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.
Understanding the mechanisms that maintain biodiversity remains a persistent challenge, and various methodologies are used to elucidate this issue. Typically, approaches focus on pairwise interactions, in which two species interact directly. However, this approach has limitations for elucidating the complex characteristics of ecosystems or for understanding interactions among multiple organisms; consequently, methods that incorporate higher-order interactions are being proposed. In particular, the intervention of neutral species has been reported to affect biodiversity in spatial systems of the rock-paper-scissors (RPS) game, a representative evolutionary game model used to describe ecosystems. As the number of neutral species that do not participate in actual competition increases, the natural birth and death of populations and their impact on existing ecosystems are obscured. We explore the effect of neutral species, which do not directly compete with existing populations, on biodiversity, considering the fundamental characteristics of individual birth and death. The presence of neutral species has been shown to have both positive and negative effects on ecosystems, and, unlike previous RPS systems, to have the potential to sustain biodiversity. Through our work, we can reconfirm the critical role of neutral species in conserving biodiversity.
The mechanisms by which biodiversity is maintained have persistently challenged and perplexed academics in the ecological and biological sciences. 1,2 To address such a complex issue, scientists have usually focused on pairwise interactions, in which two species interact directly. [3][4][5][6][7][8] Subsequently, they proposed and investigated the effects of higher-order interactions (HOIs) on biodiversity. Higher-order interactions occur when the interaction between two species is influenced by additional species or factors. 9 Over the past decades, extensive research has focused on these higher-order interactions and their effects on the maintenance of biodiversity. [10][11][12][13] In studies on biodiversity, the rock-paper-scissors game (RPS) has been a fundamental tool for examining nonhierarchical, cyclic competitive interactions, elucidating diverse mechanisms in ecosystems, including mating strategies among three side-blotched lizard populations in California 14 and the evolution of microbial populations. [15][16][17] As the study in theoretical frameworks is identified to support experimental studies that highlight the importance of mobility for biodiversity, 4 studies of cyclic competition have been extensively examined by considering various factors that affect species biodiversity from the perspective of pairwise interactions, e.g., the double-edged sword effect of cooperation on promoting coexistence, 18 species characteristics, 19,20 anthropogenic influences and the consideration of a wild refuge, 21 and the beneficial role of the weakest species in promoting coexistence. 22,23 Even if previous studies have focused on pairwise interactions, higher-order interactions have recently gained attention and have become one of the interesting issues in this field. [23][24][25][26][27] To be concrete, the authors in Ref. [25] introduced a novel framework for understanding how higher-order interactions shape the evolution of social phenotypes. Their findings indicate that a perturbed interaction network can establish coexistence equilibria among competing species, highlighting an inverse relationship between the ratio of unperturbed to perturbed species and the magnitude of perturbations. In other scenarios involving higher-order interactions with neutral species, the increase in mobility can promote species coexistence. 24 In addition, following the work, the intensity of competition among different species on spatially embedded hyper-lattices has been shown to promote species coexistence through higher-order forms of competition. 27 As the importance of higher-order interactions increases for elucidating complex phenomena in ecosystems, it is becoming necessary to consider them. In addition, within the framework of HOIs, the significant role of neutral species has been revealed as one of the beneficial factors in promoting biodiversity and maintaining coexistence [28][29][30] . Relevant works indicate that third-party species, which can be considered neutral in real ecosystems, are able to maintain system stability by modulating competitive effects. For example, the presence of a third-party tree can mitigate the direct competitive effects of neighboring trees on focal trees, thereby sustaining forest species diversity 31 . For species survival, especially coexistence, previous studies have shown that sufficient space (also regarded as carrying capacity) is required. 32 In spatial RPS systems, such spaces are generated by predation under cyclic competition, in the classic manner, and additional interactions, e.g., intraspecific competition, can substantially
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