  In this paper, we provide some topological criteria for the Poisson
Dixmier-Moeglin equivalence for $A$ in terms of the poset $({\rm P. spec A},
\subseteq)$ and the symplectic leaf or core stratification on its maximal
spectrum. In particular, we prove that the Zariski topology of the Poisson
prime spectrum and of each symplectic leaf or core can detect the Poisson
Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover,
we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for
a complex affine Poisson algebra proved in [J. Bell, S. Launois, O.L.
S\'anchez, and B. Moosa, Poisson algebras via model theory and differential
algebraic geometry, J. Eur. Math. Soc. (JEMS), 19(2017), no. 7, 2019-2049] to
the general context of a commutative differential algebra.
