Stochastic evolutionary public goods game with first and second order costly punishments in finite populations<xref rid="cpb_27_6_060203_fn1" ref-type="fn">*</xref> <fn id="cpb_27_6_060203_fn1"> <label>*</label> <p>Project supported by the National Natural Science Foundation of China (Grant Nos. 71501149 and 71231007), the Soft Science Project of Hubei Province, China (Grant No. 2017ADC122), and the Fundamental Research Funds for the Central Universities, China (Grant No. WUT: 2017VI070).</p> </fn>

Stochastic evolutionary public goods game with first and second order costly punishments in finite populations* *

Project supported by the National Natural Science Foundation of China (Grant Nos. 71501149 and 71231007), the Soft Science Project of Hubei Province, China (Grant No. 2017ADC122), and the Fundamental Research Funds for the Central Universities, China (Grant No. WUT: 2017VI070).

Quan Ji^{1, †}, Chu Yu-Qing^{2, ‡}, Liu Wei³, Wang Xian-Jia⁴, Yang Xiu-Kang¹

Ranges of (β, r) for the system reaching the three different kinds of stable states for fixed α₁ = 1 when α₂ = 0, 0.5, 0.8 respectively.