Characterization of spiraling patterns in spatial rock-paper-scissors games

The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising “rock-paper-scissors” interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial “rock-paper-scissors” competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.

Figure 1

Cartoon of the metapopulation model: $L\times L$ patches (or islands) are arranged on a periodic square lattice (of linear size $L)$. Each patch $\ell =({\ell }_1,{\ell }_2)$ can accommodate at most $N$ individuals of species $S_1,S_2,S_3$ and empty spaces denoted Ø. Each patch consists of a well-mixed population of $N_{S_1}$ [red (gray)] individuals of species $S_1,N_{S_2}$ [green (light gray)] of type $S_2,N_{S_3}$ [blue (dark gray)] of type $S_3$ and $N_{Ø}=N-N_{S_1}-N_{S_2}-N_{S_3}$ (black) empty spaces. The composition of a patch evolves in time according to the processes (1) and (2). Furthermore, migration from the focal patch (dark gray) to its four nearest neighbors (light gray) occurs according to the processes (4), see text.

Figure 2

Comparison of lattice simulations (performed using a spatial Gillespie algorithm [37]) with solutions of (5) in the bound state phase (BS), where the spiral waves are stable, near the HB point, see text of Sec. 3. Rightmost panels show the solutions of (5) while the remaining panels show results of stochastic simulations for $L^2={128}^2$ with $N=4$, 16, 64, 256, 1024 (from left to right). As in all other figures, each color (level of gray) represents one species with black dots indicating low density regions. Top panels show initial conditions while the lower panels show the domains at $t=1000$. The other parameters are $\beta =\sigma ={\delta }_D={\delta }_E=1,\zeta =0.6$, and $\mu =0.02$.

Figure 3

Four phases in the two-dimensional CGLE (6) for $c=(2.0,1.5,1.0,0.5)$ from left to right. Spiral waves of the third panel (from the left) are stable while the others are unstable, see Sec. 3. Here, the colors represent the argument of $\mathcal{A}$ encoded in hue: red (gray), green (light gray), and blue (dark gray), respectively, correspond to arguments 0, $\pi /3$, and $2\pi /3$.

Figure 4

Upper panels: Typical snapshots of the phases AI, EI, BS, and SA (from left to right) as obtained from (5) (top row) and from lattice simulations (middle row) with parameters $\sigma =\beta ={\delta }_E={\delta }_D=1,\mu =0.02,L=128,N=64$ and, from left to right, $\zeta =(1.8,1.2,0.6,0)$. The corresponding values of the CGLE parameter (7) are $c\approx (1.94,1.47,1.01,0.63)$. Lower panel: Phase diagram of the two-dimensional RPS system around the Hopf bifurcation with contours of $c=(c_{\mathrm{AI}},c_{\mathrm{EI}},c_{\mathrm{BS}})$ in the $\sigma \text{−}\zeta$ plane, see text. We distinguish four phases: spiral waves are unstable in AI, EI, and SA phases, while they are stable in BS phase. The boundaries between the phases have been obtained using (7), see Refs. [19, 20] for details.

Figure 5

Leftmost: Domain of size ${512}^2$ cut out from a numerical solution of (5) with $\beta =\sigma ={\delta }_D={\delta }_E=1,\zeta =0.3,\mu =0.02$, and $L^2={1024}^2$. The yellow frame outlines domain of size ${128}^2$ enlarged in the middle panel. Middle: Part of a spiral arm (far from the core) resembling a plane wave enlarged from the left panel. The color (gray color) depth of the right half of the image was reduced to 256 colors (levels of gray) for an easy identification of the wavelength found to be equal to 71 length units in the physical domain as measured by the yellow bar. Rightmost: Same as in the middle panel from lattice simulations with $N=64$.

Figure 6

Numerical values of ${\left|\mathcal{A}\right|}^2$ obtained from a histogram with 1000 bins (squares) and averaging (circles) with interpolation (dashed). When the traveling wave ansatz is valid (in BS and EI phases, away from the spirals' cores), ${\left|\mathcal{A}\right|}^2\approx R^2$, see text. Solid line is the theoretical Eckhaus criterion (11) obtained from the plane-wave ansatz yielding $c_{\mathrm{EI}}\approx 1.28$ marked by the dotted line. This has to be compared with the value of $c_{\mathrm{EI}}\approx 1.25$ reported in the phase diagram of the two-dimensional CGLE [29]. Spiral waves are convectively unstable in the region where $c>c_{\mathrm{EI}}$ and are stable just below that value in the BS phase, see Sec. 3b.

Figure 7

Wavelength $(\circ )$ and (rescaled) velocity $(\diamond )$ obtained from the CGLE (6) with $\delta =1$ as functions of the parameter $c=0.6$–1.5. The critical values $c_{\mathrm{BS}}$ and $c_{\mathrm{EI}}$ separating the SA and BS phases and the BS and EI phases are indicated by thin vertical dotted lines, see text.

Figure 8

Space and time development of a spiral wave solution of the CGLE (6) with $c=1.5$ and $\delta =1$ in the EI phase (argument of $\mathcal{A}$ encoded in grayscale): At time $t=700$ the spiral wave propagates with a wavelength ${\lambda }_{\mathrm{CGLE}}\approx 13.7$ (left). Subsequently, the arms start to deform $(t=800$, middle) and then a far-field breakup, due to a convective Eckhaus instability, occurs causing the spiral arms to break into an intertwining of smaller spirals $(t=900$, right), see text.

Figure 9

Wavelengths of well-developed spiral wave solutions of the CGLE (6) with $\delta =1$ in the BS and EI phases (argument of $\mathcal{A}$ encoded in grayscale). Here, the wavelengths are measured by counting pixels. Left: $c=1.0$ and spirals are stable (BS phase). The measured wavelength is $20.2$ and compares well with the theoretical predictions ${\lambda }_{\mathrm{CGLE}}\approx 20.3$ obtained from (9) with ${\left|\mathcal{A}\right|}^2\approx R^2$ measured as $0.904$. Right: $c=1.5$ and spirals waves are in the EI phase, but their arms are still unperturbed. The Eckhaus instability will cause a far-field breakup further away from the core (not shown here, see text and Fig. 8). The measured wavelength of $13.8$ is in excellent agreement with ${\lambda }_{\mathrm{CGLE}}\approx 13.7$ from (9) with ${\left|\mathcal{A}\right|}^2\approx R^2$ measured as $0.791$.

Figure 10

Staggered decay of the total core area in the solutions of the CGLE (6) with $c=0.4$ and $\delta =1$. The initial condition consists of perturbations around ${\left|\mathcal{A}\right|}^2=0$. Here, after initial transients, 10 spirals remain with a total core area of approximately 120 pixels. Subsequently, further five annihilations occur marked by the sharp decreases in the total core area until the disappearance of all spirals.

Figure 11

Spiral annihilation in the solutions of the CGLE (6) with $c=0.1$ and $\delta =1$. The square modulus ${\left|\mathcal{A}\right|}^2$ is visualized here with dark pixels representing ${\left|\mathcal{A}\right|}^2\approx 0$ while light pixels show regions where ${\left|\mathcal{A}\right|}^2\approx 1$. Snapshots are taken at times $t=(1800,2000,2200,2400,2600)$ from left to right.

Figure 12

Spatial arrangements in the EI (left) and AI (center, right) phases as obtained from lattice simulations near the Hopf bifurcation. Parameters are $\sigma =\beta ={\delta }_E={\delta }_D=1,\mu =0.02,L=128,N=64$, with $\zeta =1.2$ in the EI phase (left) and $\zeta =(1.8,2.4)$ in the AI phase (center, right). While the spatial arrangement is still characterized by (deformed) spiraling patterns in the EI phase, no spiraling arms can develop in the AI phase resulting in an incoherent spatial structure.

Figure 13

Four phases away from the HB (low mutation rate). Results of lattice simulations at low mutation rate $\mu =0.001\ll {\mu }_H\approx 0.042$ (far away from the Hopf bifurcation) and with all the other parameters kept at same values as in Fig. 4. One recognizes the AI, EI, and BS phases (from left to right) while the spiral annihilation in the SA phase (rightmost panel) are no longer observed on the same length scales and time scales as in Fig. 4, see text.

Figure 14

Dependence of ${\lambda }_{\mathrm{CGLE}}=\epsilon \lambda$ on the vanishing mutation rate $\mu$ for various values of $\zeta$: Near the Hopf bifurcation $\mu \lesssim {\mu }_H\approx 0.042$, the wavelengths $(\square )$ are obtained from the CGLE according to (9). For lower values of $\mu$, the wavelengths $(\circ )$ are measured in the solutions of (5), see text. When $\mu \to 0,\lambda$ approaches a value $\tilde{\lambda }(\sigma ,{\delta }_D,{\delta }_E)$, see text. Parameters are as follows: $\sigma =\beta ={\delta }_E={\delta }_D=1$.

Figure 15

Effects of nonlinear mobility on spiraling patterns at zero mutation rate for various values of ${\delta }_D$ at ${\delta }_E$ fixed. Lattice simulations for the metapopulation model with $N=256,L^2={512}^2,\zeta =\mu =0,\sigma =\beta =1,{\delta }_E=0.5$, and ${\delta }_D=(0.5,1,1.5,2)$ from left to right. Spiral waves are stable and form geometric patterns when ${\delta }_D={\delta }_E$ (leftmost, linear diffusion), and Eckhaus-like instability occurs when ${\delta }_D>{\delta }_E$ and cause their far-field breakup, resulting in a disordered intertwining of small spiraling patterns of short wavelengths, see text.

Figure 16

Raising $\sigma$ away from HB cause instability: Lattice simulations for the metapopulation model with $N=64,L^2={512}^2,\zeta =\mu =0,\beta =1,{\delta }_D={\delta }_E=0.5$, and $\sigma =(1,2,3,4)$ from left to right. While the spiral waves are stable and form a geometrically ordered when $\sigma =1$ (leftmost panel), Eckhaus-like instability occurs when $\sigma$ is raised and cause their far-field breakup (middle panels). When $\sigma =4$, the ordered spiraling patterns is entirely disintegrated and replaced by a disordered intertwining of spirals of small size and short wavelengths (rightmost panel), see text.