Long-term evolution of quantitative traits is classically and usefully described as the directional change in phenotype due to the recurrent fixation of new mutations. A formal justification for such continual evolution ultimately relies on the "invasion implies substitution"-principle. Here, whenever a mutant allele causing a small phenotypic change can successfully invade a population, the ancestral (or wild-type) allele will be replaced, whereby fostering gradual phenotypic change if the process is repeated. It has been argued that this principle holds in a broad range of situations, including spatially and demographically structured populations experiencing frequency- and density-dependent selection under demographic and environmental fluctuations. However, prior studies have not been able to account for all aspects of population structure, leaving unsettled the conditions under which the "invasion implies substitution"-principle really holds. In this paper, we start by laying out a program to explore and clarify the generality of the "invasion implies substitution"-principle. Particular focus is given on finding an explicit and functionally constant representation of the selection gradient on a quantitative trait. Using geometric singular perturbation methods, we then show that the "invasion implies substitution"-principle generalizes to well-mixed and scalar-valued polymorphic multispecies ecological communities that are structured into finitely many demographic (or physiological) classes. The selection gradient is shown to be constant over the evolutionary timescale and that it depends only on the resident phenotype, individual growth-rates, population steady states and reproductive values, all of which are calculated from the resident dynamics. Our work contributes to the theoretical foundations of evolutionary ecology.

中文翻译：

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入侵意味着在生态群落中被阶级结构的人口所取代。

定量性状的长期演变被经典地和有用地描述为由于新突变的反复固定而导致的表型方向改变。这种持续发展的正式理由最终取决于“入侵意味着替代”原理。在这里，每当引起小的表型变化的突变等位基因可以成功地入侵种群时，祖先的（或野生型）等位基因将被替换，从而如果重复该过程，则会促进逐渐的表型变化。有人认为，这一原则适用于广泛的情况，包括在人口和环境波动下经历频率和密度依赖选择的空间和人口结构化人口。然而，先前的研究不能解释人口结构的所有方面，而使“入侵意味着替代”的原则真正成立的条件尚未解决。在本文中，我们从设计一个程序开始，以探索和阐明“入侵隐含替代”原理的普遍性。特别关注的是找到定量特征上选择梯度的明确且功能上恒定的表示形式。然后，使用几何奇异摄动法，我们证明“入侵隐含替代”原理可以推广到结构合理且数量有限的人口（或生理）类别中的，具有充分混合和标量值的多态多物种生态群落。显示选择梯度在进化时间尺度上是恒定的，并且仅取决于居民表型，个体增长率，种群稳定状态和生殖值，所有这些都是根据居民动态来计算的。我们的工作为进化生态学的理论基础做出了贡献。