  We study the dissipative dynamics of a one-dimensional bosonic system
described in terms of the bipartite Bose-Hubbard model with alternating gain
and loss. This model exhibits the $\mathcal{PT}$ symmetry under some specific
conditions and features a $\mathcal{PT}$-symmetry phase transition. It is
characterized by an order parameter corresponding to the population imbalance
between even and odd sites, similar to the continuous phase transitions in the
Hermitian realm. In the noninteracting limit, we solve the problem exactly and
compute the parameter dependence of the order parameter. The interacting limit
is addressed at the mean-field level, which allows us to construct the phase
diagram for the model. We find that both the interaction and dissipation rates
induce a $\mathcal{PT}$-symmetry breaking. On the other hand, periodic
modulation of the dissipative coupling in time stabilizes the
$\mathcal{PT}$-symmetric regime. Our findings are corroborated numerically on a
tight-binding chain with gain and loss.
