  Characterizing the dynamics of time-evolving data within the framework of
topological data analysis (TDA) has been attracting increasingly more
attention. Popular instances of time-evolving data include flocking/swarming
behaviors in animals and social networks in the human sphere. A natural
mathematical model for such collective behaviors is a dynamic point cloud, or
more generally a dynamic metric space (DMS).
  In this paper we extend the Rips filtration stability result for (static)
metric spaces to the setting of DMSs. We do this by devising a certain
three-parameter "spatiotemporal" filtration of a DMS. Applying the homology
functor to this filtration gives rise to multidimensional persistence module
derived from the DMS. We show that this multidimensional module enjoys
stability under a suitable generalization of the Gromov-Hausdorff distance
which permits metrizing the collection of all DMSs.
  On the other hand, it is recognized that, in general, comparing two
multidimensional persistence modules leads to intractable computational
problems. For the purpose of practical comparison of DMSs, we focus on both the
rank invariant or the dimension function of the multidimensional persistence
module that is derived from a DMS. We specifically propose to utilize a certain
metric d for comparing these invariants: In our work this d is either (1) a
certain generalization of the erosion distance by Patel, or (2) a specialized
version of the well known interleaving distance. We also study the
computational complexity associated to both choices of d.
