Chaos and stability of a fractional model of the cyber ecosystem

José F. Gómez-Aguilar; Manisha Krishna Naik; Reny George; Chandrali Baishya; İbrahim Avcı; Eduardo Pérez-Careta; José F. Gómez-Aguilar; Manisha Krishna Naik; Reny George; Chandrali Baishya; İbrahim Avcı; Eduardo Pérez-Careta

doi:10.3934/math.20241077

AIMS Mathematics

2024, Volume 9, Issue 8: 22146-22173. doi: 10.3934/math.20241077

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Chaos and stability of a fractional model of the cyber ecosystem

1.
Centro de Investigación en Ingeniería y Ciencias Aplicadas (CIICAp-IICBA), Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos, México
2.
Department of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, Karnataka, India
3.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
4.
Department of Computer Engineering, Faculty of Engineering, Final International University, Kyrenia, Northern Cyprus, via Mersin 10, Turkey
5.
Universidad de Guanajuato, Dpto. Electrónica, Carretera Salamanca-Valle de Santiago, Km 3+1.8, Salamanca Gto, México

Received: 29 April 2024 Revised: 17 June 2024 Accepted: 01 July 2024 Published: 16 July 2024
MSC : 65P20, 92D40, 26A33, 34D08

The widespread use of computer hardware and software in society has led to the emergence of a type of criminal conduct known as cybercrime, which has become a major worldwide concern in the 21st century spanning multiple domains. As a result, in the present setting, academics and practitioners are showing a great deal of interest in conducting research on cybercrime. In this work, a fractional-order model was replaced by involving three sorts of human populations: online computer users, hackers, and cyber security professionals, in order to examine the online computer user-hacker system. The existence, uniqueness and boundedness were studied. To support our theoretical conclusions, a numerical analysis of the influence of the various logical parameters was conducted and we derived the necessary conditions for the different equilibrium points to be locally stable. We examined the effects of the fear level and refuge factor on the equilibrium densities of prey and predators in order to explore and understand the dynamics of the system in a better way. Using some special circumstances, the model was examined. Our theoretical findings and logical parameters were validated through a numerical analysis utilizing the generalized Adams-Bashforth-Moulton technique.
- chaos,
- fear factor,
- prey refuge,
- Caputo fractional derivative,
- cyber ecosystem
Citation: José F. Gómez-Aguilar, Manisha Krishna Naik, Reny George, Chandrali Baishya, İbrahim Avcı, Eduardo Pérez-Careta. Chaos and stability of a fractional model of the cyber ecosystem[J]. AIMS Mathematics, 2024, 9(8): 22146-22173. doi: 10.3934/math.20241077

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Abstract

The widespread use of computer hardware and software in society has led to the emergence of a type of criminal conduct known as cybercrime, which has become a major worldwide concern in the 21st century spanning multiple domains. As a result, in the present setting, academics and practitioners are showing a great deal of interest in conducting research on cybercrime. In this work, a fractional-order model was replaced by involving three sorts of human populations: online computer users, hackers, and cyber security professionals, in order to examine the online computer user-hacker system. The existence, uniqueness and boundedness were studied. To support our theoretical conclusions, a numerical analysis of the influence of the various logical parameters was conducted and we derived the necessary conditions for the different equilibrium points to be locally stable. We examined the effects of the fear level and refuge factor on the equilibrium densities of prey and predators in order to explore and understand the dynamics of the system in a better way. Using some special circumstances, the model was examined. Our theoretical findings and logical parameters were validated through a numerical analysis utilizing the generalized Adams-Bashforth-Moulton technique.

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