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One of the most common and straightforward ways to explicitly represent spatial heterogeneity in simulations is with the use of some form of a lattice. Lattices are two- or three-dimensional grids in which entities are connected using various forms of local rules. They are thus ideal for representing systems with different levels of local interactions and, thus, for exploring the processes and impacts of self-organization. Models based on lattices have found wide usage in ecology and geology and often use the same basic formalism, despite the differences in the entities being studied. Groups of models, such as cellular automata, self-organized criticality, and diffusion limited aggregation, show how complex spatial structures and temporal behaviors can arise from local interactions only in the absence of external forcing. Other models that incorporate external processes, such as percolation-based models of fire and diseases, demonstrate that self-organization can strongly affect the signal produced by exogenous disturbances. Most lattice models are best used as tools for improving understanding of the dynamics of systems under various sets of assumptions of internal dynamics and external forcing, rather than as a means for accurate predictions of actual system behaviors. Lattice models that integrate ecology and sedimentology could be used to introduce an explicit spatial component into studies of Earth system history.

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