Abstract
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.
Highlights
Evolutionary Game Theory, introduced by [1] to simulate competitions among animals, is one of the latest game theory developments, and aimed at predicting the population dynamics caused by many local interactions between individuals
Replicator dynamics, which combines evolutionary game theory with differential equations, provides an important theoretical basis to investigate how the spread of advantageous strategies is more probable to happen through imitation than inheritance [3,4]
Based on the previous research, we have found that replicator dynamics of two-community with multi-delays are rarely studied
Summary
Evolutionary Game Theory, introduced by [1] to simulate competitions among animals, is one of the latest game theory developments, and aimed at predicting the population dynamics caused by many local interactions between individuals. In the case of continuous-time delay, the authors in [8] studied the effect of time delays to the ESS in the convergence of evolutionary dynamics. The authors in [14] considered bounded continuously distributed time delay in replicator dynamics Many papers such as [15,16,17] investigated the stability of the Hopf bifurcation and its period solution in replicator dynamics with delays. We aim to discuss whether the stability of the fully mixed Nash equilibrium is affected by discrete multi-delays, and obtain some sufficient conditions that can make the equilibrium asymptotical stability in the replicator dynamics.
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