Assessing the impact of escalating attacks on soft targets by criminal gang: A modelling viewpoint using bifurcation analysis

Abstract

The emerging threat of criminal gangs in society has severe consequences on the socio-economic well-being of individuals. To examine the impact of morally upright individuals and attacks on soft targets (i.e., vulnerable members of the society) on the coexistence dynamics of the society, we employ a system of ordinary differential equations (ODE) where individuals in the community are allowed to mix and interact. In the co-dimension one bifurcation analysis, we discover the emergence of supercritical and subcritical Hopf bifurcation, transcritical bifurcation, and saddle–node bifurcation of cycle. To understand how these different attractors interact and mediate a crime-free society, we conduct a co-dimension two bifurcation analysis (i.e., by using the morally upright individuals (h1) and attacks on soft target (e) as our bifurcation parameter). The interplay between morally upright individuals and attacks on soft targets shows the presence of a global attractor (i.e., Generalized Hopf bifurcation), which serves as a synchronization centre for the model. Several studies have reported the emergence of stable limit cycles in the criminal gang model. However, we discover a transition from stable to unstable limit cycles when the attacks on soft targets are considered under distinct coexistence factors. This study revealed that morally upright individuals and attacks on soft targets positively affected the coexistence dynamics of society. Furthermore, we observed that an unstable limit cycle is a causative factor for chaotic dynamics in the non-linear system. We further noted that the positive effect of morally upright individuals plays a crucial role in mediating a stable and crime-free society. The study concluded that the recruitment rate of susceptible individuals to the criminal gang has little consequence on the peaceful coexistence of society if the society maintains high morality.