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

Abstract

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.

Appendix

Richardson arms race

With Δ≡rs−ρσ, the intersection of the null clines

is at

whose solution for Δ>0 is in the R, S<0 quadrant. The solutions relative to (S, R)=(k, l) correspond to the eigenvectors of , which has eigenvalues

with λ ₊>0, λ ₋<0. The eigenvectors are in directions (ρ+λ _±, s). Noting that ρ+λ ₊>0, ρ+λ ₋<0, we see that there is exponential (runaway) growth that converges along direction (ρ+λ ₊, s) from (k, l).

With Δ<0, the null clines intersect in the positive quadrant. Further, both eigenvalues are now negative, and the fixed point is a stable node, a stalemate.

Richardson arms race with Lanchester aimed fire

The above generalizes straightforwardly to describe Section 4.1, with s replaced by s−d and r replaced by r−c. Now with Δ≔(r−c)(s−d)−ρσ, the appendix describes for Δ>0 the escalation regime and for Δ<0 the stalemate.

Now consider the regime for an S win. If s−d<0 then Δ<0 and the fixed point is at S=(ρk+(r−c)l)/(−Δ)>0, while R=((s−d)l+σk)/(−Δ) is positive or negative respectively as s−d>−σk/l or<−σk/l.

The LRD model: stalemate and S win

That the null clines intersect at a stable fixed point, the stalemate that appears in Figure 3, can be checked by solving for the fixed point and linearizing. Alternatively, consider the null clines near the fixed point, and a small disc centred on it. It is straightforward to show that all trajectories enter this circle, and thence, by the Poincaré-Bendixson theorem in the absence periodic orbits, that the fixed point is stable.

As s increases through=−kl/σ, the points of intersection of the null clines with the S axis cross and the fixed point enters the positive quadrant. In this regime, some trajectories approach the fixed point, while others, with sufficiently large initial S, lead to an S win. They are separated by the trajectory, which passes through the point R=0,S=−l/s.

Two positive-quadrant fixed points in the LRD model

In the LRD model, the positive quadrant typically contained one or no fixed point. However, we noted in Section 4.2 the possibility, under a highly constrained regime of the parameters, of two fixed points. Here we derive the conditions for this to occur.

First we note that the two null clines intersect at

or equivalently

Thus for two real roots, positive in both R and S, we need from (A.4) that ρk+(r−c)l<0 or r−c<−ρk/l, and (since r−c<0) from (A.5) that s>−σl/k. The existence of two positive-quadrant fixed points then further requires that, with Δ=(r−c)s−σρ,

(and we note that both right-hand sides are positive). The resulting fixed points are a stalemate and a saddle point.

Generalizing the LRD model

The condition (16) is for the null clines (14, 15) to intersect, then

and thus (with Δ=(r−c)s−ρσ)

However, θ=0 at S=0, whereas the right-hand side is positive. Thus an intersection is guaranteed if Δ<0, while for Δ>0 there will be an intersection if θ grows faster than linearly in S. With θ≈dR ^u S ^v this occurs if u+v>1.