A two-population insurgency in Colombia: Quasi-predator-prey models–A trend towards simplicity

A sequence of analytic mathematical models has been developed in the context of the “low-level insurgency” in Colombia, from 1993 to the present. They are based on generalizations of the two-population “predator-prey” model commonly applied in ecological modeling, and interestingly, the less sophisticated models yield more insight into the problem than the more complicated ones, but the formalism is available to adapt the model “upwards” in the event that more data becomes available, or as the situation increases in complexity. Specifically, so-called “forcing terms” were included initially in the coupled differential equations to represent the effects of government policies towards both the narco-terrorist or insurgent (II) and susceptible (SS) populations. These terms are in general functions of time, since it is to be expected that changes in policy will occur as the outcomes of previously implemented policies are recognized. Both continuous and discontinuous forcing functions can be appropriate for each population. Although nonhomogeneous systems are discussed for both populations, the majority of the analysis focused on a system with forcing terms for the terrorist population only.Two categories of models emerged: 1 — in which the time-dependent forcing terms were independent of the two populations, and 2 — in which these terms were directly proportional to the respective populations. Model 2 in fact, can be considered as a generalization of the unforced (or homogeneous) version of model 1, and as such is implemented to describe the data obtained for the insurgent population from 1993–2003. This provides some restrictions on the unknown parameters in the model. Because of new government policy towards the insurgent population as a result of the election of President Uribe in 2002, a slight but significant modification of this model is used to describe the II-population from 2003 to the present time, again resulting in useful relationships between the various model parameters. The data suggest that a further simplification in this model is appropriate beyond 2003, and an analytic solution was found for both populations. A further simplification results from examining the related decoupled system of equations, yielding a straightforward predictive model for the behavior of the SS-population in particular. Finally, in the Appendix, detailed analysis of model 2 yields a general analytic solution of the system for a wide range of sub-models, expressible in terms of confluent hypergeometric functions.