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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. [ABSTRACT FROM AUTHOR]

Rumor spreading is the circulation of doubtful messages on the social network. Fact retrieving process that aims at preventing the spread of the rumor, appears to have a significant global impact. In this research, we have investigated a mathematical model projecting rumor spread by considering six groups of individuals namely ignorant, exposed, intentional rumor spreader, unintentional rumor spreader, stifler, and fact retriever. To represent the current abnormal and fast pattern of the message spread around various platforms, in the projected model, we have implemented the fractional derivative in the Caputo context. Using the existing theory of the fractional derivative, we have examined the theoretical aspects such as the existence and uniqueness of the solutions, the existence and stability of the rumor-free and rumor equilibrium points, and the global stability of the rumor-free equilibrium point. Computing basic reproduction numbers, we have analyzed the existence and stability of points of equilibrium. The sensitivity of basic reproduction numbers is also examined. Importance of the fact retrieving drive is highlighted by relating it to the basic reproduction number. Finally, by applying the Adams-Bashforth-Moulton method, we have presented the numerical results by capturing the profile of each of the groups under the influence of fractional derivative and investigated the impact of rumor verification rate and contact rate in controlling and preventing the rumor. With the Caputo fractional operator in the projected model, the current research highlights the significance of the fact retriever and the curb in individual contact and captures the relevant consequences. [ABSTRACT FROM AUTHOR]

In the marine ecosystem, phytoplankton plays a vital role as the primary supplier of oxygen, contributing to 50% of the total oxygen production. Not only does it serve as a significant source of food for other species, but it also sustains life in the ocean. However, the rising ocean temperatures caused by global warming have severely hindered the ability of phytoplankton to generate oxygen. Furthermore, fishes are crucial consumers of oxygen within the marine ecosystem. This research paper presents a model that intricately connects the dynamics of oxygen, phytoplankton, zooplankton, and fish using the Caputo fractional derivative. The model aims to examine the impact of global warming on the collective dynamics by considering the relationship between the rate of oxygen generation, temperature, and time. The paper establishes the existence and uniqueness of solutions and also analyzes the stability of equilibrium points. Numerical simulations are conducted to demonstrate the impact of fractional derivatives and global warming on oxygen depletion and species extinction. [ABSTRACT FROM AUTHOR]

Investigation of the dynamical behavior related to environmental phenomena has received much attention across a variety of scientific domains. One such phenomenon is global warming. The main causes of global warming, which has detrimental effects on our ecosystem, are mainly excess greenhouse gases and temperature. Looking at the significance of this climatic event, in this study, we have connected the ACT-like model to three climatic components, namely, permafrost thaw, temperature, and greenhouse gases in the form of a Caputo fractional differential equation, and analyzed their dynamics. The theoretical aspects, such as the existence and uniqueness of the obtained solution, are examined. We have derived two different sliding mode controllers to control chaos in this fractional-order system. The influences of these controllers are analyzed in the presence of uncertainties and external disturbances. In this process, we have obtained a new controlled system of equations without and with uncertainties and external disturbances. Global stability of these new systems is also established. All the aspects are examined for commensurate and non-commensurate fractional-order derivatives. To establish that the system is chaotic, we have taken the assistance of the Lyapunov exponent and the bifurcation diagram with respect to the fractional derivative. To perform numerical simulation, we have identified certain values of the parameters where the system exhibits chaotic behavior. Then, the theoretical claims about the influence of the controller on the system are established with the help of numerical simulations.