  This is Part 2 of our work aimed at classifying the long-time behavior of the
solution to a free boundary problem with monostable reaction term in space-time
periodic media. In Part 1 (see \cite{ddl}) we have established a theory on the
existence and uniqueness of solutions to this free boundary problem with
continuous initial functions, as well as a spreading-vanishing dichotomy. We
are now able to develop the methods of Weinberger \cite{w1, w2} and others
\cite{fyz,lyz,lz1,lz2,lui} to prove the existence of asymptotic spreading speed
when spreading happens, without knowing a priori the existence of the
corresponding semi-wave solutions of the free boundary problem. This is a
completely different approach from earlier works on the free boundary model,
where the spreading speed is determined by firstly showing the existence of a
corresponding semi-wave. Such a semi-wave appears difficult to obtain by the
earlier approaches in the case of space-time periodic media considered in our
work here.
