Applications of Spatial Models to Ecology and Social Systems

Abstract

Interacting particle systems have been applied to model the spread of infectious diseases and opinions, interactions between competing species, and evolution of forest landscapes. In this thesis, we study three spatial models arising from from ecology and social sciences. First, in a model introduced by Schelling in 1971, in which families move if they have too many neighbors of the opposite type, we study the phase transition between a randomly distributed and a segregated equilibrium. Second, we consider a combination of the contact process and the voter model and study the asymptotics of the critical value of the contact part as the rate of the voting term goes to infinity. Third, we consider a family of attractive stochastic spatial models, one of which is introduced by Staver and Levin to describe the coverage of forest. We prove that the mean-field ODE gives the asymptotically sharp phase diagram for existence of stationary distributions, while for Staver and Levin model there can still be non-trivial stationary distributions even when the absorbing fixed point of the mean-field ODE is stable.