How can we model global phenomenon based on local interactions? Agent-Based (AB) models are local rule-based discrete method that can be used to simulate complex interactions of many agents. Unfortunately, the relative ease of implementing the computational model is often counter-balanced by the difficulty of performing rigorous analysis to determine emergent behaviors. Calculating existence of fixed points and their stability is not tractable from an analytical perspective and can become computationally expensive, involving potentially millions of simulations. To construct meaningful analysis, we need to create a framework to approximate the emergent, global behavior. Our research has been devoted to developing a framework for approximating AB models that move via random walks and undergo state transitions. First, we developed a general method to estimate the density of agents in each state for AB models whose state transitions are caused by neighborhood interactions between agents. Second, we extended previous random walk models of instantaneous state changes by adding a cumulative memory effect. In this way, our research seeks to answer how memory properties can also be incorporated into continuum models, especially when the memory properties effect state changes on the agents. The state transitions in this type of AB model is primarily from the agents’ interaction with their environment. These modeling frameworks will be generally applicable to many areas and can be easily extended.

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