  We study the time evolution of the entanglement entropy after quantum
quenches in Lifshitz free scalar theories, with the dynamical exponent $z>1$,
by using the correlator method. For quantum quenches we consider two types of
time-dependent mass functions: end-critical-protocol (ECP) and
cis-critical-protocol (CCP). In both cases, at early times the entanglement
entropy is independent of the subsystem size. After a critical time ($t_c$),
the entanglement entropy starts depending on the subsystem size significantly.
This critical time $t_c$ for $z = 1$ in the fast ECP and CCP has been explained
well by the fast quasi-particle of the quasi-particle picture. However, we find
that for $z > 1$ this explanation does not work and $t_c$ is delayed. We
explain why $t_c$ is delayed for $z>1$ based on the quasiparticle picture: in
essence, it is due to the competition between the fast and slow quasiparticles.
At late times, in the ECP, the entanglement entropy slowly increases while, in
the CCP, it is oscillating with a well defined period by the final mass scale,
independently of $z$. We give an interpretation of this phenomena by the
correlator method. As $z$ increases, the entanglement entropy increases, which
can be understood by long-range interactions due to $z$.
