Public goods games (PGGs) describe situations in which individuals contribute to a good at a private cost, but others can free-ride by receiving a share of the public benefit at no cost. The game occurs within local neighbourhoods, which are subsets of the whole population. Free-riding and maximal production are two extremes of a continuous spectrum of traits. We study the adaptive dynamics of production and neighbourhood size. We allow the public good production and the neighbourhood size to coevolve and observe evolutionary branching. We explain how an initially monomorphic population undergoes evolutionary branching in two dimensions to become a dimorphic population characterized by extremes of the spectrum of trait values. We find that population size plays a crucial role in determining the final state of the population. Small populations may not branch or may be subject to extinction of a subpopulation after branching. In small populations, stochastic effects become important and we calculate the probability of subpopulation extinction. Our work elucidates the evolutionary origins of heterogeneity in local PGGs among individuals of two traits (production and neighbourhood size), and the effects of stochasticity in two-dimensional trait space, where novel effects emerge.

公共物品游戏（PGG）描述了个人以私人成本为商品做出贡献的情况，而其他人则可以免费获得公共利益的一部分来搭便车。博弈发生在当地居民区中，当地居民区是整个人口的子集。搭便车和最大化产量是连续性状的两个极端。我们研究生产和邻里规模的适应性动态。我们允许公共产品生产和邻里规模共同发展并观察进化分支。我们解释了最初的单态种群如何在两个维度上经历进化分支，从而变成以特征值频谱的极端特征为特征的双态种群。我们发现，人口规模在确定人口的最终状态方面起着至关重要的作用。少数人群可能不会分支或可能在分支后灭绝亚种群。在少数人群中，随机效应变得很重要，我们计算了亚群灭绝的可能性。我们的工作阐明了具有两个特征（生产和邻域大小）的个体之间局部PGGs异质性的进化起源，以及在出现新效应的二维特征空间中随机性的影响。