The Rock-Paper-Scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three state Lotka-Volterra or the four state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. With the objective of reducing the impact of the use of a discrete lattice, in this paper we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum euclidean radius $R$. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three and four state formulations.
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。
Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model
物种的循环优势已被确定为维持生物多样性的一种潜在机制，见例如B。Kerr，M. a.莱利，M。W. Feldman和B. J. M. Bohannan [Nature {\bf 418}，171（2002）]和B. Kirkup和M. a. Riley [Nature {\bf 428}，412（2004）].通过数值模拟支持的分析方法，我们解决了这个问题，通过研究一个典型的非空间三种群随机系统，即"石头剪刀布"或循环洛特卡-沃尔泰拉模型的属性。虽然确定性的方法（速率方程）预测的物种的共存导致定期（但中性稳定）振荡的人口密度，我们表明，在系统中产生的波动与\emph{finite number of agents}大大改变了这一局面，并负责灭绝：经过足够长的时间后，三个物种中的两个灭绝。作为主要发现，我们提供了在给定时间的灭绝概率的解析估计和数值计算。我们还讨论了我们的研究结果的影响，一类广泛的竞争人口系统。
Topologically robust zero-sum games and Pfaffian orientation -- How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra equation
