The Stability of Two-Community Replicator Dynamics with Discrete Multi-Delays

• Main Authors: Jinxiu Pi, Hui Yang, Yadong Shu, Chongyi Zhong, Guanghui Yang
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Mathematics
• Subjects: replicator dynamics; asymptotical stability; two-community; discrete multi-delays
• Online Access: https://www.mdpi.com/2227-7390/8/12/2120

This article investigates the stability of evolutionarily stable strategy in replicator dynamics of two-community with multi-delays. In the real environment, players interact simultaneously while the return of their choices may not be observed immediately, which implies one or more time-delays exists. In addition to using the method of classic characteristic equations, we also apply linear matrix inequality (i.e., LMI) to discuss the stability of the mixed evolutionarily stable strategy in replicator dynamics of two-community with multi-delays. We derive a delay-dependent stability and a delay-independent stability sufficient conditions of the evolutionarily stable strategy in the two-community replicator dynamics with two delays, and manage to extend the sufficient condition to n time delays. Lastly, numerical trials of the Hawk–Dove game are given to verify the effectiveness of the theoretical consequences.