We derive how directional and disruptive selection operate on scalar traits in a heterogeneous group-structured population for a general class of models. In particular, we assume that each group in the population can be in one of a finite number of states, where states can affect group size and/or other environmental variables, at a given time. Using up to second-order perturbation expansions of the invasion fitness of a mutant allele, we derive expressions for the directional and disruptive selection coefficients, which are sufficient to classify the singular strategies of adaptive dynamics. These expressions include first- and second-order perturbations of individual fitness (expected number of settled offspring produced by an individual, possibly including self through survival); the first-order perturbation of the stationary distribution of mutants (derived here explicitly for the first time); the first-order perturbation of pairwise relatedness; and reproductive values, pairwise and three-way relatedness, and stationary distribution of mutants, each evaluated under neutrality. We introduce the concept of individual k-fitness (defined as the expected number of settled offspring of an individual for which $k-1$ randomly chosen neighbors are lineage members) and show its usefulness for calculating relatedness and its perturbation. We then demonstrate that the directional and disruptive selection coefficients can be expressed in terms individual k-fitnesses with $k=1,2,3$ only. This representation has two important benefits. First, it allows for a significant reduction in the dimensions of the system of equations describing the mutant dynamics that needs to be solved to evaluate explicitly the two selection coefficients. Second, it leads to a biologically meaningful interpretation of their components. As an application of our methodology, we analyze directional and disruptive selection in a lottery model with either hard or soft selection and show that many previous results about selection in group-structured populations can be reproduced as special cases of our model.

我们推导了针对通用模型模型的异质群体结构人群中标量性状的定向选择和破坏选择。尤其是，我们假设总体中的每个组都可以处于有限数量的州之一，在给定的时间，州可以影响组的大小和/或其他环境变量。使用突变等位基因入侵适应性的最高二阶扰动展开，我们导出了方向性和破坏性选择系数的表达式，这些表达式足以对自适应动力学的奇异策略进行分类。这些表达包括个体适应度的一阶和二阶扰动（个体产生的定居后代的预期数量，可能包括通过生存获得的自我）；突变体平稳分布的一阶扰动（首次在此明确推导）；成对相关性的一阶扰动；生殖价值，成对和三向相关性以及突变体的平稳分布，每个都在中性下评估。我们介绍个人的概念k-适应度（定义为一个个体的预期后代数量）$ķ--\mathrm{1个}$随机选择的邻居是沿袭成员），并显示出其在计算相关性和扰动方面的有用性。然后，我们证明了方向性的和破坏性的选择系数可以在各个方面来表达ķ与-fitnesses$ķ=\mathrm{1个}，2，3$只要。这种表示有两个重要的好处。首先，它允许显着减少描述突变动力学的方程组的维数，为明确评估两个选择系数需要解决这些动力学问题。其次，它导致对其成分的生物学意义的解释。作为我们方法学的一种应用，我们在具有硬性选择或软性选择的彩票模型中分析了方向性和破坏性选择，并表明许多先前关于群体结构人群中选择的结果都可以作为我们模型的特例再现。