscillation as a result of
self-consistent $e^{\pm}$ pair production may be the source of coherent radio
emission from pulsars. Direct Particle-in-Cell (PIC) simulations of this
process have shown that the screening of the parallel electric field by this
pair cascade manifests as a limit cycle, as the parallel electric field is
recurrently induced when pairs produced in the cascade escape from the gap
region. In this work, we develop a simplified time-dependent kinetic model of
$e^{\pm}$ pair cascades in pulsar magnetospheres that can reproduce the
limit-cycle behavior of pair production and electric field screening. This
model includes the effects of a magnetospheric current, the escape of
$e^{\pm}$, as well as the dynamic dependence of pair production rate on the
plasma density and energy. Using this simple theoretical model, we show that
the power spectrum of electric field oscillations averaged over many limit
cycles is compatible with the observed pulsar radio spectrum.
