Our model of the interaction between the media and terrorist groups is similar in spirit to Frey and Rohner (2007) and Pfeiffer (2012b). However, their models are static – although Frey and Rohner (2007) consider a dynamic extension in the appendix – and formulated in terms of controls rather than states. We also explicitly introduce a third sector, namely sponsors of terrorism, to motivate our hypothesis that there exists a feedback loop between the *share* of media capacity devoted to terrorism and terrorist activity. On the other hand, the model remains *ad hoc* as we specify the behavior of agents in our model parametrically rather than derive it from optimization. To do the latter, one would need to solve a differential game between terrorists (and the media sector) that contains non-linearities. Beckmann and Reimer (2014) review the theoretical problems this entails and the consequent limitations for formal conflict economics.

The model contains three state variables, viz. the resources *r*_{t} available to the terrorist group, the public attention *a*_{t} for terrorism and for other news items (*b*_{t} ). Suppressing time indices, the latter two are supposed to change over time according to the following linear differential equations:

$$\dot{a}=\frac{\u03f5sr}{p}-{\delta}_{1}a$$(1)

$$\dot{b}=\tilde{n}-{\delta}_{2}b$$(2)

In equation (1), *s* represents the share of available resources that terrorist groups devote to carrying out attacks, *p* are the costs of a single attack, and *ϵ* measures the effectiveness of terrorist attacks, e.g. the average number of casualties per attack.

Public interest in a news item wanes as time goes by. We capture this by including depreciation of *a* and *b* at the rates *δ*_{1} and *δ*_{2}, respectively. “Other” news is assumed to arrive at a random rate $\tilde{n};$ in numerical simulation, we typically assume this to be uniformly distributed over some range [0, *n*_{max}].

$$\dot{r}=\frac{aw}{a+b}-{\delta}_{3}r-sr$$(3)

The terrorist sector receives funding from sponsoring nations, groups and individuals, whose willingness to pay depends on the parameter *w* and the *share* of reporting on terrorist events in the media (Pfeiffer, 2012b). The fraction $\frac{a}{a+b}$ represents the familiar ratio conflict success function (Hirshleifer, 2001), where the scale parameter has been set to unity. Terrorist resources can be spent on attacks (*sr*) or saved, in which case they depreciate at a rate *δ*_{3}. This assumption reflects the hypothesis that terrorist groups find it difficult to put their capital to productive use in the official economy, and that some resources will be destroyed by anti-terror efforts.

There is no stationary state proper in this model due to the random stream of general news $\tilde{n}.$ However, if *n* were a constant and equal to its expectation, a stationary state could be computed. Letting $\dot{a}=\dot{b}=\dot{r}=0,$ two solutions to our system of equations can be obtained. The first one is a corner solution where both the number of attacks and the terrorist resources are zero, and the second one is

$${a}^{\ast}=\frac{\u03f5sw}{{\delta}_{1}p\mathrm{(}{\delta}_{3}+s\mathrm{)}}-\frac{\text{E}\tilde{n}}{{\delta}_{2}}$$(4)

$${b}^{\ast}=\frac{\text{E}\tilde{n}}{{\delta}_{2}}$$(5)

$${r}^{\ast}=\frac{w}{s+{\delta}_{3}}-\frac{{\delta}_{1}p\text{E}\tilde{n}}{{\delta}_{2}\u03f5s}$$(6)

Besides imposing obvious restrictions on the values of our parameters required to ensure that the stationary state lies in the positive orthant, these results do not offer any surprises. In order to determine the stability properties of the system, we compute its Jacobian

$$\mathcal{J}=\mathrm{(}\begin{array}{ccc}-{\delta}_{1}& 0& \frac{\u03f5s}{p}\\ 0& -{\delta}_{2}& 0\\ \frac{bw}{{\mathrm{(}a+b\mathrm{)}}^{2}}& -\frac{aw}{{\mathrm{(}a+b\mathrm{)}}^{2}}& -{\delta}_{3}-s\end{array}\mathrm{)}$$(7)

the eigenvalues of which are *λ*_{1}=–*δ*_{2}, ${\lambda}_{2}=\frac{\sqrt{{A}^{2}+4b\u03f5sw}-\mathrm{(}a+b\mathrm{)}\sqrt{p}\mathrm{(}{\delta}_{3}+s+{\delta}_{1}\mathrm{)}}{2\mathrm{(}a+b\mathrm{)}\sqrt{p}}$ and ${\lambda}_{3}=\frac{-\sqrt{{A}^{2}+4b\u03f5sw}-\mathrm{(}a+b\mathrm{)}\sqrt{p}\mathrm{(}{\delta}_{3}+s+{\delta}_{1}\mathrm{)}}{2\mathrm{(}a+b\mathrm{)}\sqrt{p}},$ where *A* represents the expression $A=\mathrm{(}a+b\mathrm{)}\sqrt{p}\mathrm{(}{\delta}_{3}+s-{\delta}_{1}\mathrm{}\mathrm{)}\mathrm{.}$ As *A*^{2}+4*bϵw*>0, all three eigenvalues will be elements of the real line. Furthermore, it is obvious that *λ*_{1}<0 and *λ*_{3}<0. The sign of *λ*_{2} is indeterminate, although numerical experimentation shows that it is positive only for small values of *a* and *b*. Consequently, the stationary point – if it exists at all in the positive orthant –, is either asymptotically stable or saddle point stable, with the former case being more likely. Figure 1 presents a representative simulation run of the model.
^{3}

Figure 1: An example simulation of the theoretical model. (A) Vector field, (B) time path of media attention and the number of incidents.

As the simulated time series in Figure 1 illustrates, the model predicts terrorist acts and their share of reporting to move together. All three stocks are endogenous, in particular, there is a causal link from terrorism to media reports as well as the reverse causality. We therefore state the following two hypotheses:

**Hypothesis 1.** *Terrorist activity Granger-causes the proportion of media reports devoted to terrorism.*

**Hypothesis 2.** *The intensity of reporting in the media Granger-causes the volume of terrorist activity.*

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