When Lanchester met Richardson, the outcome was stalemate: A parable for mathematical models of insurgency

Many authors have used dynamical systems to model asymmetric war. We explore this approach more broadly, first returning to the prototypical models such as Richardson’s arms race, Lanchester’s attrition models and Deitchman’s guerrilla model. We investigate combinations of these and their generalizations, understanding how they relate to assumptions about asymmetric conflict. Our main result is that the typical long-term outcome is neither annihilation nor escalation but a stable fixed point, a stalemate. The state cannot defeat the insurgency by force alone, but must alter the underlying parameters. We show how our models relate to or subsume other recent models. This paper is a self-contained introduction to 2D continuous dynamical models of war, and we intend that, by laying bare their assumptions, it should enable the reader to critically evaluate such models and serve as a reminder of their limitations.