We consider systems of many spatially distributed phase oscillators that
interact with their neighbors. Each oscillator is allowed to have a different
natural frequency, as well as a different response time to the signals it
receives from other oscillators in its neighborhood. Using the ansatz of Ott
and Antonsen (Ref. \cite{OA1}) and adopting a strategy similar to that employed
in the recent work of Laing (Ref. \cite{Laing2}), we reduce the microscopic
dynamics of these systems to a macroscopic partial-differential-equation
description. Using this macroscopic formulation, we numerically find that
finite oscillator response time leads to interesting spatio-temporal dynamical
behaviors including propagating fronts, spots, target patterns, chimerae,
spiral waves, etc., and we study interactions and evolutionary behaviors of
these spatio-temporal patterns