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This document summarizes the key findings from the predator-prey cellular automaton experiments exploring the Hydra effect and self-organized criticality.
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## Phase 1: Critical Point Identification
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### Objective
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#### Phase 1: Critical Point Identification
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Locate the critical prey mortality rate where the population dynamics undergo a qualitative transition and test whether the system exhibits signatures of self-organized criticality at this point.
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### Methods
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We performed a parameter sweep over ```prey_death_rate``` $\in [0, 0.2]$ while holding other parameters fixed.
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Each simulation ran for 1000 warmup steps followed by 1000 measurement steps. The mean populations and cluster statiustics were recorded on the final grid states.
The log-log plot shows the probability distribution of prey cluster sizes at the critical point. Visual inspection suggests approximate power law scaling with $$\alpha \approx 2.29$$.
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Our results showed $\alpha = 2.29$ and $R = -1.3$. The negative $R$ value indicates that a lognormal distribution provides a better fit to the cluster size data than a power-law. This is evidence against true self-organized criticality. This might suggest that the system is near critical but not at a true critical point.
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## Phase 2: Evolutionary SOC Analsyis
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####Phase 2: Evolutionary SOC Analsyis
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### Objective
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Test whether predator-prey system exhibits SOC by allowing prey mortality rates to evolve. Under the SOC hypothesis, evolution should drive the system toward the transition point identified in Phase 1.
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### Methods
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We enabled per-cell evolution of the prey death parameter. Each prey individual carries its own ```prey_death_rate``` value. The offsprings inherit their parents value with Gaussian mutation. Evolution occurs during the warmup steps, then frozen for the measurement.
**Right panel**: The evolved prey death rate as a function of initial prey death rate.
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The negative $R$ value is even more extreme than at the Phase 1 critical point ($R = -1.3$), indicating the evolved state is **further from criticality**, not closer.
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These results represent a significant puzzle. The system appears to self-organize. However, this convergence does not take place toward the transition point. At the moment, we do not have a definitive explanation for this behavior. A possible explanation could relate to evolution optimizing for individual fitness, not population level-properties like criticality.
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## Phase 3: Finite-size Scaling
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### Objective:
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#### Phase 3: Finite-size Scaling
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Investigate whethr the near-critical behavior observed in Phase 1 is a finite-size artifact. If the system is critical, scaling analysis across different system sizes should reveal universal behavior. If not, larger system should show stronger deviation from power-law scaling.
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### Methods
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We ran simulation at the critical point ($d_1 = 0.0955$) across six grid sizes $L \in \{50, 100, 250, 500, 1000, 2500\}$ with fixed parameters:
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For each system size, we aggregated all prey cluster sizes across replicates, fitted a lognormal distrbution using the maximum likelihood estimation, and computed power-law vs. lognormal likelihood ratio $R$.
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---
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#### NOTE: Lognormal Distribution
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The lognormal distribution describes a variable whose logarith is normally distributed:
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The lognormal parameters were fitted using `scipy.stats.lognorm.fit()` with location fixed at zero.
The figure shows prey cluster size distributions at the critical point across all system sizes with lognormal fit parameters and likelihood ratios.
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**Observations:**
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**1. Lognormal parameters are stable across system sizes**
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The fitted $\mu$ and $\sigma$ values converge to approximately $\mu \approx 1.36$ and $\sigma \approx 1.39$ for $L \geq 250$. This consistency indicates that the lognormal form is a robust description of the cluster size distribution, not a finite-size artifact.
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**2. Evidence against power-law grows with system size**
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This is the crucial result: as $L$ increases, $R$ becomes *more negative*, not less:
If the system were truly critical with finite-size scaling corrections, we would expect $R \rightarrow 0$ as $L \rightarrow \infty$.
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**3. Distributions do not collapse**
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At a true critical point, rescaling cluster sizes by $L^{d_f}$ (where $d_f$ is the fractal dimension) should collapse all distributions onto a universal scaling function. The cluster size distribution is lognormal, which becomes more apparent with increasing system size. This result suggests that the underlying process involves multiplicative growth rather than the scale free branching processes that generate power laws.
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#### Phase 4 & 5: Sensitivity Analysis and Directed Hunting
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We map the hydra effect across the full 4D parameter space to understand the conditions under which the Hydra effect occurs, the strength of the effect across different parameter regimes, and whether directed hunting alters the effect.
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Phase 4 uses random neighbor selection while Phase 6 uses directed hunting where predators preferentially target prey neighbors.
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We performed as sweep across the following model parameters:
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```
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prey_birth
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prey_death
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predator_birth
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predator_death
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```
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For each parameter combination, we computed the equilibrium prey density curve and detected the Hydra effect by checking fro regions with a positive derivative:
A curve exhibits the Hydra effect is prey density increases with mortality over any portion of the ```prey_death``` range.
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To measure the stength of the Hydra effect, we fitted a qudratic function to the rising portion of each curve (truncated at the max derivative point):
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$$\rho(d_1) = a \cdot d_1^2 + b \cdot d_1 + c$$
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The quadratic coefficient $a$ captures how sharply the prey density rises and falls. Larger $|a|$ indicates a more pronounced Hydra effect with sharper peak. This apporach filters to only the Hydra effect portion, captures curvature, and provides a single scalar metric for comparison across parameter space.
**Left panel**: All parameter combinations exhibiting the Hydra effect.
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**Right panel**: Curves truncated at the maximum derivative point, isolating the rising (Hydra) portion. Color indicates the maximum positive derivative—darker blue corresponds to steeper initial increases.
**Left panel (Strong Hydra, coef > 15)**: Extremely sharp peaks with quadratic coefficients reaching 49–98. However, the curves are notably noisier than the undirected case, with irregular trajectories.
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**Right panel (Moderate Hydra, coef 5–10)**: More variable behavior compared to undirected hunting. Some curves show multiple peaks or irregular shapes.
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The results indicate that directed hunting amplifies the Hydra effect but also introduces instability. Smarter predators paradoxically create conditions for stronger Hydra effects.
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