  We investigate the evolution of entanglement within an open, strongly coupled
system interacting with a heat bath as its environment, in the frameworks of
both the doubly holographic model and the Sachdev-Ye-Kitaev (SYK) model.
Generally, the entanglement within the system initially increases as a result
of internal interactions; however, it eventually dissipates into the
environment. In the doubly holographic setup, we consider an end-of-the-world
brane in the bulk to represent an eternal black hole hermalized by holographic
matters. The reflected entropy between the bipartition of a large black hole
exhibits a ramp-plateau-slump behavior, where the plateau arises due to the
phase transition of the entanglement wedge cross-section before the Page time.
In quantum mechanics, we consider a double copy of the SYK-plus-bath system in
a global thermofield double state, resembling an eternal black hole interacting
with an environment. The R\'enyi mutual information within the double-copied
SYK clusters exhibits a ramp-plateau-slope-stabilizing behavior. The dynamic
behaviors of the entanglement quantities observed in these two models are
attributable to the competition between the internal interaction of the system
and the external interaction with the baths. Our study provides a fine-grained
picture of the dynamics of entanglement inside black holes before their Page
time.
