🧬 biology

Modulating biodiversity through higher-order interactions and intraspecific competition in rock-paper-scissors dynamics

This study demonstrates that in rock-paper-scissors dynamics, resource-mediated higher-order interactions promote the formation of compact spatial patterns but simultaneously reduce system resilience by lowering the mobility threshold required for species extinction.

Original authors: Chunpeng Du, Haoshu Wang, Yikang Lu, Lijuan Qin, Junpyo Park

Published 2026-03-03

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Chunpeng Du, Haoshu Wang, Yikang Lu, Lijuan Qin, 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 ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed.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 where three rival gangs—let's call them Rock, Paper, and Scissors—are constantly fighting for territory. In this city, Rock beats Scissors, Scissors beats Paper, and Paper beats Rock. This is the classic "Rock-Paper-Scissors" game, a model scientists use to understand how different species in nature (like bacteria, lizards, or plants) coexist without one of them wiping out the others.

Usually, scientists thought this balance was simple: if the gangs move around too much, they mix up and chaos ensues; if they stay put, they form neat, swirling patterns that keep everyone alive.

But this new study adds a twist: The environment itself changes the rules of the fight.

The New Rule: "The Hunger Factor"

In this study, the researchers introduced a variable called food. Imagine that the gangs only fight each other fiercely when they are hungry. If a gang member is surrounded by plenty of food, they are less aggressive toward their own kind. But if food is scarce, they turn on each other with a vengeance.

This is what the scientists call a "Higher-Order Interaction." It's not just a one-on-one fight anymore; the outcome of a fight depends on the whole neighborhood around the fighters. It's like a boxing match where the crowd's energy (the food) decides how hard the boxers punch.

What Happened in the Simulation?

The researchers ran thousands of computer simulations to see how this "hunger factor" changed the city's layout. Here is what they found, explained simply:

1. The "Compact City" Effect

When the gangs are influenced by this food-based rule (strong higher-order interactions), the swirling patterns they form become tighter and more compact.

Analogy: Imagine a group of dancers in a circle. Without the food rule, they dance in a wide, loose circle. With the food rule, they huddle closer together, forming a tight, efficient spiral. The "territory" each gang controls shrinks, but the pattern becomes more organized.

2. The Double-Edged Sword of Moving Around

In the old model, moving around (mobility) was the main thing that could destroy the balance.

Low Movement: The gangs stay in their neighborhoods, forming beautiful, stable spirals. Everyone survives.
High Movement: The gangs run around too fast, mix everything up, and one gang eventually takes over the whole city, wiping out the others.

The New Discovery: The "food rule" makes the gangs much more sensitive to movement.

The Metaphor: Think of the gangs as a house of cards. In the old model, you could blow a little wind (low movement) and the house stands. In the new model, the "food rule" makes the cards sticky and the structure more fragile. Now, even a tiny breeze (much less movement than before) can knock the whole house down.
Result: Because the gangs are so sensitive to their environment, they lose their balance much faster if they start moving around too much. The "safe zone" where they can all coexist gets smaller.

3. The Empty Lots

Interestingly, even when one gang takes over, the city isn't 100% full. There are always some empty lots left over.

Why? Because the "hunger rule" creates a constant cycle of fighting and retreating. Even the winning gang can't fill every single spot because they are busy fighting their own kind when food is low. These empty spots are actually crucial—they are the "breathing room" that keeps the ecosystem from becoming a stagnant, dead zone.

The Big Takeaway

This study teaches us that nature is more fragile than we thought.

We used to think that as long as species didn't move too fast, they would be fine. But this research shows that if you add in complex factors like resource scarcity (food) and how it changes how species treat their own kind, the system becomes much more delicate.

Good News: These complex rules help create beautiful, organized patterns (like tight spirals) that look very stable.
Bad News: These same rules make the system crash much faster if the environment changes (like if the animals start moving around more).

In plain English: Nature is like a complex dance. Adding a "hunger" rule makes the dancers move in tighter, more beautiful circles, but it also means that if they trip or move too fast, the whole dance falls apart much quicker than before. This helps conservationists understand that protecting biodiversity isn't just about stopping animals from moving; it's about understanding how their food and local environment change the way they interact with each other.

1. Problem Statement

The preservation of biodiversity in ecosystems governed by cyclic competition (e.g., Rock-Paper-Scissors or RPS dynamics) is a central challenge in ecology. Traditional RPS models typically assume:

Pairwise interactions: Competition occurs only between two individuals.
Homogeneous parameters: Competition rates and resource availability are constant.
Static intraspecific competition: Competition within the same species is often modeled as a fixed rate.

However, real-world ecological systems are influenced by higher-order interactions (HOI), where the outcome of a pairwise interaction is modulated by the collective state of the surrounding neighborhood (e.g., local resource abundance). Furthermore, empirical evidence suggests that fluctuating resource conditions can exacerbate intraspecific competition, driving strategic shifts. The authors aim to investigate how resource-mediated higher-order interactions regulate intraspecific competition and how this mechanism, in conjunction with individual mobility, affects species coexistence, spatial pattern formation, and extinction risks.

2. Methodology

The study employs a spatial evolutionary game theory approach using a modified Rock-Paper-Scissors model on a square lattice ( $N = L \times L$ ).

Model Dynamics

The system evolves via asynchronous random sequential updates involving three species ( $A, B, C$ ) and empty sites ( $\emptyset$ ). The interactions include:

Interspecific Competition (Cyclic Dominance): $A \to B \to C \to A$ (Rate $\sigma$ ).
Reproduction: Species colonize adjacent empty sites (Rate $\mu$ ).
Migration (Mobility): Individuals exchange positions with neighbors (Rate $r$ , defined by mobility $M$ ).
Intraspecific Competition: Individuals of the same species compete (Rate $P$ ).

Key Innovation: Resource-Mediated Higher-Order Interaction

Unlike previous models with constant rates, the intraspecific competition rate $P$ is dynamically regulated by local resource availability ( $\rho$ ) via a higher-order mechanism:
$P = \alpha \exp\left(\frac{\rho}{6\delta}\right)$

$\alpha$ : Baseline intraspecific competition rate.
$\rho$ : Local food resource abundance, defined as the count of food items in the 6 nearest-neighbor sites surrounding the competing pair.
$\delta$ : The higher-order modulation factor. A negative $\delta$ implies that higher local resource density increases the intensity of intraspecific competition (context-dependent regulation).
Mechanism: The outcome of a pairwise interaction is co-determined by the collective state of the immediate neighborhood (6 surrounding sites), representing a microscopic, individual-based HOI.

Simulation Parameters

Lattice Size: $300 \times 300$ (with periodic boundary conditions).
Mobility ( $M$ ): Varied from low ( $10^{-5}$ ) to high ( $10^{-3}$ ).
HOI Strength ( $\delta$ ): Varied from $0$ (no HOI) to $-1$ (strong HOI).
Analysis Metrics:
- Temporal evolution of species densities.
- Spatial pattern analysis (spiral waves, domain size).
- Basin Entropy ( $S_b$ ): Quantifies the unpredictability of final states and spatial disorder.
- Characteristic Wavelength ( $L$ ): Measures the spatial scale of spiral waves.
- Extinction Probability ( $P_{ext}$ ): Calculated over 800 independent realizations.

3. Key Contributions

Novel Mechanism: Introduces a specific mathematical framework where intraspecific competition is dynamically regulated by local resource abundance through higher-order interactions, moving beyond static pairwise assumptions.
Dual Role of HOI: Demonstrates that HOI has a dual effect: it promotes local pattern formation (structural tightening) but simultaneously reduces the system's resilience to dispersal (mobility).
Quantitative Link: Establishes a quantitative relationship between HOI strength ( $\delta$ ), spatial wavelength, and basin entropy, showing that stronger HOI leads to more compact, ordered domains.
Extinction Threshold Shift: Reveals that strong HOI shifts the critical extinction threshold toward lower mobility values, meaning biodiversity is lost more easily in highly mobile systems when HOI is strong.

4. Key Results

Microscopic Dynamics (Spatial Patterns)

Low/Intermediate Mobility ( $M \le 10^{-4}$ ):
- The system maintains stable species coexistence characterized by spiral wave patterns.
- Increasing HOI strength (decreasing $\delta$ ) causes the spatial wavelength to decrease exponentially. The spiral domains become more compact and refined.
- Basin Entropy ( $S_b$ ) decreases linearly with stronger HOI, indicating that the system becomes more predictable and spatially ordered.
- Empty sites persist at low levels, providing necessary space for cyclic recovery.
High Mobility ( $M = 10^{-3}$ ):
- The system becomes unstable regardless of $\delta$ , leading to biodiversity loss and single-species dominance.
- Strong HOI accelerates this process. For example, at $\delta = -1$ , species $A$ dominates the entire space, whereas at $\delta = 0$ , species $C$ might dominate.
- The "structural tightening" caused by HOI makes the system more sensitive to the homogenizing effect of high mobility.

Macroscopic Dynamics (Extinction Risk)

Extinction Probability ( $P_{ext}$ ):
- In classic RPS models, high mobility is the primary driver of extinction.
- In this model, stronger HOI shifts the extinction threshold to lower mobility values. Even at moderate mobility levels, strong HOI can trigger a transition from coexistence to extinction.
- The probability of extinction drops sharply as the baseline competition rate ( $\alpha$ ) increases, but the presence of strong HOI ( $\delta < 0$ ) generally increases the risk of extinction compared to the baseline ( $\delta = 0$ ) at the same mobility level.

5. Significance and Implications

Theoretical Advancement: The study challenges the traditional view that higher-order interactions always stabilize ecosystems. Instead, it shows that while HOI can enhance local structural order (compact domains), it can destabilize the global system by making it more fragile to dispersal.
Ecological Insight: The findings suggest that resource-mediated competition is a critical, often overlooked factor in biodiversity maintenance. Fluctuating resources that intensify intraspecific competition can inadvertently drive species to extinction if dispersal rates are not carefully managed.
Conservation Strategy: The results imply that biodiversity conservation strategies must account for the interplay between resource distribution, local competition intensity, and species mobility. Simply increasing connectivity (mobility) in systems with strong resource-mediated competition could accelerate extinction rather than prevent it.
Methodological Contribution: The use of basin entropy and characteristic wavelength provides robust quantitative tools for analyzing the stability and spatial organization of complex ecological networks beyond simple density measurements.

In summary, this paper provides a rigorous theoretical framework demonstrating that higher-order interactions act as a "structural tightening" force that refines spatial patterns but simultaneously lowers the resilience threshold of the ecosystem against mobility-induced extinction.