Segregation process and phase transition in cyclic predator-prey models with even number of species

We study a spatial cyclic predator-prey model with an even number of species (for n=4, 6, and 8) that allows the formation of two defective alliances consisting of the even and odd label species. The species are distributed on the sites of a square lattice. The evolution of spatial distribution is governed by iteration of two elementary processes on neighboring sites chosen randomly: if the sites are occupied by a predator-prey pair then the predator invades the prey’s site; otherwise the species exchange their site with a probability $X$. For low $X$ values a self-organizing pattern is maintained by cyclic invasions. If $X$ exceeds a threshold value then two types of domains grow up that formed by the odd and even label species, respectively. Monte Carlo simulations indicate the blocking of this segregation process within a range of X for n=8.

💡 Research Summary

The paper investigates a spatial cyclic predator‑prey model that contains an even number of species (n = 4, 6, 8) and examines how a simple exchange process between neighboring sites controls the emergence of large‑scale patterns and a phase‑transition‑like segregation. The system is defined on a two‑dimensional square lattice of size L × L. Each lattice site is occupied by one of the n species, and the species interact according to a strict cyclic dominance rule: species i preys on species i + 1 (mod n) and is preyed upon by species i − 1. The dynamics proceeds by repeatedly selecting a random nearest‑neighbor pair and applying one of two elementary processes: (1) if the pair forms a predator‑prey relationship, the predator invades the prey’s site (the classic “rock‑paper‑scissors” invasion); (2) if the pair does not belong to a predator‑prey pair, the two occupants exchange their positions with probability X. The exchange probability X therefore acts as a tunable diffusion‑like parameter that mixes species without direct competition.

Monte‑Carlo simulations were performed, where one Monte‑Carlo step (MCS) consists of N = L² random pair selections. The authors measured time‑dependent quantities such as species densities, spatial correlation functions, domain size distributions, and the length of interfaces separating different domains. By varying X from 0 to 0.2 and observing the long‑time stationary states for each n, two qualitatively distinct regimes were identified.

In the low‑X regime (X < Xc), predator‑prey invasions dominate. The system self‑organizes into a dynamic mosaic of small rotating spirals or wave fronts, reminiscent of the well‑known patterns in three‑species rock‑paper‑scissors models. The characteristic domain size remains bounded, and the system exhibits a statistically stationary state where all species coexist with roughly equal average densities.

When X exceeds a critical value Xc, a different macroscopic structure emerges. Because the exchange process does not change predator‑prey relationships, species with even labels (2, 4, …) and those with odd labels (1, 3, …) become internally neutral: they never invade each other directly. Consequently, each parity class forms a “defective alliance” that behaves as a single effective entity. Within an alliance, only the exchange process operates, leading to rapid internal mixing, while at the interface between the two alliances predator‑prey invasions still occur. This creates sharp domain walls that move outward, causing the two alliances to coarsen. The coarsening speed grows with X, and eventually one alliance completely dominates the lattice, a state the authors refer to as domain domination.

The most intriguing behavior appears for n = 8. In a narrow interval of X (approximately 0.06 ≤ X ≤ 0.08) the coarsening process stalls: domain growth slows dramatically and the system remains trapped in a mixed configuration for an extended period. The authors interpret this “blocking” phenomenon as a metastable state arising from the competition between multiple interface geometries and the enhanced combinatorial possibilities of forming alliances when more species are present. This effect is absent for n = 4 and n = 6, indicating that the number of species critically influences the topology of the phase space.

From these observations the authors draw several conclusions. First, the exchange probability X serves as a control parameter that tunes the system between a cyclic‑invasion dominated, self‑organized phase and an alliance‑segregated phase. Second, even‑odd parity provides a natural mechanism for spontaneous symmetry breaking in even‑species cyclic games, leading to the formation of two macroscopic “teams.” Third, the blocking region observed for n = 8 suggests the existence of non‑continuous or metastable transitions that are not captured by simple mean‑field or pair‑approximation theories. Finally, the results highlight how a minimal diffusion‑like process can dramatically reshape the collective dynamics of ecological or evolutionary systems, offering a tractable framework for studying pattern formation, segregation, and phase transitions in multi‑species interactions.

The paper contributes a clear computational methodology, systematic analysis across different numbers of species, and a novel insight into how simple stochastic exchanges can induce rich macroscopic behavior. Future work could extend the model to irregular networks, asymmetric interaction strengths, or time‑dependent environmental perturbations, thereby bridging the gap between idealized lattice models and real ecological communities.