Stability analysis of a simple mathematical model for school bullying

Abstract

School bullying is a highly concerned problem due to its effect on students’ academic achievement. The effect might go beyond that to develop health problems, school drop out and, in some extreme cases, commit suicide for victims. On the other hand, adolescents who continuously bully over time are at risk of becoming involved in gang membership and other types of crime. Therefore, we propose a simple mathematical model for school bullying by considering two variables: the number of victims students and the number of bullies students. The main assumption employed to develop the mathematical model is that school policy bans bullying and expels students who practice this behavior to maintain a constructive educational environment within the school. We show that the model has two equilibrium points, and that both equilibrium points are locally and globally asymptotically stable under certain conditions. Also, we define a threshold parameter with a new criterion called the bullying index. Furthermore, we show that the model exhibits the phenomena of transcritical bifurcation subject to the bullying index. All the findings are supported with numerical simulations.