  An agent-based model (ABM) is a computational model in which the local
interactions of autonomous agents with each other and with their environment
give rise to global properties within a given domain. As the detail and
complexity of these models has grown, so too has the computational expense of
running several simulations to perform sensitivity analysis and evaluate
long-term model behavior. Here, we generalize a framework for mathematically
formalizing ABMs to explicitly incorporate features commonly found in
biological systems: appearance of agents (birth), removal of agents (death),
and locally dependent state changes. We then use our broader framework to
extend an approach for estimating long-term behavior without simulations,
specifically changes in population densities over time. The approach is
probabilistic and relies on treating the discrete, incremental update of an ABM
via "time steps" as a Markov process to generate expected values for agents at
each time step. As case studies, we apply our extensions to both a simple ABM
based on the Game of Life and a published ABM of rib development in
vertebrates.
