Adaptive movement strategy may promote biodiversity in the rock-paper-scissors model (2203.12101v2)

Abstract: We study the role of the adaptive movement strategy in promoting biodiversity in cyclic models described by the rock-paper-scissors game rules. We assume that individuals of one out of the species may adjust their movement to escape hostile regions and stay longer in their comfort zones. Running a series of stochastic simulations, we calculate the alterations in the spatial patterns and population densities in scenarios where not all organisms are physically or cognitively conditioned to perform the behavioural strategy. Although the adaptive movement strategy is not profitable in terms of territorial dominance for the species, it may promote biodiversity. Our findings show that if all individuals are apt to move adaptively, coexistence probability increases for intermediary mobility. The outcomes also show that even if not all individuals can react to the signals received from the neighbourhood, biodiversity is still benefited, but for a shorter mobility range. We find that the improvement in the coexistence conditions is more accentuated if organisms adjust their movement intensely and can receive sensory information from longer distances. We also discover that biodiversity is slightly promoted for high mobility if the proportion of individuals participating in the strategy is low. Our results may be helpful for biologists and data scientists to understand adaptive process learning in system biology.

• The paper demonstrates that adaptive kinesis in species 1 alters spiral spatial patterns by adjusting movement based on local predator-prey densities.
• The paper reveals that although individual species 1 benefits by occupying favorable areas, this adaptive strategy can lead to local overexploitation and reduced population sustainability.
• The paper finds that adaptive movement promotes biodiversity at intermediate mobility levels, while high levels of adaptiveness may destabilize coexistence in the system.

This paper investigates the impact of an adaptive movement strategy on biodiversity in a spatial rock-paper-scissors model. The standard rock-paper-scissors game simulates cyclic dominance (species 1 beats 2, 2 beats 3, 3 beats 1), often leading to characteristic spiral patterns in spatial implementations. Typically, movement is modeled as random diffusion with constant mobility. This research introduces a variation where individuals of one species (species 1 in this case) can adjust their movement speed (kinesis) based on their local environment, specifically the densities of their prey (species 2) and predator (species 3).

The core idea is that adaptive individuals can sense their surroundings within a specified perception radius ($R$). They calculate a local "attractiveness" ($\mathcal{A}$) score, defined as the difference between the local density of species 2 (prey, attractive) and species 3 (predator, hostile). Their mobility probability ($m_1$) is then adjusted using a tanh function: $$m_1 = \frac{m}{2} [1 + \tanh(-\epsilon \mathcal{A})]$$. This formula means mobility is lower than the baseline $m/2$ in attractive areas ($\mathcal{A} > 0$) and higher in hostile areas ($\mathcal{A} < 0$). The parameter $\epsilon$ controls the responsiveness intensity to these cues.

A key novelty of this work is the introduction of a conditioning factor ($\alpha$), which represents the proportion of individuals in species 1 that are capable of performing this adaptive movement strategy. The remaining $1-\alpha$ proportion of species 1 individuals move with the same constant mobility ($m/2$) as species 2 and 3. The model is implemented using stochastic simulations on a square lattice with periodic boundary conditions, simulating selection, reproduction, and mobility events following standard rules for spatial biological models.

Implementation and Simulation Details:

The simulation proceeds in timesteps, where $\mathcal{N}$ interaction attempts constitute one generation ($\mathcal{N}$ being the total number of grid points). At each attempt, a random occupied site (an individual) is chosen. One of the three interaction types (selection, reproduction, mobility) is chosen based on fixed probabilities $s$, $r$, and $m_{max}$ (where $s+r+m_{max}=1$). A random neighbor is chosen.

• Selection: If the chosen individual is species $i$ and the neighbor is species $i+1$, the neighbor is removed (becomes empty).
• Reproduction: If the chosen individual is species $i$ and the neighbor site is empty, the neighbor site is filled with an offspring of species $i$.
• Mobility: If the chosen individual is species $i$ and the neighbor site is occupied by species $j$ or is empty, the individual attempts to move to the neighbor site.
  • Species 2 and 3 attempt to move with probability $m_2 = m_3 = m_{max}/2$.
  • Species 1: With probability $\alpha$, the individual calculates its local attractiveness $\mathcal{A}$ within radius $R$ and attempts to move with probability $$m_1 = \frac{m_{max}}{2} [1 + \tanh(-\epsilon \mathcal{A})]$$. With probability $1-\alpha$, the individual attempts to move with probability $m_{max}/2$.
  • If a move attempt is successful (determined by the calculated probability), the individual exchanges positions with the neighbor (whether it's another individual or an empty space).

Simulations typically start from random initial conditions with equal proportions of species and empty sites on lattices ranging from $100^2$ to $500^2$ grid points. Results are often averaged over multiple simulation runs (e.g., 100 to 1000) to account for stochasticity and analyze ensemble properties like mean species densities, spatial correlation, and coexistence probability. Parameters $s, r, m$, $\alpha, \epsilon, R$ are varied to explore their effects.

Practical Findings and Implications:

1. Spatial Structure Alteration: The adaptive movement significantly alters the spatial patterns compared to the standard model. Instead of symmetric spirals, the adaptive species (1) forms irregular, narrower spiral arms. This breaks the balance in spatial domain sizes, which is quantified by the autocorrelation function. The adaptive species' domains shrink, while the domains of the species it preys upon (species 2) tend to expand.
2. Individual Advantage vs. Population Disadvantage: While adapting movement allows individuals of species 1 to spend more time in favorable areas (rich in species 2, poor in species 3), this strategy is disadvantageous for species 1 at the population level. By effectively avoiding areas with predators (species 3) and staying in areas with prey (species 2), species 1 overexploits species 2 locally. This leads to a decline in species 2, which in turn limits the future growth of species 1. This highlights a potential conflict between individual-level adaptive benefits and population-level sustainability, a common theme in ecological modeling.
3. Biodiversity Promotion: The most notable finding is the effect on biodiversity, measured by the probability of all three species coexisting in the long term.
   • At low mobility ($m \leq 0.28$), adaptive movement has minimal effect on coexistence.
   • At intermediate mobility ($0.28 < m \leq 0.60$), adaptive movement promotes coexistence. This effect is strongest when all individuals of species 1 are adaptive ($\alpha = 1.0$) and when they have higher responsiveness ($\epsilon$) and larger perception radii ($R$). If only a fraction of species 1 is adaptive ($\alpha < 1.0$), the benefit to coexistence is still present but for a smaller range of intermediate mobility.
   • At high mobility ($m > 0.60$), adaptive movement generally jeopardizes biodiversity if a significant proportion of species 1 is adaptive ($\alpha \geq 0.6$). The system becomes more unstable, leading to extinctions. However, counter-intuitively, if only a small fraction of species 1 is adaptive ($\alpha=0.3$), high mobility might slightly promote coexistence.
4. Relevance to AI and Robotics: The model connects to artificial intelligence and robotics by modeling agents (animats or robots) that can sense their environment and adapt their movement. The paper demonstrates how such adaptive behaviors, while potentially enhancing an individual agent's ability to navigate or exploit resources, can have complex, sometimes detrimental, impacts on the collective dynamics and stability of a multi-agent system interacting in a competitive environment. It suggests that designing adaptive behaviors for multi-agent systems requires considering not just individual performance but also emergent collective outcomes like stability and diversity.

Implementation Considerations:

• Computational Resources: Simulating spatial agent-based models on large lattices ($500^2$ sites) over many generations ($5000-10000$) is computationally intensive, especially when averaging over hundreds or thousands of runs. Efficient data structures for the lattice and optimized interaction loops are crucial.
• Parameter Tuning: The behavior of the system is highly sensitive to parameters like $m, \alpha, \epsilon, R$. Extensive parameter sweeps are necessary to understand the different regimes (low, intermediate, high mobility) and the influence of adaptiveness factors.
• Boundary Conditions: Periodic boundary conditions are standard for minimizing finite-size effects, but other boundary types might yield different results.
• Neighborhood Definition: The paper uses the Moore neighborhood (8 nearest neighbors). Using different neighborhoods (e.g., Von Neumann, hexagonal) or interactions beyond immediate neighbors could alter spatial dynamics.
• Adaptive Strategy Complexity: The kinesis strategy here is relatively simple (adjusting speed based on local densities). More complex adaptive strategies (e.g., taxis, learning, memory) could lead to richer dynamics.

In essence, the research provides a practical simulation framework to explore how simple adaptive movement rules in one species can profoundly affect the spatial ecology and long-term biodiversity of a multi-species cyclic system, offering insights applicable to both biological ecosystems and synthetic multi-agent systems.