Sui Xiukai, Wu Bin, Wang Long
Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China.
School of Science, Beijing University of Posts and Communications, Beijing 100876, China.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Dec;92(6):062124. doi: 10.1103/PhysRevE.92.062124. Epub 2015 Dec 14.
The likelihood that a mutant fixates in the wild population, i.e., fixation probability, has been intensively studied in evolutionary game theory, where individuals' fitness is frequency dependent. However, it is of limited interest when it takes long to take over. Thus the speed of evolution becomes an important issue. In general, it is still unclear how fixation times are affected by the population structure, although the fixation times have already been addressed in the well-mixed populations. Here we theoretically address this issue by pair approximation and diffusion approximation on regular graphs. It is shown (i) that under neutral selection, both unconditional and conditional fixation time are shortened by increasing the number of neighbors; (ii) that under weak selection, for the simplified prisoner's dilemma game, if benefit-to-cost ratio exceeds the degree of the graph, then the unconditional fixation time of a single cooperator is slower than that in the neutral case; and (iii) that under weak selection, for the conditional fixation time, limited neighbor size dilutes the counterintuitive stochastic slowdown which was found in well-mixed populations. Interestingly, we find that all of our results can be interpreted as that in the well-mixed population with a transformed payoff matrix. This interpretation is also valid for both death-birth and birth-death processes on graphs. This interpretation bridges the fixation time in the structured population and that in the well-mixed population. Thus it opens the avenue to investigate the challenging fixation time in structured populations by the known results in well-mixed populations.
在进化博弈论中，突变体在野生种群中固定的可能性，即固定概率，已得到深入研究，其中个体的适应度取决于频率。然而，当突变体需要很长时间才能占据主导时，其研究意义有限。因此，进化速度成为一个重要问题。一般来说，尽管在均匀混合种群中已经讨论过固定时间，但固定时间如何受到种群结构的影响仍不清楚。在这里，我们通过对规则图进行配对近似和扩散近似，从理论上解决了这个问题。结果表明：（i）在中性选择下，无条件和有条件的固定时间都会随着邻居数量的增加而缩短；（ii）在弱选择下，对于简化的囚徒困境博弈，如果收益成本比超过图的度数，那么单个合作者的无条件固定时间比中性情况下要慢；（iii）在弱选择下，对于有条件的固定时间，有限的邻居规模会减弱在均匀混合种群中发现的违反直觉的随机减速现象。有趣的是，我们发现我们所有的结果都可以解释为在具有变换后的收益矩阵的均匀混合种群中的情况。这种解释对于图上的生死过程和出生死亡过程也同样有效。这种解释架起了结构化种群中的固定时间与均匀混合种群中的固定时间之间的桥梁。因此，它为通过均匀混合种群中的已知结果来研究结构化种群中具有挑战性的固定时间开辟了道路。
