🧬 biology

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

This study demonstrates that incorporating natural birth and death processes into spatial rock-paper-scissors models reveals how neutral species can either suppress or enhance competing species' densities depending on their interference effects, thereby enabling the effective maintenance of biodiversity even in high-mobility regimes.

Original authors: Yikang Lu, Wenhao She, Xiaofang Duan, Junpyo Park

Published 2026-02-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yikang Lu, Wenhao She, Xiaofang Duan, Junpyo Park

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a bustling city park where three rival gangs—let's call them Rock, Paper, and Scissors—are constantly fighting for territory.

In the classic version of this story (known in science as the "Rock-Paper-Scissors" game), these gangs chase each other in a circle. Rock beats Scissors, Scissors beats Paper, and Paper beats Rock. Usually, if they move around too fast, they accidentally run into each other too often, and the whole system collapses into chaos or one gang takes over completely. But if they move just right, they form beautiful, swirling patterns that keep everyone alive.

Now, imagine a fourth character enters the park: The Neutral Ghost.

This ghost doesn't fight anyone. It doesn't attack Rock, Paper, or Scissors, nor does it get attacked. It just wanders around, occupying space, eating a snack, and then maybe dying of old age or being born again.

This paper asks a simple but profound question: What happens to the three fighting gangs when this Neutral Ghost is hanging around, especially when we add real-life rules like "people get tired and die" and "babies are born"?

Here is the breakdown of their findings, using some everyday analogies:

1. The "Ghost" Changes the Rules of the Fight

The researchers discovered that the Neutral Ghost isn't just a bystander; it acts like a traffic cop or a weather system that changes how the gangs interact.

The "Bad Ghost" (Negative Interference): Imagine the Ghost is grumpy and constantly bumps into the gangs, making them angry and aggressive. In this scenario, the Ghost actually helps the gangs fight harder against each other. This sounds bad, but it turns out to be good for diversity! The Ghost gets so busy causing trouble that it ends up taking up most of the park, leaving less room for the gangs to kill each other off. It's like a bouncer at a club who is so busy stopping fights that the actual clubbers (the gangs) can't get to the dance floor to destroy each other.
The "Good Ghost" (Positive Interference): Imagine the Ghost is friendly and calms the gangs down. If the Ghost is too effective at calming things down, the gangs stop fighting, but they also stop moving. This can lead to a stalemate where the Ghost takes over the whole park, pushing the gangs out.

2. The "Birth and Death" Twist

Previous studies often ignored the fact that living things are born and die. This paper adds that realism back in.

The Analogy: Think of the gangs as a game of musical chairs. If you add a rule that chairs disappear (death) and new ones appear (birth) randomly, the game changes.
The Finding: When you add natural birth and death, the Neutral Ghost becomes even more powerful. Even if the gangs are moving very fast (which usually causes them to crash and burn), the presence of the Ghost can actually save them. It's like having a safety net. The Ghost absorbs some of the chaos, allowing the gangs to coexist even when they are running around like crazy.

3. The "High-Speed" Miracle

In the old version of this game, if the gangs moved too fast, they would destroy biodiversity (one gang would win, and the others would die out).

The New Discovery: With the Neutral Ghost and natural birth/death, the gangs can move super fast and still survive together. The Ghost acts as a buffer, breaking up the dangerous collisions that usually happen at high speeds. It's like adding a layer of foam padding to a rollercoaster; even if the ride gets faster, the passengers (the species) don't get hurt.

4. The "Super-Baby" Scenario

The researchers also asked: "What if the gangs have way more babies than usual?" (Maybe humans gave them extra food).

The Result: When the gangs have super-high birth rates, the system changes again. Instead of a messy scramble, they form perfect, beautiful spiral galaxies (spiral waves) on the screen.
The Catch: This only works if the "Neutral Ghost" has a slightly different death rate than the gangs. If the Ghost dies too easily, it disappears, and the gangs crash. If the Ghost lives just long enough, it helps the gangs form these stable, swirling patterns that keep everyone alive.

The Big Takeaway

Nature is messy. Species don't just fight; they are born, they die, and they are influenced by neighbors who aren't even fighting them.

This paper tells us that biodiversity is more resilient than we thought. Even when species are moving fast or reproducing wildly, the presence of "neutral" species (like plants that don't compete with animals, or animals that don't eat each other) can act as a glue that holds the ecosystem together. They don't need to be the strongest to save the day; sometimes, just being there and occupying space is enough to stop the strong from eating the weak, ensuring that everyone gets to stay in the park.

In short: The Neutral Ghost is the unsung hero of the ecosystem. By getting in the way, it accidentally saves the world from becoming a one-gang dictatorship.

1. Problem Statement

The maintenance of biodiversity in ecosystems is a central challenge in ecology. While the Rock-Paper-Scissors (RPS) game is a fundamental model for understanding cyclic competition and species coexistence, previous studies often rely on pairwise interactions and neglect two critical ecological realities:

Natural Birth and Death: Most theoretical models assume closed systems or simplified dynamics, ignoring the fundamental mechanisms of natural mortality and reproduction that drive population changes.
Higher-Order Interactions (HOIs) involving Neutral Species: Real ecosystems contain "neutral" species that do not directly compete but influence the environment. Previous studies often treated these species as static or ignored their dynamic impact on the interaction strength of competing species.

The core problem addressed is: How do neutral species, acting through higher-order interactions and coupled with natural birth/death processes, affect the biodiversity, spatial patterns, and extinction risks of a cyclically competing (RPS) system?

2. Methodology

The authors developed a spatial stochastic model on a square lattice ( $N = L \times L$ ) incorporating four species ( $A, B, C, D$ ) and empty sites ( $\emptyset$ ).

System Dynamics:
- Cyclic Competition (A, B, C): Follows standard RPS rules where $A$ preys on $B$ , $B$ on $C$ , and $C$ on $A$ , leaving an empty site.
- Neutral Species (D): Does not participate in direct predation but occupies space and interacts via Higher-Order Interactions (HOIs).
- Birth and Death Processes:
  - Reproduction (Birth): All species (including $D$ ) reproduce into neighboring empty sites at rate $q$ .
  - Natural Death: All species die naturally at rate $f$ , creating empty sites.
  - Mobility: Species exchange positions with neighbors at rate $\epsilon$ (derived from random walk theory).
Higher-Order Interaction Mechanism:
- The predation rate $p$ is not constant but modulated by the local density of the neutral species ( $\rho_i$ ).
- The rate is defined as $p = \exp(K\rho_i)$ $p = exp (K ρ_{i})$ , where $K$ $K$ is a sensitivity coefficient.
  - $K > 0$ : Neutral species enhances competition among $A, B, C$ .
  - $K < 0$ : Neutral species suppresses competition (interferes positively with coexistence).
Simulation Approach:
- Monte Carlo Simulations: Performed on a $300 \times 300$ lattice.
- Parameters: Varied mobility ( $M$ ), intervention intensity ( $K$ ), and reproduction/death rates ( $q, f$ ).
- Scenarios: Standard reproduction rates vs. high reproduction rates (simulating anthropogenic intervention) with asymmetric mortality rates between competitive and neutral species.

3. Key Contributions

Integration of Natural Demography: The study explicitly integrates natural birth and death processes into the RPS-HOI framework, moving beyond idealized models.
Quantification of Neutral Species Impact: It establishes a mathematical link between the local density of a neutral species and the interaction strength of competitors via the sensitivity coefficient $K$ .
Discovery of Coexistence Mechanisms: It reveals that neutral species can paradoxically support biodiversity by suppressing competitive interactions, even in regimes where high mobility typically leads to extinction.
Asymmetric Mortality Analysis: The paper explores how differences in mortality rates between competitive and neutral species alter spatial patterns (e.g., spiral waves) and extinction probabilities under high-reproduction scenarios.

4. Key Results

A. Pattern Formation and Biodiversity (Standard Rates)

Negative Interference ( $K < 0$ ): When the neutral species suppresses competition, species $D$ tends to dominate the available space. This "crowding" effect reduces the frequency of direct predation events among $A, B,$ and $C$ , thereby preventing the extinction of weaker species and maintaining diversity.
Positive Interference ( $K > 0$ ): Enhanced competition leads to higher densities of $A, B,$ and $C$ but reduces the proportion of empty sites.
Mobility Effects: Unlike classic RPS models where high mobility often causes extinction, the presence of neutral species allows biodiversity to be maintained even in high-mobility regimes. The neutral species acts as a buffer against the destabilizing effects of rapid movement.

B. Extinction Probability ( $P_{ext}$ )

Sensitivity Dependence: Extinction probability drops to zero when the sensitivity coefficient $K$ is sufficiently high (positive) or when $K > -1$ .
Low Sensitivity Risk: For negative $K$ values (strong suppression), extinction probability can peak at intermediate mobility levels before declining. However, extremely low or high mobility values generally support coexistence.
Robustness: The system demonstrates robustness against extinction across a wide range of parameters when neutral species are present, challenging the notion that high mobility is always detrimental to biodiversity.

C. High Reproduction and Asymmetric Mortality

Spiral Waves: Under high reproduction rates ( $q=6$ ), the system transitions from scattered clusters to stable spiral wave patterns, which are known to promote coexistence.
Mortality Asymmetry:
- When the neutral species has a lower death rate ( $f_2 < f_1$ ), it can survive in high numbers, disrupting the spiral waves and potentially leading to the extinction of competitive species (specifically $A$ and $C$ ), leaving $B$ dominant.
- Conversely, if the neutral species has a higher death rate, its population is suppressed, allowing the competitive species to form stable spiral waves and coexist effectively.
Synergistic Effects: Heatmaps of extinction probability show that increasing mobility amplifies the negative impact of high mortality rates on biodiversity, expanding the "extinction region" in the parameter space.

5. Significance

Theoretical Advancement: The study bridges the gap between evolutionary game theory and realistic ecological dynamics by incorporating birth/death processes and HOIs. It demonstrates that "neutral" species are not passive observers but active regulators of ecosystem stability.
Conservation Implications: The findings suggest that preserving neutral species or managing their population densities could be a viable strategy for maintaining biodiversity, particularly in ecosystems facing high mobility or environmental stress.
Management of Anthropogenic Impact: By simulating high reproduction rates (mimicking human intervention), the paper provides insights into how altering mortality rates (e.g., through conservation efforts or pest control) can stabilize or destabilize ecosystems. It highlights that tuning the differential mortality between competitive and neutral groups is a critical lever for preserving species coexistence.

In conclusion, the paper reaffirms that neutral species play a critical, active role in preserving biodiversity. By modulating interaction strengths and occupying space, they can prevent the collapse of cyclic competition systems, even under conditions (like high mobility) that would otherwise lead to species extinction.