Department of Mathematics, University of Vienna, Austria; Vienna Graduate School of Population Genetics, Austria.
Department of Biology, Stanford University, Stanford, CA, USA.
Theor Popul Biol. 2020 Dec;136:12-21. doi: 10.1016/j.tpb.2020.10.001. Epub 2020 Nov 19.
In the evolutionary biology literature, it is generally assumed that for deterministic frequency-independent haploid selection models, no polymorphic equilibrium can be stable in the absence of variation-generating mechanisms such as mutation. However, mathematical analyses that corroborate this claim are scarce and almost always depend upon additional assumptions. Using ideas from game theory, we show that a monomorphism is a global attractor if one of its alleles dominates all other alleles at its locus. Further, we show that no isolated equilibrium exists, at which an unequal number of alleles from two loci is present. Under the assumption of convergence of trajectories to equilibrium points, we resolve the two-locus three-allele case for a fitness scheme formally equivalent to the classical symmetric viability model. We also provide an alternative proof for the two-locus two-allele case.
在进化生物学文献中，一般认为对于确定性的、与频率无关的单倍体选择模型，如果没有产生变异的机制（如突变），则不存在稳定的多态性平衡。然而，支持这一说法的数学分析很少，而且几乎总是依赖于其他假设。我们利用博弈论的思想表明，如果一个等位基因在其位置上优于所有其他等位基因，那么单态性就是一个全局吸引子。此外，我们还表明，在两个位点的等位基因数量不相等的孤立平衡点是不存在的。在轨迹收敛到平衡点的假设下，我们解决了一个适应度方案与经典对称生存力模型形式等效的两个位点三个等位基因的情况。我们还为两个位点两个等位基因的情况提供了另一种证明。
