Interfaces with internal structures in generalized rock-paper-scissors models

In this work we investigate the development of stable dynamical structures along interfaces separating domains belonging to enemy partnerships in the context of cyclic predator-prey models with an even number of species $N\ge 8$. We use both stochastic and field theory simulations in one and two spatial dimensions, as well as analytical arguments, to describe the association at the interfaces of mutually neutral individuals belonging to enemy partnerships and to probe their role in the development of the dynamical structures at the interfaces. We identify an interesting behavior associated with the symmetric or asymmetric evolution of the interface profiles depending on whether $N/2$ is odd or even, respectively. We also show that the macroscopic evolution of the interface network is not very sensitive to the internal structure of the interfaces. Although this work focuses on cyclic predator-prey models with an even number of species, we argue that the results are expected to be quite generic in the context of spatial stochastic May-Leonard models.

Figure 1

Illustration of the cyclic predator-prey rule in the case with eight species.

Figure 2

The ${512}^2$ stochastic network simulations of models with $N=4$ (top left), $N=6$ (top right), $N=8$ (bottom left), and $N=10$ (bottom right). The snapshots were taken after 5000 generations.

Figure 3

The ${512}^2$ mean field theory simulations of models with $N=4$ (top left), $N=6$ (top right), $N=8$ (bottom left), and $N=10$ (bottom right). The snapshots were taken after 5000 generations.

Figure 4

Time evolution of ${\varphi }_0$ (solid black line) and ${\varphi }_i$ (colored lines) at the interface for the model with four species. No prominent dynamical structures are formed at the interfaces in this case.

Figure 5

Time evolution of ${\varphi }_0$ (solid black line) and ${\varphi }_i$ (colored lines) at the interface for the model with six species. Transient dynamical structures are formed at the interfaces, but they are rapidly damped.

Figure 6

Time evolution of ${\varphi }_0$ (solid black line) and ${\varphi }_i$ (colored lines) at the interface for the model with eight species. The oscillation amplitudes of ${\varphi }_i$ increase until they reach a constant value, indicating the stability of the dynamical structures at the interface.

Figure 7

Time evolution of ${\varphi }_0$ (solid black line) and ${\varphi }_i$ (colored lines) for the model with ten species. Similar to the $N=8$ case, the results indicate the stability of the dynamical structures at the interface. However, the oscillatory behavior of ${\varphi }_0$ is much more pronounced for $N=10$ than for $N=8$.

Figure 8

Interface profiles ${\varphi }_0\left(x\right)$ for different values of the time $t$, showing a symmetric or asymmetric evolution, depending on whether $N/2$ is odd [second ($N=10$) and fourth ($N=14$) panels] or even [first ($N=8$) and third ($N=12$) panels], respectively.

Figure 9

Late time behavior of the length scale $L$, computed from ensembles of twenty ${1024}^2$ (top) stochastic network and (bottom) mean field theory simulations, for models with $N=4$, 6, 8, and 10 species.