On the multi-cluster flocking of the fractional Cucker–Smale model

Hyunjin Ahn; Hyunjin Ahn

doi:10.3934/mine.2024024

Mathematics in Engineering

2024, Volume 6, Issue 4: 607-647. doi: 10.3934/mine.2024024

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Research article

On the multi-cluster flocking of the fractional Cucker–Smale model

Hyunjin Ahn ^,

Department of Data Technology, Myongji University, Seoul 03674, Korea

Received: 26 January 2024 Revised: 02 July 2024 Accepted: 14 August 2024 Published: 19 August 2024

This paper demonstrates several sufficient frameworks for the multi-cluster flocking behavior of the fractional Cucker–Smale (CS) model. For this, we first employ the Caputo fractional derivative instead of the usual derivative to propose the fractional CS model with the memory effect. Then, using mathematical tools based on fractional calculus, we present suitable sufficient conditions in terms of properly separated initial data close to the multi-cluster, and well-prepared system parameters for the multi-cluster flocking of the fractional system to emerge. Finally, we offer several numerical simulations and compare them with the analytical results.

Keywords:

Citation: Hyunjin Ahn. On the multi-cluster flocking of the fractional Cucker–Smale model[J]. Mathematics in Engineering, 2024, 6(4): 607-647. doi: 10.3934/mine.2024024

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The short communication “Graphene derivatives potentiate the activity of antibiotics against Enterococcus faecium, Klebsiella pneumoniae and E. coli” written by Butler and colleagues is an important contribution to the fight against the alarming increase in the number of key priority pathogens developing resistance against antibiotics. The antimicrobial effect of graphene, graphite and graphene oxide is well-known. In this article, the combination of these compounds with three clinically relevant antibiotics displaying different modes of activity, is described. This strategy might constitute a new approach to prevent resistant bacterial infections.

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The green microalga Chlorella vulgaris is a popular food additive due to its nutrient-content. In addition, it can be used in the cosmetics industry or to feed aquaculture. In their contribution “Improved growth and harvesting of microalgae Chlorella vulgaris on textile fabrics as 2.5 D substrates”, Brockhagen and co-workers investigated novel methods to cultivate the microalga on different natural and non-natural fabrics.

Last but not least, Tsivileva and colleagues report on the effect of different synthetic acridones on the mushroom Lentinula edodes (shiitake). Apart from their technical use as dyes, many natural-occurring acridones were reported to display anti-inflammatory or anti-cancer effects. Thus, possible unfavorable ecological effects of these compounds must be clarified. The authors analyzed several metabolic substances in the fungus that could be attributed to the presence of the particular acridone.

Taken together, the reader gets new insights into the behavior of small organisms in unusual environments during the visit of this special zoo.

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