We introduce a new model for planar point point processes, with the aim of capturing the structure of point interaction and spread in persistence diagrams. Persistence diagrams themselves are a key tool of TDA (topological data analysis), crucial for the delineation and estimation of global topological structure in large data sets. To a large extent, the statistical analysis of persistence diagrams has been hindered by difficulties in providing replications, a problem that was addressed in an earlier paper, which introduced a procedure called RST (replicating statistical topology). Here we significantly improve on the power of RST via the introduction of a more realistic class of models for the persistence diagrams. In addition, we introduce to TDA the idea of bagplotting, a powerful technique from non-parametric statistics well adapted for differentiating between topologically significant points, and noise, in persistence diagrams. Outside the setting of TDA, our model provides a setting for fashioning point processes, in any dimension, in which both local interactions between the points, along with global restraints on the general point cloud, are important and perhaps competing.