The Richardson Arms Race model was developed by English physicist Lewis Fry Richardson(1881-1953), who was troubled by WWI and WWII because of his Quaker beliefs. Based on the assumption that having a large available arsenal makes a given nation more likely to engage in conflicts, Richardson conjectured that an arms race was often a prelude to war. The ultimate goal of his model is to examine the stability (or lack thereof) of an arms race between two nations in order to predict whether a small incident could potentially start a large conflict.

The model consists of a system of two linear differential equations which are meant to address the following assumptions:

Due to the nature of the equations, solutions do not depend as much on the values of the constants as on their relative magnitude and the signs of r and s, the grievance terms.

The actual equations are:

The constants a and b are thought of as the fear constants, m and n are the restraint constants, and as mentioned, r and s are the grievance terms. Note that only r and s are allowed to assume negative values. Elements of interest when graphing this series are the optimal lines (where dx/dt = 0 and dy/dt = 0), the equilibrium point (x*,y*) where the optimal lines intersect, and the dividing line L* for cases where the ultimate outcome depends on the starting point. Trajectories that approach the origin or the x- or y-axis are said to be going towards disarmament. Those that head towards infinity are said to be runaway arms races. There are four general cases that the situation can resemble (try these in the demo):

Scroll down for instructions

Instructions:
To graph a series of equations, enter the parameters into their corresponding fields on the applet and click Graph. You may need to adjust the window size (Max X and Max Y) to achieve the best view. The optimal lines are automatically graphed for you. Check the Show L* box if you want the dividing line to be displayed.

To plot a trajectory on the graph, just click on the point where you would like the trajectory to start. Note that trajectories that approach a stable point may take a while to completely draw, especially if delta-T is very small. You can draw up to 50 trajectories on the graph before you must clear it or graph a new set of equations.

To clear all trajectories without re-graphing, just click the Clear button.

To zoom in or out on the graph, enter new values in the Max X and/or Max Y fields and click Apply to make the changes. This will also redraw all trajectories to match the new dimensions.

To change the value of Delta T used in computing the trajectories, enter a new value in the Delta T field and click Apply to make the changes. Note that a very small Delta T will cause the trajectories to be drawn very slowly, and you can't draw another while the previous one is still being drawn. Alternatively, a very large Delta T will cause only a few points to be plotted and won't produce a very useful graph.