Neutral species facilitate coexistence among cyclically competing species under birth and death processes
School of Statistics and Mathematics, Yunnan University of Finance and Economics（云南财经大学统计与数学学院）；School of Mathematics and Statistics, Xidian University（西安电子科技大学数学与统计学院）；Department of Applied Mathematics, Colleage of Applied Sciences, Kyung Hee University（韩国庆熙大学应用科学学院应用数学系）；G-LAMP NEXUS Institute, Kyung Hee University（韩国庆熙大学G-LAMP NEXUS研究所）
Natural birth and death are fundamental mechanisms of population dynamics in ecosystems and have played pivotal roles in shaping population dynamics. Nevertheless, in studies of cyclic competition systems governed by the rock-paper-scissors (RPS) game, these mechanisms have often been ignored in analyses of biodiversity. On the other hand, given the prevalence and profound impact on biodiversity, understanding how higher-order interactions (HOIs) can affect biodiversity is one of the most challenging issues, and thus HOIs have been continuously studied for their effects on biodiversity in systems of cyclic competing populations, with a focus on neutral species. However, in real ecosystems, species can evolve and die naturally or be preyed upon by predators, whereas previous studies have considered only classic reaction rules among three species with a neutral, nonparticipant species. To identify how neutral species can affect the biodiversity of the RPS system when species' natural birth and death are assumed, we consider a model of neutral species in higher-order interactions within the spatial RPS system, assuming birth-and-death processes. Extensive simulations show that when neutral species interfere positively, they dominate the available space, thereby reducing the proportion of other species. Conversely, when the interference is harmful, the density of competing species increases. In addition, unlike traditional RPS dynamics, biodiversity can be effectively maintained even in high-mobility regimes. Our study reaffirms the critical role of neutral species in preserving biodiversity.
Exact fixation probabilities for the Birth-Death and Death-Birth frequency-dependent Moran processes on the star graph
Broom and Rychtásness [Proc. R. A（2008）464，2609- 2627]发现了结构化群体的Moran过程的固定概率的精确解，其中个体之间的相互作用结构由所谓的星形图给出，即一个中心顶点和$n$叶，叶仅连接到中心。我们通过允许个体的适应度依赖于种群频率，以及允许繁殖和死亡抽签顺序的可能变化来推广他们的解决方案。在他们引用的论文中，Broom和Rychtásniness考虑了出生-死亡（BD）过程，其中在每个时间步骤中，一个个体首先被吸引进行繁殖，然后一个个体被选择死亡。在死亡-出生（DB）过程中，抽签的顺序是颠倒的。可以看出，抽签的顺序在注视概率上有很大的不同。我们的解决方案方法适用于BD和DB的情况。正如预期的那样，固定概率的精确公式是复杂的。我们还将通过一些示例来说明它们，并提供当图中的叶子数量$n$趋于无穷大时固定概率的渐进行为的结果。
