Differential equations

Models composed of differential equations (DEs) have also been widely used to obtain insights into social systems. In contrast to ABMs, in which the interest is often on individuals, the dependent variable in a DE-based model of a social system is often taken to be some attribute associated with a group of individuals. DE-based models are therefore typically used for more aggregated scenarios than ABMs (although, there are exceptions: DE-based models are employed with individual perspectives in Liebovitch et al. (2008) and Curtis and Smith (2008) and agent-based models are employed with aggregated perspectives in Cederman (2003)).

There are many examples of DE-based models being applied to study civil vio- lence in a modern setting. Classical models, however, are typically concerned with the actions of two or more adversaries during more conventional forms of conflict or warfare. More recently, some of these have been adapted to consider modern conflicts, including civil violence and insurgencies. For this reason, attention is initially given to models of conventional conflict that have more recently been adapted and applied to civil violence.

In many cases, the dependent variable of a DE conflict model is taken to be the number of individuals on each side of a conflict. In an early example, Lanchester (1916) uses DEs to model different types of attritional warfare between two adversaries. He

considers how new technologies, such as the introduction of aircraft, might change military strategy. He does this by proposing two models: one in which the rate of loss of each adversary is proportional to the size of their opponent, representing aimed firepower, and one in which the loss of each adversary is proportional to the product of the size of their opponent and the size of themselves, corresponding to unaimed fire. In the case of aimed fire, the model suggests that a better advantage comes from having a larger army, rather than training the army to be more effective. For unaimed fire, the benefit associated with being more effective is equivalent to a benefit brought about by a numerical advantage, suggesting that effective training is likely to be just as successful as recruitment during warfare.

There have been a number of studies that follow Lanchester in modelling the change in the size of adversaries with DEs. Deitchman (1962), for example, proposes a Lanchester-type model of Guerrilla warfare. This model is further developed in In- triligator and Brito (1988), who incorporate a predator-prey framework to examine the impact of civilians, and in Kress and MacKay (2014), who generalise the model to ac- count for military intelligence as well as diminishing numbers of insurgents. Atkinson et al. (2011) also use Lanchester-type models to investigate insurgent warfare, in which they compare a DE model to a number of modern conflicts.

Another class of DE-based models stems from the work of Richardson (1960a) on the actions of nations during the lead up to war. In this case, the dependent variable is not the size of each adversary but the level of military spending. The key assumption in Richardson’s model is that the extent of a nation’s military defences, denoted by

p, reacts to the military defences of their adversary, given by q, at a rate proportional

to q. The adversary behaves similarly and reacts to the defence p. This reciprocal

action-reaction process can result in an escalating arms race between two adversaries. A nation may react to the military defences of its rival both as a defensive measure, in order to provide protection from the threat posed by their opponent, as well as an aggressive measure, to exert threat over their opponent.

Richardson believed this process on its own was not enough to model how arms races might evolve and so included two more factors which influenced the military defences of a nation: its own level of expenditure, which was hypothesised would diminish the change in defences as measured by the model, and also exogenous effects,

which were termed ‘grievances’. The model was first presented as a two-dimensional linear system of ordinary differential equations given by

dp

dt = ˙p =−σ1p + ρ1q + ǫ1 (2.3)

dq

dt = ˙q = ρ2p− σ2q + ǫ2,

whereσ1andσ2are parameters that specify the strength of inhibition from each nation’s

own expenditure on the model, ρ1 and ρ2 are parameters that specify the strength of

interaction between adversaries, andǫ1 andǫ2 are parameters that specify the external

grievances of each adversary.

Measuring the ‘defence’ of a nation—the dependent variable considered in this model—is difficult to achieve empirically. Richardson initially operationalised the dependent variablesp and q by considering military expenditures of two adversaries.

However, there are complications encountered by defining the variable in this way, as some have pointed out (Brauer, 2002). Richardson’s primary objective was to demon- strate how modelling simple interactions can shed light on the resolution of conflict, and was not necessarily on the quantification of military defences. As a result, he also allowed the possibility for negative values ofp and q. Although difficult to comprehend

in terms of military expenditure, it was argued that negative values might correspond to some measure of cooperation between the two nations, which might, for example, be measured via trade.

In the first application of the model in equation 2.3, Richardson (1960a) shows how the increase in military expenditure of four nations—Russia, Germany, France and Austria-Hungary—on two sides of a conflict in the years prior to the First World War very closely follows a pattern that would have been predicted by the model. A figure from Richardson (1960a) is reproduced in Figure 2.1 that shows the straight line expected from the model, against the data Richardson gathers for the years shown. The equation for the straight line is obtained by summing the two equations in 2.3 and assuming that both sides of the conflict react to their own defences and the defences of their opponent at the same rate, so thatσ1 = σ2andρ1 = ρ2.

Perhaps as a consequence of the very close fit between the model and the small dataset in Figure 2.1, Richardson’s arms race model has been applied to various scenar- ios around the world which have been considered to exhibit ‘arms race’-type behaviour.

190 200 210 220 230 240 250 260 270 Total expenditure 0 10 20 30 40 50 60 C ha ng e in ex pe nd it ur e 1909 10 1910 11 1911 12 1912 13

Figure 2.1: The change in the sum of defence budgets against the sum of defence

budgets for four nations during the four years prior to the First World War. The

four nations are Russia, Germany, France and Austria-Hungary and the values plotted represent the sum of defence budgets over these nations. Defence expenditure data was gathered from various sources by Richardson, and the line represents the best fit of what would be expected from the model in equation 6.1, assuming thatσ1 = σ2 = σ

andρ1 = ρ2 = ρ. This figure is reproduced from Richardson (1960a). The gradient is

given by ρ− σ and is estimated by Richardson to be 0.73. An ordinary least squares

In many of these cases, however, when using modern estimation techniques with large datasets, the model has been unable to reproduce the empirical data to such a close ex- tent. In fact, much of the time, the model prediction is found to be a poor fit to the data. Dunne and Smith (2007) give an overview of some of the econometric applications of Richardson’s arms race model. They discuss the mixed results when the model is ap- plied to the India-Pakistan arms race from 1960. In particular, using purely temporal vector autoregression methods, they apply the Richardson model to arms expenditure data for India and Pakistan for the period between 1960 and 2003. They find that, for some time periods, action-reaction type dynamics present in the Richardson model can be observed in empirical data; however, for other time periods, no such consistencies can be found.

Brauer (2002) reviews applications of the model to the Greco-Turkish arms race, and points out several issues associated with fitting such models to arms race data. Some of the issues Brauer points out are relevant to many applications of differential equation-based models to social systems. For example, problems are often encountered with data availability, leading to complications in defining appropriate dependent vari- ables from the data, which are required in order to validate the model. In the case of arms expenditure, for example, decisions regarding whether to take the dependent vari- able as the absolute expenditure on defence for each nation, or the relative amount of expenditure on defence as a proportion of that nation’s GDP, can lead to varying levels of success of the fit of the model.

Parameter estimation can also be compromised as, in social systems in particular, parameter values can change very quickly. As Saperstein (2007) points out, the param- eters of the original Richardson model in equation 2.3 are assumed to remain constant for timescales over which the dependent variables change. Since decisions regarding military expenditure can be made by reacting to a single event that can occur on very short timescales, there may be many scenarios in which this assumption is not valid. Saperstein (2007) goes on to define nonlinear extensions of the model in which the parameters of the system change according to the strategic aims of each nation.

Studies reporting difficulties in matching the model to empirical data sometimes overlook the principal reason for such discrepancies: the model is very simplistic. There are mechanisms not present in the model which may well play an important

role. Richardson’s model is a useful descriptive tool to understand the possible states of an international system, and how the system might transition between these states. It was not intended to be used as a predictive tool to forecast defence budgets (Zinnes and Muncaster, 1984). Indeed, proponents of Richardson’s model will argue that the simplicity of the model is a virtue: it can be easily analysed, understood, and be used to explain the outcomes of different scenarios, and how transitions might occur between them.

Similarly to Lanchester’s model of combat, Richardson’s model has been extended in a number of ways to consider the dynamics of different types of conflict. In one example, asymmetrical conflict is investigated by considering what might occur if a smaller adversary is unlikely to directly compete with a larger one, instead choosing to change its tactics by, for example, submitting to the larger nation’s threats or attempting to undermine the larger nation by employing different strategies rather than directly competing by increasing the size of their own defences. In Richardson (1951) and Richardson (1960a), the model in equation 2.3 is extended to consider the possibility of submission of a nation in an arms race if the lead became too large. This model is given by:

˙p =_−σ1p + ρ1q (1− υ1(q− p)) + ǫ1 (2.4)

˙q =_−σ2q + ρ2p (1− υ2(p− q)) + ǫ2,

whereυ1, υ2 ≥ 0 are additional parameters that Richardson termed ‘submissiveness’,

whilst all other parameters have the same interpretation as in equation 2.3. The param- etersυ1 andυ2 determine the extent to which the reaction terms are diminished pro-

portional to the opponent’s lead in defences. Their inclusion has the effect of enabling scenarios in which, once a sufficient lead develops for one nation, their opponent will slowly begin to react less and eventually begin to reduce their defences, as they concede their position in the arms race.

Asymmetric dynamics can also occur during insurgent warfare and other types of civil violence (Ryan, 2006). In this case, whilst it is difficult to measure the dependent variable in terms of military expenditure, there may be other measures that determine the level of threat or cooperation between opponents, such as the amount of public support for either side, or the likelihood of one side initiating conflict against the other.

Karmeshu et al. (1990) consider an extension of the Richardson model that can be applied to domestic political conflict in order to investigate the interactions between a ruling and a challenger group. Similarly, models proposed by Jackson et al. (1978), Intriligator and Brito (1988) and Blank et al. (2008) are all reminiscent of Richardson’s model since they incorporate dynamical processes of escalation and inhibition, as well as various extensions that might be relevant in civil violence scenarios.

The ability for models of escalation processes to be applied to a range of different types of conflict, from arms races to insurgencies, suggests that they can be interpreted as models of general conflict between two adversaries. Indeed, recently, a number of authors have taken this perspective, and proposed models of general conflict processes that build upon theories of conflict developed in psychology. The authors argue that such models can be applied to nations, groups or individuals who interact in a conflict with an opponent. No constraints are placed upon the range of situations to which the model may be applied. For example, Liebovitch et al. (2008) present the dynamical properties of the model given by:

˙p =_−σ1p + ρ1tanh q + ǫ1 (2.5)

˙q =_−σ2q + ρ2tanh p + ǫ2.

The relationship between this model and Richardson’s model in equation 2.3 is clear: the interaction terms have become nonlinear functions bounded in (_−ρ1, ρ1) and

(−ρ2, ρ2), respectively. Further extensions have recently been explored in Qubbaj and

Muneepeerakul (2012) and Rojas-Pacheco et al. (2013) by adding time delays to these reaction terms.

Perhaps surprisingly, given how important the consideration of space is in various conflict processes, there have been few spatial extensions of DE-based conflict mod- els. Borrowing techniques from ecology (see, for example, Malchow et al. (2008)), some spatially-explicit models have been proposed using reaction-diffusion equations to specify how a dependent variable of interest varies in space. For example, Keane (2011a) presents a spatially extended version of the Lanchester equations and demon- strates how strategic manoeuvring of combat units can be incorporated into a spatially continuous model. Spatial Lanchester models are explored further in Gonz´alez and Villena (2011), in which they are derived from first principles based on assumptions

of the movement dynamics of troops. Another example is Brantingham et al. (2012) who present a spatially extended version of the Lotka-Volterra equations to model the geographic evolution of gang boundaries in Los Angeles. They observe consisten- cies between the model and the real world system, demonstrating that events cluster in space in a way predicted by the model. Reaction-diffusion models have also been used extensively in models of urban crime (Short et al., 2010a,b; Pitcher, 2010; Ro- driguez and Bertozzi, 2010; Berestycki and Nadal, 2010). Some have argued however, that reaction-diffusion models may not be the most appropriate method of accounting for spatial dependency since such models can lack a clear theoretical argument for the continuous diffusion of the dependent variable in space (Gonz´alez and Villena, 2011; Ilachinski, 2004).

Another approach to modelling spatial dependencies with DEs is through the use of spatial interaction models. Spatial interaction models specify how the value of a dependent variable at one location interacts with the dependent variable at another. They can be readily employed within differential equations, which typically specify the change in that variable over time, taking into account any spatial interaction. Davies et al. (2013), for example, present a DE-based model of the London riots that employs a spatial interaction model to account for spatial dependency in contagion processes associated with rioting. The authors use their model to investigate policing strategies— in particular concerning police deployment strategies—in an effort to understand how these might affect outcomes during such extreme events.