Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 940465, 21 pages
http://dx.doi.org/10.1155/2012/940465
Research Article

Fixation Probabilities on Complete Star and Bipartite Digraphs

Chunming Zhang,1 Yingjiang Wu,1 Wanping Liu,2 and Xiaofan Yang2

1School of Information Engineering, Guangdong Medical College, Dongguan 523808, China
2College of Computer Science, Chongqing University, Chongqing 400044, China

Received 31 January 2012; Accepted 18 April 2012

Academic Editor: Zuo Nong Zhu

Copyright © 2012 Chunming Zhang et al. 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

This paper exactly formulates the kth-order fixation probabilities on complete star digraphs (CSDs), which extend the results from Broom and Rychtář (2008). By applying these probability formulae, some asymptotic properties on CBDs are analyzed, and certain CSDs are determined to be amplifiers of selection for arbitrary relative fitness larger than 1, while all the CSDs are proved to be amplifiers of selection for fixed relative fitness slightly larger than 1. A numerical method for fixed population structure (by solving a linear system) is developed to calculate the fixation probabilities on complete bipartite digraphs (CBDs), and some conjectures are finally given through simulations.