Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 735248, 13 pages
http://dx.doi.org/10.1155/2011/735248
Research Article

Collective Dynamics of Swarms with a New Attraction/Repulsion Function

Hong Shi1 and Guangming Xie2,3

1Department of Mathematics and Physics, Beijing Institute of Petrochemical Technology, Beijing 102617, China
2State Key Laboratory of Turbulence and Complex Systems, Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China
3School of Electrical and Electronics Engineering, East China Jiaotong University, Nanchang 330013, China

Received 10 March 2011; Accepted 19 June 2011

Academic Editor: Jitao Sun

Copyright © 2011 Hong Shi and Guangming Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We specify an “individual-based” continuous-time model for swarm aggregation in 𝑛 -dimensional Euclidean space. We show that the swarm is completely stable, and the center of the swarm is stationary. Numerical simulations indicate that the individuals will form a stable and cohesive swarm, and under the attraction/repulsion function, the bound of the swarm size will increase as the number of individuals increases.