Flocking and invariance of velocity angles

Le  Li; Lihong  Huang; Jianhong Wu

doi:10.3934/mbe.2015007

Article Contents

2016, Volume 13, Issue 2: 369-380. Doi: 10.3934/mbe.2015007

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Flocking and invariance of velocity angles

1.
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082
2.
College of Mathematics and Econometrics, Hunan University & Hunan Women's University, Changsha, Hunan, 410004
3.
Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3

Received: October 31, 2014

Revised: April 30, 2015

Published: December 2015

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Abstract

Motsch and Tadmor considered an extended Cucker-Smale model to investigate the flocking behavior of self-organized systems of interacting species. In this extended model, a cone of the vision was introduced so that outside the cone the influence of one agent on the other is lost and hence the corresponding influence function takes the value zero. This creates a problem to apply the Motsch-Tadmor and Cucker-Smale method to prove the flocking property of the system. Here, we examine the variation of the velocity angles between two arbitrary agents, and obtain a monotonicity property for the maximum cone of velocity angles. This monotonicity permits us to utilize existing arguments to show the flocking property of the system under consideration, when the initial velocity angles satisfy some minor technical constraints.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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