Let the populations of the owners, criminals, and guards be
Consider a society in which the criminal rate is
, where ξ is a proportionality constant, and the rate of arrests by guards is
The equations that describe the model are:
In the absence of criminals and guards, the population of owners obeys a logistic-type dynamic (with maximum capacity 1). The presence of criminals hinders the growth of
and the existence of security forces represents a cost for the owners.
The criminal population increases at the rate
and it decreases because of the predatory action of guards, the retirement of criminals, and the interspecific competition of criminals (the quadratic last term).
The growth of the security forces is proportional to the crimes committed and decreases as a consequence of the casualties suffered in encounters with criminals and of natural decay.
For more details, see J. C. Nuño, M. A. Herrero, and M. Primicerio, "A Triangle Model of Criminality," Physica A
(12), 2008 pp. 2926–2936.