Neutral species facilitate coexistence among cyclically competing species under birth and death processes

TL;DR

This paper investigates how a neutral third species and higher-order interactions influence biodiversity in a spatial rock-paper-scissors system under birth–death processes. It introduces a four-species lattice model where neutral $D$ occupies empty sites and modulates interspecific competition via a density-dependent parameter $K$ with $p=\exp(K \rho_i)$ and mobility $\varepsilon = 2 M N$, and executes Monte Carlo simulations on an $N\times N$ grid. Key findings show that positive $K$ can sustain coexistence by maintaining spatial patterns and total $A,B,C$ density, while negative $K$ allows $D$ to dominate and reshapes patterns, with high mobility still supporting biodiversity under suitable HOIs; and under high reproduction rates with asymmetric mortality, spiral waves can emerge and stabilize coexistence. Together, these results highlight the role of neutral species and HOIs in biodiversity maintenance and offer theoretical insight into how external perturbations and mobility influence coexistence in cyclic competition.

Abstract

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.

Figures (12)

• Figure 1: Species $A$, $B$ and $C$ exhibit cyclic predation following RPS rule, as indicated by the solid black arrows. Higher-order interactions induced by the neutral species $D$ significantly modulate the predation rates among the competing species, as illustrated by the dashed lines.
• Figure 2: Characteristic snapshots under different parameter conditions of $(M, K)$: (a) $(M, K)=(10^{-4}, -3)$, (b) $(10^{-4}, 3)$, (c) $(10^{-3}, -3)$, and $(10^{-3}, 3)$. Colors in each panel, green, blue, red, yellow, and white, represent species $A$, $B$, $C$, $D$, and $\varnothing$, respectively. As the sensitivity $K$ of species $D$ increases, the total density of species $A$, $B$, and $C$ increases.
• Figure 3: Synergistic effects of $K$ and $M$ on the change in the density of (a) the neutral species $D$, (b) the total density of the three species excluding $D$, and (c) the density of empty sites. (a) The density of species $D$ decreases with increasing $K$ and reaches a steady state regardless of $M$. (b) On the other hand, the total density of the three species can increase as $K$ increases overall. Notably, as the proportion of empty sites reaches a peak, as shown in (c), the densities decline at specific values of $K$ and $M$: $K = \pm 1$ for $M = 10^{-5}$. The shaded region denotes the 95% confidence interval estimated from 100 independent simulations. For each $M$, this decline can be relaxed and reversed, leading to an increase. (c) Overall, the change in empty sites shows the opposite pattern from the density of total species.
• Figure 4: The extinction probability $P_{\textrm{ext}}$ as a function of: (a) $M$ for various values of $K$ and (b) $K$ for different values of $M$. In the context of higher-order interactions, increased mobility promotes coexistence, and a higher value of $K$ further enhances species coexistence.
• Figure 5: Characteristic snapshots of the system at high reproduction rates for different combinations of death rates for the competitive and neutral species: $(f_1, f_2)$: (a) $(1,1)$, (b) $(1,0.5)$, (c) $(0.8,1)$, and (d) $(0.8,0.5)$. The color scheme for each species is consistent with that in Fig. \ref{['fig1']}. Differences in mortality between the competitive and neutral species give rise to either spiral-wave patterns or extinction of the competitive species.