  We study positive Toeplitz operators on the Bergman spaces having the fast
decreasing weight functions in a certain class. We give the characterizations
for the boundedness and compactness of Toeplitz operators in terms of their
Berezin transforms and averaging functions. In particular, we show the
equivalent conditions in order that Toeplitz operators on $A^2(\omega)$ belong
to the Schatten class $S_p$, $0<p<\infty$.
