Abstract The average abundance function reflects the level of cooperation in the population. So it is important to analyze how to increase the average abundance function in order to facilitate the proliferation of cooperative behavior. The characteristics of average abundance function based on multi-player threshold public goods evolutionary game model under redistribution mechanism have been explored by analytical analysis and numerical simulation in this article. The main research findings contain four aspects. Firstly, we deduce the concrete expression of expected payoff function. In addition, we obtain the intuitive expression of average abundance function by taking the detailed balance condition as the point of penetration. Secondly, we obtain the approximate expression of average abundance function when selection intensity is sufficient small. In this case, average abundance function can be simplified from composite function to linear function. In addition, this conclusion will play a significant role when analyzing the results of the numerical simulation. Thridly, we deduce the approximate expression of average abundance function when selection intensity is large enough. Because of this approximation expression, the range of summation will be reduced, the number of operations for average abundance function will be reduced, and the operating efficiency for numerical simulation will be improved. Fourthly, we explore the influences of parameters (the size of group d, multiplication factor r, cost c, aspiration level α and the proportion of income redistribution τ) on the average abundance function through numerical simulation. Also the corresponding results have been explained based on the expected payoff function and function h(i, ω). It can be concluded that when selection intensity ω is small, the effects of parameters (d, r, c, α and τ) on average abundance function is slight. When selection intensity ω is large, there will be five conditions. (1) Average abundance function will decrease with d regardless of whether threshold m is small or large. (2) Average abundance function will decrease at first and then increase with r when threshold m is small. Average abundance function will increase with r when threshold m is large. (3) Average abundance function will basically remain unchanged with c regardless of whether threshold m is small or large. (4) Average abundance function will remain stable at first and then increase with α when threshold m is small. Average abundance function will remain stable at first and then decrease with α when threshold m is large. It should be noted that average abundance function will get close to 1/2 when α is large enough. (5) Average abundance function will increase with τ regardless of whether threshold m is small or large.

中文翻译：

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再分配机制下多人门槛公共物品演化博弈模型的平均丰度函数特征

摘要 平均丰度函数反映了种群的合作程度。因此，分析如何增加平均丰度函数以促进合作行为的扩散很重要。本文通过解析分析和数值模拟，探讨了再分配机制下基于多参与者阈值公共物品演化博弈模型的平均丰度函数的特征。主要研究成果包括四个方面。首先推导出期望收益函数的具体表达式。此外，我们以详细的平衡条件为切入点，得到了平均丰度函数的直观表达。第二，we obtain the approximate expression of average abundance function when selection intensity is sufficient small. 在这种情况下，平均丰度函数可以从复合函数简化为线性函数。此外，该结论在分析数值模拟结果时也将发挥重要作用。Thridly, we deduce the approximate expression of average abundance function when selection intensity is large enough. 由于这个近似表达式，求和的范围会缩小，平均丰度函数的运算次数会减少，数值模拟的运算效率会提高。第四，我们探讨了参数（组 d 的大小、乘法因子 r、成本 c、通过数值模拟对平均丰度函数的期望水平 α 和收入再分配比例 τ)。相应的结果也基于预期收益函数和函数 h(i, ω) 进行了解释。It can be concluded that when selection intensity ω is small, the effects of parameters (d, r, c, α and τ) on average abundance function is slight. When selection intensity ω is large, there will be five conditions. (1) 无论阈值 m 是小还是大，平均丰度函数都会随 d 减小。(2)当阈值m较小时，平均丰度函数随r先减小后增大。当阈值 m 较大时，平均丰度函数将随 r 增加。(3) 无论阈值m是大是小，平均丰度函数与c基本保持不变。(4) 当阈值m较小时，平均丰度函数先保持稳定，然后随α增加。当阈值 m 较大时，平均丰度函数将先保持稳定，然后随 α 减小。需要注意的是，当 α 足够大时，平均丰度函数将接近 1/2。(5) 无论阈值 m 是小还是大，平均丰度函数都会随着 τ 增加。需要注意的是，当 α 足够大时，平均丰度函数将接近 1/2。(5) 无论阈值 m 是小还是大，平均丰度函数都会随着 τ 的增加而增加。需要注意的是，当 α 足够大时，平均丰度函数将接近 1/2。(5) 无论阈值 m 是小还是大，平均丰度函数都会随着 τ 增加。