Stability analysis of a simple mathematical model for school bullying

H. A. Ashi; H. A. Ashi

doi:10.3934/math.2022274

AIMS Mathematics

2022, Volume 7, Issue 4: 4936-4945. doi: 10.3934/math.2022274

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Research article

Stability analysis of a simple mathematical model for school bullying

H. A. Ashi ^,

Department of Mathematics, King AbdulAziz University, Jeddah 21589, Saudi Arabia

Received: 13 October 2021 Revised: 07 December 2021 Accepted: 08 December 2021 Published: 28 December 2021
MSC : 34D20

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.
- nonlinear stability theory,
- local stability,
- global stability,
- transcritical bifurcation,
- numerical simulation,
- Lyapunov function
Citation: H. A. Ashi. Stability analysis of a simple mathematical model for school bullying[J]. AIMS Mathematics, 2022, 7(4): 4936-4945. doi: 10.3934/math.2022274

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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.

References

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