A Stochastic Spatial Model for Invasive Plants and A General Theory of Monotonicity for Interaction Map Particle Systems

Abstract

Awareness of biological invasions is becoming widespread and several mathematical tools have been used to study this problem. Interacting particle systems, specifically the contact process, have been used to study systems with invasion/infection type dynamics. The Propp-Wilson algorithm is a method for exact sampling from the stationary distribution of an ergodic monotone Markov chain using a method called coupling from the past. The contact process is monotone so we can sample exactly from the stationary distribution of a modified finite grid version using the Propp-Wilson algorithm. In order to study an invasion, we would like to include at least 2 species; however, monotonicity is not well defined for contact processes with more than 2 particle types. Here we develop a general theory of monotonicity for interaction map particle systems, which are interacting particle systems with contact process type dynamics. This allows us to create monotone models with any number of particles and to use the Propp-Wilson algorithm for not only sampling from the stationary distribution, but analyzing the path of invasion leading to equilibrium. Virtual particle invasion models that fall into this new theoretical framework, which we develop here, present a wide range of biological dynamics. Computer simulation of the stochastic system and mean field analysis are two powerful tools that we use for analyzing these types of models. Statistics gathered along the path to invasion help us understand the spatial dynamics of this ecological process and what the stationary behavior looks like. This allows us to understand when the invasion is successful or if coexistence occurs and how these depend on the transition rates and interactions within the process.