Percolation Operators and Related Models

Abstract

Here we review the theory and methods of the two basic stochastic models with absorbing states: percolation operators and contact processes. We explore connections between them by studying discrete-time approximations of a continuous-time contact processes. In particular, we look at the approximations based on both synchronous and asynchronous updating schemes. Additionally, we go on to discuss several individual-based models, which are commonly used to model different biological phenomena. Specifically, we focused on models with absorbing states that, have spatially non-homogenous stationary states, or that have shown to be bi-stable. More generally, we aim to demonstrate the challenges associated with reconciling different mathematical descriptions of natural phenomena.