File S1 - Apparent Strength Conceals Instability in a Model for the Collapse of Historical States

Mathematical details. This can be read as an extended methods section. All of the supporting information including Sections S1–2 and Figures S1–7 are contained within. Section S1 derives the ordinary and partial differential equation models (Models 1a and 1b). These allow calculation of the dynamics of power in cooperation and defection phases (Section S1.1). From this we can obtain defection thresholds for each faction (Section S1.2). Bounds for the duration of cooperation and defection (Section S1.3) are calculated by assuming that all non-leading factions have the same power. Some equivalent reformulations (Section S1.4) are provided. Section S2 describes each of the model extensions described in Section, with Figures S1–7 showing samples from each of these models in the same format as Figure 2a. (PDF)