We study the phenomenon of cyclic dominance in the paradigmatic Rock--Paper--Scissors model, as occurring in both stochastic individual-based models of finite populations and in the deterministic replicator equations. The mean-field replicator equations are valid in the limit of large populations and, in the presence of mutation and unbalanced payoffs, they exhibit an attracting limit cycle. The period of this cycle depends on the rate of mutation; specifically, the period grows logarithmically as the mutation rate tends to zero. We find that this behaviour is not reproduced in stochastic simulations with a fixed finite population size. Instead, demographic noise present in the individual-based model dramatically slows down the progress of the limit cycle, with the typical period growing as the reciprocal of the mutation rate. Here we develop a theory that explains these scaling regimes and delineates them in terms of population size and mutation rate. We identify a further intermediate regime in which we construct a stochastic differential equation model describing the transition between stochastically-dominated and mean-field behaviour.
Heteroclinic cycling and extinction in May-Leonard models with demographic stochasticity
May和Leonard（SIAM J. Appl. Math 1975）提出了一个具有异宿循环的三种群Lotka-Volterra型种群模型。而不是产生一个周期性的极限环，轨迹需要越来越长的时间来完成每个“周期”，越来越接近不稳定的固定点，其中一个人口占主导地位，其他人接近零。非周期异宿动态随后在生态系统中进行了研究（侧斑蜥蜴;大肠杆菌E。大肠杆菌）、免疫系统、神经信息处理模型（"无赢家竞争"）和神经中枢模式发生器的模型中。然而，正如梅和伦纳德所观察到的那样："从生物学上讲，（由模型产生的）行为是毫无意义的。一旦承认变量代表动物，因此不能低于1，很明显，系统将在几个周期后收敛于某个单一种群，消灭其他两个种群。"在这里，我们探索不同的方式引入离散随机动力学的基础上，五月和伦纳德的ODE模型，应用生态种群动力学，和神经运动中枢模式发生器系统。我们研究了几个数量上不同的渐近行为的例子，包括所有物种的灭绝，灭绝到一个单一的物种，和持久的周期优势有限的平均周期长度。
