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Artzrouni and Komlos's 1996 spatial model visually represents territorial dynamics in Europe from 500 to 1800 AD using a grid system. The model underscores the influence of a state's border position and suggests coastal countries form more predictably than inland ones. However, it highlights the limitations of solely using geopolitical mechanisms to predict empire dynamics. Turchin believes other factors, like Ibn Khaldun's concept of "asabiyyah" (collective solidarity), play a significant role in empire rise and fall.

Peter Turchin utilizes the Lotka-Volterra (predator-prey) equation, originally designed to model population dynamics between predators and their prey, to understand the complexities of medieval agrarian states. These states, according to Turchin, can be viewed as oscillating systems influenced by variables like territory size and military success. Drawing from Randall Collins' geopolitical theory, Turchin identifies key parameters such as geopolitical resources, logistic loads, and peripheral position. The interplay of these variables results in non-linear relationships between territory size and rate of change, suggesting there's an equilibrium point beyond which territorial expansion becomes inefficient for the state.

Introduced by Peter Turchin in 2003, cliodynamics uses mathematical models to analyze long-term historical trends. Drawing from the concept of “asabiyyah” (social solidarity), Turchin focused on medieval agrarian societies, using differential equations and multi-agent modeling. He identified various growth patterns in state dynamics and emphasized the need for negative feedback in models, highlighting the cyclical nature of historical growth and decline.