Labyrinthine clustering in a spatial rock-paper-scissors ecosystem

The spatial rock-paper-scissors ecosystem, where three species interact cyclically, is a model example of how spatial structure can maintain biodiversity. We here consider such a system for a broad range of interaction rates. When one species grows very slowly, this species and its prey dominate the system by self-organizing into a labyrinthine configuration in which the third species propagates. The cluster size distributions of the two dominating species have heavy tails and the configuration is stabilized through a complex spatial feedback loop. We introduce a statistical measure that quantifies the amount of clustering in the spatial system by comparison with its mean-field approximation. Hereby, we are able to quantitatively explain how the labyrinthine configuration slows down the dynamics and stabilizes the system.

Figure 1

Spatial self-organization in the rock-paper-scissors game. (a) The three species interact cyclically. Species $i$ invades its prey at rate $v_i$. (b)–(d) Snapshots of the steady-state spatial organization of the three species when (b) all species grow at the same rate and $L=300$; (c) species 3 grows five times faster than 1 and 2 and $L=300$; (d) species 1 grows 1250 times more slowly than 2 and 3 and $L=12800$. (e),(f) Zooms of the system in (d).

Figure 2

Species abundances and correlations for small $v_1$. (a) When the growth rate of species 1 is decreased, species 3 becomes less abundant. The mean-field theory provides a good approximation to the abundances. (b) The spatial correlation between different species are much lower than predicted by the mean-field theory due to clustering. (c) The ratio between the predicted mean-field correlations and the observed correlations are equal for all species. This ratio defines $\chi$. (d) When the growth rate of species 1 is decreased, it begins to form labyrinthine clusters. Thus, the path distance between two points $i$ and $j$ in the same cluster becomes much longer than the geodesic distance. This ratio is described by $\xi$. (e) When $v_3\gg v_1,v_2$, the ratio $\chi$ diverges corresponding to the large clustering in Fig. 1c. When one species grows much more slowly than the others, $\chi$ approaches 5, which gives rise to the labyrinthine clustering in Fig. 1d.

Figure 3

Cluster size distributions. (a) When all species grow at the same rate, all clusters consist of fewer than 5000 nodes. Here $v_1=v_2=v_3=1$. (b) When the growth rate of species 1 is decreased, species 3 becomes less abundant and large clusters of species 1 and 2 become more likely. Here $v_1\approx 0.5$ and $v_2=v_3=1$. (c) In the limit $v_1\to 0$ the cluster size distributions of species 1 and 2 become heavy tailed with a cutoff set by the system size. Here $v_1\le 0.007$ and $v_2=v_3=1$. For all plots $L=2048$.