Cooperative investments in social dilemmas can spontaneously diversify into stably coexisting high and low contributors in well-mixed populations. Here we extend the analysis to emerging diversity in (spatially) structured populations. Using pair approximation, we derive analytical expressions for the invasion fitness of rare mutants in structured populations, which then yields a spatial adaptive dynamics framework. This allows us to predict changes arising from population structures in terms of existence and location of singular strategies, as well as their convergence and evolutionary stability as compared to well-mixed populations. Based on spatial adaptive dynamics and extensive individual-based simulations, we find that spatial structure has significant and varied impacts on evolutionary diversification in continuous social dilemmas. More specifically, spatial adaptive dynamics suggests that spontaneous diversification through evolutionary branching is suppressed, but simulations show that spatial dimensions offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Even though spatial adaptive dynamics is unable to capture these new modes, they can still be understood based on an invasion analysis. In particular, population structures alter invasion fitness and can open up new regions in trait space where mutants can invade, but that may not be accessible to small mutational steps. Instead, stochastically appearing larger mutations or sequences of smaller mutations in a particular direction are required to bridge regions of unfavorable traits. The net effect is that spatial structure tends to promote diversification, especially when selection is strong.

对社会困境的合作投资可以自发地多样化，在混合良好的人群中稳定共存的高贡献者和低贡献者。在这里，我们将分析扩展到（空间）结构人群中新兴的多样性。使用配对近似，我们推导了结构化群体中稀有突变体的入侵适应性的解析表达式，从而产生了空间自适应动力学框架。这使我们能够根据单一策略的存在和位置，以及与混合良好的种群相比的收敛性和进化稳定性来预测种群结构引起的变化。基于空间适应性动力学和广泛的基于个体的模拟，我们发现空间结构对持续社会困境中的进化多样化具有显着且不同的影响。更具体地说，空间适应性动力学表明，通过进化分支的自发多样化受到抑制，但模拟表明，空间维度提供了新的多样化模式，这些模式是由有限大小的突变和种群结构的相互作用驱动的。尽管空间自适应动力学无法捕获这些新模式，但仍然可以根据入侵分析来理解它们。特别是，种群结构改变了入侵适应性，并可以在性状空间中开辟突变体可以入侵的新区域，但这可能无法通过小的突变步骤进入。相反，需要在特定方向上随机出现较大的突变或较小的突变序列来桥接不利性状的区域。最终的效果是，空间结构往往会促进多样化，尤其是在选择很强的情况下。