Veller Carl, Rajpaul Vinesh
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Oct;86(4 Pt 1):041907. doi: 10.1103/PhysRevE.86.041907. Epub 2012 Oct 10.
We introduce and analyze a purely competitive dynamics for the evolution of an infinite population subject to a three-strategy game. We argue that this dynamics represents a characterization of how certain systems, both natural and artificial, are governed. In each period, the population is randomly sorted into pairs, which engage in a once-off play of the game; the probability that a member propagates its type to its offspring is proportional only to its payoff within the pair. We show that if a type is dominant (obtains higher payoffs in games with both other types), its "pure" population state, comprising only members of that type, is globally attracting. If there is no dominant type, there is an unstable "mixed" fixed point; the population state eventually oscillates between the three near-pure states. We then allow for mutations, where offspring have a nonzero probability of randomly changing their type. In this case, the existence of a dominant type renders a point near its pure state globally attracting. If no dominant type exists, a supercritical Hopf bifurcation occurs at the unique mixed fixed point, and above a critical (typically low) mutation rate, this fixed point becomes globally attracting: the implication is that even very low mutation rates can stabilize a system that would, in the absence of mutations, be unstable.
我们引入并分析了一种适用于受三策略博弈影响的无限种群演化的纯竞争动力学。我们认为这种动力学代表了某些自然和人工系统运行方式的一种特征描述。在每个时期，种群被随机配对，进行一次性博弈；成员将其类型遗传给后代的概率仅与其在配对中的收益成正比。我们表明，如果一种类型占主导地位（在与其他两种类型的博弈中获得更高收益），其仅由该类型成员组成的“纯”种群状态是全局吸引的。如果不存在占主导地位的类型，则存在一个不稳定的“混合”不动点；种群状态最终会在三种近纯状态之间振荡。然后我们考虑突变情况，即后代有非零概率随机改变其类型。在这种情况下，占主导地位类型的存在使得其纯状态附近的一个点是全局吸引的。如果不存在占主导地位的类型，在唯一的混合不动点处会发生超临界霍普夫分岔，并且在高于某个临界（通常较低）突变率时，这个不动点会变为全局吸引：这意味着即使非常低的突变率也能使一个在没有突变时不稳定的系统稳定下来。
