Interacting Particle Systems on Lattices and on Graphs

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This project concerns spatial models for ecological and social interactions motivated by various applications. The theme of this research is to see how predictions change when systems previously studied under the assumption that each individual interacts with all the others are made more realistic by incorporating space. The four main examples are the following: (i) the Staver-Levin forest model, which predicts that forest and savannah (grassland with isolated trees) are alternative stable states; (ii) evolutionary games, which have long been used in ecology to explain phenomena such as the persistence of altruistic behavior; (iii) Axelrod's model, which studies the spread of opinions when individuals interact with a probability based on the number of the number of opinions they share; (iv) the latent voter model, which studies the spread of technology in a social network when consumers who have just acquired a new product will wait some time before they are willing to purchase a new one. The general goal of studying these idealized models is to understand how properties of the equilibrium of the system depend on the details of the interactions. When each individual interacts with all the others, the system is an ordinary differential equation and is easily studied. However, when space is explicitly taken into account the problems become very difficult. This project has the following specific goals: (i) show that in the Staver-Levin model, the direction of movement of a boundary between forest and savannah indicates the one state that is the true equilibrium in the spatial model; (ii) show that evolutionary games have three separate weak selection regimes that can lead to a PDE, ODE, or a regime in which Tarnita's formulas are valid; (ii) complete Junchi Li's thesis work studying Axelrod's model in the situation in which there are a large number of issues about which there are a large number of opinions (this would provide the first rigorous result for that model in two dimensions); (iv) show that even if latent period is brief, it changes the dynamics so that there is only one nontrivial stationary distribution, in contrast to the one parameter family in the voter model without latency.