Percolation and clustering in ecological communities: A dynamical theory

Abstract: Ecological communities with structured interactions exhibit collective phenomena such as percolation and clustering of occupied sites. While these effects have been documented in experiments and simulations, systematic analytical understanding has remained limited. In this paper, we develop a dynamical theory of these phenomena for competitive ecological systems defined on random interaction graphs. We introduce a discrete version of the generalized Lotka-Volterra model that preserves key macroscopic features of continuous ecological dynamics while enabling analytical treatment. Within this framework, we characterize the emergence of percolating clusters and describe the spatial organization of surviving sites. Our analysis uncovers which equilibria can be reached by the dynamics and shows how this dynamical accessibility governs the onset of clustering and percolation. In doing so, our framework complements classical Lotka-Volterra theory by providing a dynamical perspective on the collective organization of structured communities.

• The paper introduces a discrete gLV model that maps ecological dynamics onto random graphs to reveal distinct percolation and clustering phase transitions.
• It employs the Backtracking Dynamical Cavity Method and a tailored message-passing formalism to accurately compute key observables like extinction fraction and largest cluster size.
• Empirical validations confirm sharp phase transitions and the emergence of rare attractors, offering insights for restoration strategies in fragmented ecosystems.

Dynamical Analysis of Percolation and Clustering in Ecological Communities

Introduction and Model Formulation

The paper "Percolation and clustering in ecological communities: A dynamical theory" (2606.09494) establishes an analytical framework to study large-scale spatial phenomena—specifically, percolation and clustering, in ecological systems governed by competitive interactions. The focal point is a discrete variant of the generalized Lotka-Volterra (gLV) model mapped onto random graphs with locally tree-like structure, enabling rigorous characterizations of spatial organization and phase transitions.

The discrete gLV model defines local biomass variables $N_i \in \mathbb{N}$ at each site, evolving synchronously according to a locally interacting deterministic rule. The growth rate incorporates carrying capacity and neighbor competition, with the interaction strengths modulated by $\alpha_{ij}$. Uniform random $d$-regular graphs and uniform couplings $\alpha/d$ serve as the principal topology, but the phase diagram remains robust under heterogeneity in both network structure and interaction strength.

Phase Diagram and Collective Phenomena

A detailed dynamical analysis uncovers a sequence of sharp phase transitions as the interaction strength $\alpha$ increases, delineating distinct ecological regimes:

• Fully Occupied Phase ($\alpha < \alpha_\text{ext}$): All sites are at positive biomass, yielding a spatially uniform ecosystem.
• Percolating Phase ($\alpha_\text{ext} < \alpha < \alpha_\text{perc}$): Vacant sites emerge but a macroscopic cluster connects a finite fraction of occupied sites, corresponding to ecosystem fragmentation yet with extensive connectivity.
• Non-Percolating Phase ($\alpha > \alpha_\text{perc}$): Only sub-extensive clusters (patches) survive, leading to isolated occupancy—reminiscent of vegetation patchiness and imminent desertification transitions.
• Linear Cluster Regime ($\alpha > \alpha_\text{lin}$): Surviving clusters acquire strictly linear topology; higher-order connectivity vanishes.

  Figure 1: Spatial phase diagram of the discrete gLV model on a $d$-regular graph, with critical thresholds $\alpha_{ij}$0, $\alpha_{ij}$1, and $\alpha_{ij}$2, graphically depicting transitions between fully occupied, percolating, non-percolating, and linear cluster phases.

The dynamical accessibility of equilibria is the organizing principle: only attractors with large basins reached from random initial conditions manifest in practice. Smaller basins—subdominant attractors—often require finely tuned initial configurations.

Analytical Framework: BDCM and Message-Passing Formalism

The Backtracking Dynamical Cavity Method (BDCM) is deployed to enumerate and characterize dominant attractors, using message-passing equations specialized to locally tree-like random graphs. The method allows calculation of observables such as extinction fraction $\alpha_{ij}$3, largest cluster size $\alpha_{ij}$4, and higher-order structure functions $\alpha_{ij}$5, capturing intricate spatial statistics of the steady state.

BDCM is extended with a cluster-specific message-passing formalism for percolation observables. Unlike classical percolation (independent edge/site occupation), the dynamical nature introduces correlated edge occupations. This is formalized through cavity-generated probability distributions over entire dynamical trajectories and recursive generating functions for cluster size distributions.

Figure 2: BDCM accurately captures typical extinction and cluster statistics for $\alpha_{ij}$6, $\alpha_{ij}$7, showing perfect correspondence between theoretical predictions and empirical simulations as $\alpha_{ij}$8 varies.

Empirical Validation and Robustness

Extensive simulations confirm analytical predictions:

• Discrete gLV dynamics converges rapidly; typical attractors are dominated by 2-cycles.
• The framework is validated across topologies: regular graphs, grids, and networks with heterogeneous couplings.
• Large $\alpha_{ij}$9 produces results closely matching continuous gLV dynamics; fragmentation and percolation transitions persist under interaction heterogeneity.

  Figure 3: Discrete gLV simulations on a grid displaying fragmentation (random couplings) and ring-like structures (uniform couplings), reinforcing robustness against interaction heterogeneity and topology.

  

Figure 4: Comparison between discrete and continuous gLV models on a random 3-regular graph; both exhibit congruent trajectories and cluster statistics across interaction strengths.

Figure 5: Occupied fraction $d$0 and largest-cluster fraction $d$1 for discrete gLV under heterogeneous interactions, showing qualitative agreement across coupling ensembles.

Subdominant, Rare Attractors and Restoration Implications

Atypical, subdominant attractors—such as fully occupied configurations existing beyond the extinction threshold $d$2—are computed via BDCM. These attractors, though exponentially rare, are accessible given carefully engineered initial conditions. The gap between $d$3 (maximum $d$4 supporting full occupancy for some initialization) and $d$5 is robust, persistent for $d$6 and larger $d$7.

Figure 6: Full occupied entropy and atypical threshold $d$8 versus extinction threshold $d$9, revealing parametric regions where rare initializations prevent extinction.

Implications for restoration ecology are clear: spatial arrangement and seeding density can profoundly influence the long-term survival outcome, an insight supported by both theory and empirical ecological studies.

Theoretical and Practical Implications

• Analytical Tractability: Provides systematic methodology for quantifying collective dynamics of competitive ecosystems, generalizable to other domains (social contagion, epidemics, cellular automata) where dynamical-induced dependencies complicate percolation analyses.
• Phase Transition Characterization: Offers deterministic predictions for percolation thresholds in correlated systems, complementing earlier numerical and empirical work.
• Dynamical Accessibility vs. Raw Equilibria: Distinguishes between typical and most numerous attractors, emphasizing the necessity of dynamical analysis for predicting observable ecological outcomes.
• Restoration and Intervention Strategies: Points towards harnessing rare basin attractors through carefully tailored initializations—relevant for ecosystem restoration and robustness enhancement in engineered networks.

Conclusion

This work advances the analytical study of ecological spatial organization under competitive dynamics, introducing a discrete gLV model and leveraging BDCM and generalized message-passing for rigorous phase diagram determination. The findings bridge microscopic local interaction rules with emergent macroscopic percolation, fragmentation, and clustering phenomena. The methodological innovations are broadly applicable within statistical physics, network science, and applied ecology, and the results underscore the value of dynamical systems approaches for both predictive understanding and intervention design in complex systems.