  The evolution of non-interacting bosons in the presence of repeated
projective measurements is studied. Following the established approach, this
monitored evolution is characterized by the first detected return and the first
detected transition probabilities. We show that these quantities are directly
related to the entanglement entropy and to the entanglement spectrum of a
bipartite system. Calculations with specific values for the number of bosons,
the number of measurements and the time step between measurements reveal a
sensitive and often strongly fluctuating entanglement entropy. In particular,
we demonstrate that in the vicinity of special values for the time steps the
evolution of the entanglement entropy is either stationary or performs
dynamical switching between two or more stationary values. In the entanglement
spectrum, on the other hand, this complex behavior can be associated with level
crossings, indicating that the dominant quantum states and their entanglement
respond strongly to a change of the system parameters. We discuss briefly the
role of time averaging to remove the fluctuations of the entanglement entropy.
