Models of evolution by natural selection often make the simplifying assumption that populations are infinitely large. In this infinite population limit, rare mutations that are selected against always go extinct, whereas in finite populations they can persist and even reach fixation. Nevertheless, for mutations of arbitrarily small phenotypic effect, it is widely believed that in sufficiently large populations, if selection opposes the invasion of rare mutants, then it also opposes their fixation. Here, we identify circumstances under which infinite-population models do or do not accurately predict evolutionary outcomes in large, finite populations. We show that there is no population size above which considering only invasion generally suffices: for any finite population size, there are situations in which selection opposes the invasion of mutations of arbitrarily small effect, but favours their fixation. This is not an unlikely limiting case; it can occur when fitness is a smooth function of the evolving trait, and when the selection process is biologically sensible. Nevertheless, there are circumstances under which opposition of invasion does imply opposition of fixation: in fact, for the \(n\)-player snowdrift game (a common model of cooperation) we identify sufficient conditions under which selection against rare mutants of small effect precludes their fixation—in sufficiently large populations—for any selection process. We also find conditions under which—no matter how large the population—the trait that fixes depends on the selection process, which is important because any particular selection process is only an approximation of reality.

自然选择的进化模型通常会做出简单的假设，即种群是无限大的。在这个无限的种群限制中，被选中的稀有突变总是会灭绝，而在有限的种群中，它们可以持续存在，甚至达到固定。然而，对于任意小的表型效应的突变，人们普遍认为，在足够大的人群中，如果选择反对稀有突变体的入侵，那么它也反对它们的固定。在这里，我们确定了无限种群模型可以或不能准确预测大型有限种群进化结果的情况。我们表明，没有高于其的种群规模，仅考虑入侵通常就足够了：对于任何有限的种群规模，在某些情况下，选择反对任意小效应突变的入侵，但有利于它们的固定。这不是一个不太可能的限制情况；当适应度是进化特征的平滑函数，并且选择过程在生物学上是合理的时，它就会发生。尽管如此，在某些情况下，入侵的反对确实意味着固定的反对：事实上，对于\(n\)-玩家雪堆游戏（一种常见的合作模型）我们确定了充分条件，在这种条件下，针对小效应的稀有突变体的选择会阻止它们在足够大的种群中固定用于任何选择过程。我们还发现了在何种条件下——无论种群有多大——固定的特征取决于选择过程，这很重要，因为任何特定的选择过程只是现实的近似。