Dynamical analysis of a model of social behavior: Criminal vs non-criminal population

In this paper, we construct a model motivated by the well known predator-prey model to study the interaction between criminal population and non-criminal population. Our aim is to study various possibilities of interactions between them. First we model it using simple predator-prey model, then we modify it by considering the logistic growth of non-criminal population. We clearly deduce that the model with logistic growth is better than classical one. More precisely, the role of carrying capacity on the dynamics of criminal minded population is discussed. Further, we incorporate law enforcement term in the model and study its effect. The result obtained suggest that by incorporating enforcement law, the criminal population reduces from the very beginning, which resembles with real life situation. Our result indicates that the criminal minded population exist as long as coefficient of enforcement lc does not cross a threshold value and after this value the criminal minded population extinct. In addition, we also discuss the occurrence of saddle-node bifurcation in case of model system with law enforcement. Numerical examples and simulations are presented to illustrate the obtained results.