A Hierarchical Bayesian Agent Based Model for Binary Spatio-Temporal Spread: Theory, PDE Scaling Limit, and an Application to Predator Prey Cycles

We describe a statistical agent-based model (SABM) for binary spatio-temporal data in which the occupancy of each cell evolves as a Bernoulli mixture of three mechanistically distinct processes: local persistence, anisotropic neighborhood dispersal, and long-distance dispersal. The model is embedded in a hierarchical Bayesian framework with conjugate Beta full-conditionals for the persistence and long-distance parameters and a Dirichlet prior on the directional dispersal kernel. A nonstationary extension links the dispersal kernel to a latent habitat-suitability surface through directional gradients of a Gaussian process. We show that, in the small-step regime, the Lagrangian recurrence for the dispersal kernel scales to a classical two-dimensional advection-diffusion partial differential equation whose drift and dispersion coefficients are the first and second moments of the dispersal probabilities. We provide an MCMC algorithm exploiting the exact full-conditionals and demonstrate parameter recovery and PDE-scaling agreement in a simulated example.