Dynamic Models of Conflict

Abstract

This chapter provides a review of traditional models that are widely used for studying the dynamics of fighting. They include the combat models derived by Lanchester and the Lotka-Volterra models.

Appendix

5.1.1 A. Proof of Propositions 1, 2, 3

Combining (5.29a, 5.29b), the steady-state R* is obtained as a positive root of the second-order equation:

$$ \alpha \delta {x}^2-\left(\alpha \gamma +{\theta}_1\delta -{\theta}_2\beta \right)x+{\theta}_1\gamma =0 $$

(5.34)

Let us denote the discriminant as:

$$ \mathfrak{D}={\left(\alpha \gamma +{\theta}_1\delta -{\theta}_2\beta \right)}^2-4\alpha \gamma \delta {\theta}_1 $$

(5.35a)

Two, one or none positive solutions exist, depending on whether the above expression is positive, zero or negative respectively. The discriminant is expanded as:

$$ \mathfrak{D}={\left(\alpha \gamma \right)}^2-2\left(\alpha \gamma \right)\left({\theta}_2\beta +{\theta}_1\delta \right)+{\left({\theta}_2\beta -{\theta}_1\delta \right)}^2 $$

(5.35b)

Setting x = (αγ), the above can be considered as a second-order equation:

$$ {x}^2-2\left({\theta}_2\beta +{\theta}_1\delta \right)x+{\left({\theta}_2\beta -{\theta}_1\delta \right)}^2=0 $$

(5.36)

It is trivial to show that the two roots $\left({z}_1,\;{z}_2\right)$ are given by:

$$ {z}_1={\left[\sqrt{\theta_i\beta }-\sqrt{\theta_1\delta}\right]}^2 $$

(5.37a)

and

$$ {z}_2={\left[\sqrt{\theta_2\beta }+\sqrt{\theta_1\delta}\right]}^2 $$

(5.37b)

The discriminant in (5.34) is positive if $\alpha \gamma <{z}_1$ or $\alpha \gamma >{z}_2$, and this leads to two real roots of (5.32). It is negative if ${z}_1<\alpha \gamma <{z}_2$, and in this case no real root is obtained. Finally if $\alpha \gamma ={z}_1\;\mathrm{or}\;\alpha \gamma ={z}_2,$ the discriminant is zero and the system has a single real equilibrium. Substituting $\left({z}_1,\;{z}_2\right)$ from (5.36, 5.37a), the three Propositions are readily obtained.

5.1.2 B. Wrong Lotka-Volterra Models

To demonstrate the inadequacies of autonomous Lotka-Volterra models with lagged interactions, the two types of models employed by Francisco (2009) are examined: one is used to study the dynamics of protests versus repression, and the other the dynamics of a civil war. Both of them render improbable behaviour, as shown in the examples below.

5.1.2.1 The Civil Conflict Model: Francisco (2009, Table 2.4, p. 17)

The interacting populations are assumed to represent protesters (P) and repression (R) in France for the period 1980–1995. Keeping the same notation, the system of interactions between adversaries is described as:

$$ {P}_t={\alpha P}_{t-1}-g{P}_{t-1}{R}_{t-1} $$

(5.38a)

$$ {R}_t={bR}_{t-1}+{hP}_{t-1}{R}_{t-1} $$

(5.38b)

Estimated parameter values are reported as a = 0.0161, b = 0.0738, g = 0.234 × 10⁻³, h = 0.2217 × 10⁻⁵. For these values, it is easily found that the system has four equilibria (P*, R*) at (0,0), (417772,0), (0,−4205) and (417772,−4205). Of those, only the trivial equilibrium is meaningful and has two stable eigenvalues at (a, b). These are different from the pair {0.107,−0.00162) reported in Francisco (2009, p. 17) without any further details.

However, even the trivial equilibrium is problematic. Setting as initial conditions P(0) = 1000 and R(0) = 1000, the simulation gives P(1) = −218, R(1) = 76, and the negative value is improbable. Moreover, the low modulus of eigenvalues suggests that the system converges too quickly to the trivial equilibrium, and thus is not suitable to capture long-term dynamics.

5.1.2.2 The Civil War Model: Francisco (2009, Table 5.2, p. 72)

Populations now are assumed to represent battle-deaths (R, S) for the two sides fighting in the US Civil War 1861–1868. Keeping the author’s notation, the Lotka-Volterra model is described as:

$$ {R}_t={\alpha S}_{t-1}-m{S}_{t-1}{R}_{t-1} $$

(5.39a)

$$ {S}_t={bR}_{t-1}-n{S}_{t-1}{R}_{t-1} $$

(5.39b)

Estimated parameter values are reported as a = 0.1559, b = 0.2629, m = 0.349 × 10⁻⁴, n = 0.868 × 10⁻⁵.

Setting R _t  = R _t-1 = R* and S _t  = S _t-1 = S* and cross-substituting (5.38a) and (5.38b) we get

$$ \left(1- ab\right){R}^{*}= - \left( an+m\right){R}^{*}{S}^{*} $$

(5.40a)

$$ \left(1- ab\right){S}^{*}=-\left(bm+n\right){R}^{*}{S}^{*} $$

(5.40b)

The system (5.40a, 5.40b) has two equilibria. One of them is trivial:

$$ {R}^{*}=0,\;{S}^{*}=0 $$

(5.41)

The non-trivial equilibrium is:

$$ {R}^{*}=-\frac{1- ab}{bm+n},\;{S}^{*}=-\frac{1- ab}{an+m} $$

(5.42)

Clearly this is negative and meaningless for the estimated parameter values.

For the trivial equilibrium (5.40a), the Jacobian is given as

$$ J=\left[\begin{array}{cc}\frac{\partial {R}_t}{\partial {R}_{t-1}}& \frac{\partial {R}_t}{\partial {S}_{t-1}}\\ {}\frac{\partial {S}_t}{\partial {R}_{t-1}}& \frac{\partial {S}_t}{\partial {S}_{t-1}}\end{array}\right]=\left[\begin{array}{cc}0& a\\ {}b& 0\end{array}\right],\;\mathrm{with}\;\mathrm{eigenvalues}\pm \sqrt{ab} $$

(5.43)

The eigenvalues are calculated as (−0.20, +0.20) and are very different from the pair {0.1559, −0.50 × 10⁻⁴) reported by Francisco (2009, p. 72) without details.

Setting as initial conditions R(0) = 10,000 and S(0) = 10,000 the simulation gives R(1) = −1931, S(1) = 1761, which are improbable. Again, eigenvalues have a very low modulus, and dynamics die away after a couple of periods. The model is clearly unable to portray the dynamics of a 7-year war.

The reason of both cases failing to realistically express conflict situations lies with the wrong structure of the model, as discussed in Sect. 5.3.6 of the present chapter.