  We conduct a numerical investigation of the dynamics of the central spin
model in the presence of measurement processes. This model holds promise for
experimental exploration due to its topology, which facilitates the natural
distinction of a central particle and the quantum bath as different subsystems,
allowing for the examination of entanglement phase transitions. To characterize
the measurement-induced phase transition in this system, we employ a recently
developed method based on second R\'enyi entropy in dual space. Our simulations
account for decoherence, energy relaxation, and gate errors. We determine
critical measurement rates and demonstrate that they significantly differ from
those predicted by a simple approach based on mutual entropy.
