Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada.
Proc Biol Sci. 2012 Nov 22;279(1747):4596-603. doi: 10.1098/rspb.2012.1408. Epub 2012 Sep 12.
We study the evolution of a pair of competing behavioural alleles in a structured population when there are non-additive or 'synergistic' fitness effects. Under a form of weak selection and with a simple symmetry condition between a pair of competing alleles, Tarnita et al. provide a surprisingly simple condition for one allele to dominate the other. Their condition can be obtained from an analysis of a corresponding simpler model in which fitness effects are additive. Their result uses an average measure of selective advantage where the average is taken over the long-term--that is, over all possible allele frequencies--and this precludes consideration of any frequency dependence the allelic fitness might exhibit. However, in a considerable body of work with non-additive fitness effects--for example, hawk-dove and prisoner's dilemma games--frequency dependence plays an essential role in the establishment of conditions for a stable allele-frequency equilibrium. Here, we present a frequency-dependent generalization of their result that provides an expression for allelic fitness at any given allele frequency p. We use an inclusive fitness approach and provide two examples for an infinite structured population. We illustrate our results with an analysis of the hawk-dove game.
我们研究了在具有非加性或“协同”适合度效应的结构种群中，一对竞争行为等位基因的进化。在弱选择的一种形式下，并且在一对竞争等位基因之间具有简单的对称性条件下，Tarnita 等人提供了一个令人惊讶的简单条件，即一个等位基因可以支配另一个等位基因。他们的条件可以从对具有加性适合度效应的相应更简单模型的分析中获得。他们的结果使用了选择性优势的平均度量，其中平均值是在长期内（即在所有可能的等位基因频率上）取的，这排除了对等位基因适合度可能表现出的任何频率依赖性的考虑。然而，在具有非加性适合度效应的大量工作中 - 例如鹰鸽和囚徒困境博弈 - 频率依赖性在建立稳定的等位基因频率平衡的条件中起着至关重要的作用。在这里，我们提出了他们的结果的频率依赖性推广，该推广提供了在任何给定等位基因频率 p 下的等位基因适合度的表达式。我们使用适合度综合方法，并为无限结构种群提供了两个示例。我们通过对鹰鸽博弈的分析来说明我们的结果。
