  Mathematical models come in many forms across biological applications. In the
case of complex, spatial dynamics and pattern formation, stochastic models also
face two main challenges: pattern data is largely qualitative, and model
realizations may vary significantly. Together these issues make it difficult to
relate models and empirical data -- or even models and models -- limiting how
different approaches can be combined to offer new insights into biology. These
challenges also raise mathematical questions about how models are related,
since alternative approaches to the same problem -- e.g., cellular Potts
models; off-lattice, agent-based models; on-lattice, cellular automaton models;
and continuum approaches -- treat uncertainty and implement cell behavior in
different ways. To help open the door to future work on questions like these,
here we adapt methods from topological data analysis and computational geometry
to quantitatively relate two different models of the same biological process in
a fair, comparable way. To center our work and illustrate concrete challenges,
we focus on the example of zebrafish-skin pattern formation, and we relate
patterns that arise from agent-based and cellular automaton models.
