Tuning Agent-Based Predator-Prey Models Toward Lotka-Volterra Dynamics

Technical Summary: Tuning Agent-Based Predator-Prey Models Toward Lotka-Volterra Dynamics

Problem Statement
Recent increases in computational power have enabled the use of large-scale agent-based models (ABMs) to simulate complex adaptive systems. However, a central difficulty remains: ABMs often contain numerous local rules and parameters where minor adjustments can lead to runaway behaviors, population collapse, or artificial saturation. The challenge lies in constraining these dynamics to produce stable, meaningful population-level structures without removing the local interactions that generate complexity. Specifically, the authors investigate whether the microscopic parameters of a continuous predator-prey system can be tuned so that the resulting macroscopic population dynamics resemble the classical, smooth oscillations of Lotka-Volterra (LV) equations.

Methodology
The study employs a continuous two-dimensional predator-prey system where sheep (prey) and wolves (predators) are modeled as self-propelled active agents. The system is implemented using ABMax, a JAX-based framework that enables efficient batched simulation on hardware accelerators (NVIDIA A100 GPUs).

• Agent Dynamics: Agents possess internal energy, local sensing capabilities (ray-casting with occlusion), and are controlled by Continuous-Time Recurrent Neural Networks (CTRNNs) based on a modern Wilson-Cowan interpretation. Agents move in a Cartesian plane, gaining energy from a central grass patch (sheep) or by consuming prey (wolves), while incurring metabolic costs.
• Population Turnover: Birth and death are threshold-based. Agents reproduce if energy exceeds a threshold for a specific duration and a slot is available; they die if energy remains below a threshold for a duration. Offspring inherit controller parameters and half the parent's energy.
• Optimization Strategy: The optimization occurs in two distinct stages:
  1. Controller Optimization: Using Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES), the authors first optimize the neural network controllers for sheep and wolves separately. Wolves are trained against random sheep controllers, and vice versa, to establish robust local behaviors.
  2. ABM-Level Tuning: With controllers fixed (either as the evolved versions or as random initializations), the authors optimize ecological parameters (e.g., grass intake rates, predation transfer rates, metabolic costs, and birth/death thresholds).
• Objective Function: The tuning targets a feature-based loss function designed to enforce LV-like dynamics. The loss ($L$) is a weighted sum of several components:
  • Correlation Loss ($L_{corr}$): Uses Pearson correlation to reward the phase lag between prey and predator populations (prey leading predator) and the alignment of their growth/decline slopes. It also encodes local LV pressure (high prey abundance driving predator growth, and vice versa).
  • Stability Losses: Includes terms to prevent extinction ($L_e$), prevent saturation at artificial population caps ($L_c$), ensure sufficient oscillation amplitude ($L_a$), enforce non-monotonic trajectories ($L_t$), penalize trajectories that do not cross their mean frequently enough ($L_{cross}$), and minimize long-term drift in population means ($L_d$).

Key Results

• Convergence to LV Dynamics: The optimization successfully guided the ABM parameters toward regimes exhibiting sustained, coupled oscillations between sheep and wolf populations. The loss function decreased significantly, particularly when using evolved controllers compared to random ones.
• Controller Influence:
  • Evolved Controllers: Produced "noisier" and less idealized oscillations that remained well-bounded and avoided extreme saturation. This suggests that learned local behaviors, when paired with tuned ecological parameters, yield more behaviorally grounded dynamics.
  • Random Controllers: Produced smoother, more regular oscillations but tended to approach the artificial population ceiling, indicating that without learned behavioral constraints, the system relies heavily on the tuned parameters to maintain stability.
• Spatial Organization: When visualized, the evolved agents exhibited clear spatial structures: wolves aggregated near the grass patch (where prey density was higher), while sheep dispersed more broadly. The authors note that this spatial behavior emerged from the controller optimization, while the ABM-level tuning ensured this behavior remained viable over long time horizons by balancing energy flows.

Significance and Claims
The paper claims that a desired macroscopic ecological regime (Lotka-Volterra dynamics) can serve as an explicit top-down optimization target for searching the parameter space of a microscopic agent-based model. The authors emphasize that their work does not imply that ABMs automatically discover LV dynamics, nor that a unique microscopic model can be inferred from a macroscopic trajectory.

Instead, the significance lies in demonstrating that macroscopic constraints can be used to identify parameter regimes where complex, individual-based systems remain dynamically well-behaved. The study highlights a distinction between behavioral structure (generated by local controller optimization) and population-level stabilization (achieved by tuning global ecological parameters). This approach offers a method for studying open-ended multi-agent systems by using ecological constraints to keep the system within viable bounds while preserving non-trivial, emergent dynamics. The work also validates the utility of the ABMax framework for parallelizing and optimizing large-scale agent simulations.