Zealotry promotes coexistence in the rock-paper-scissors model of cyclic dominance

Cyclic dominance models, such as the classic rock-paper-scissors (RPS) game, have found real-world applications in biology, ecology, and sociology. A key quantity of interest in such models is the coexistence time, i.e., the time until at least one population type goes extinct. Much recent research has considered conditions that lengthen coexistence times in an RPS model. A general finding is that coexistence is promoted by localized spatial interactions (low mobility), while extinction is fostered by global interactions (high mobility). That is, there exists a mobility threshold which separates a regime of long coexistence from a regime of rapid collapse of coexistence. The key finding of our paper is that if zealots (i.e., nodes able to defeat others while themselves being immune to defeat) of even a single type exist, then system coexistence time can be significantly prolonged, even in the presence of global interactions. This work thus highlights a crucial determinant of system survival time in cyclic dominance models.

Figure 1

Heat maps of (a) rock fraction $x$, (b) paper fraction $y$, and (c) scissors fraction $z$ vs the zealot fractions $r,p,s$. The top corner point of the simplex is the all-scissors point $(x,y,z)=(0,0,1)$, the bottom right corner point is the all-rock point $(1,0,0)$, and the bottom left corner point is the all-paper point $(0,1,0)$.

Figure 2

Phase plane view of the simplex for (a) $p=0$, (b) $p=0.2$, (c) $p=0.4$, and (d) $p=0.6$, with $r=s=0$. Red points indicate the corner points of the simplex or the coexistence point $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. The top corner point of the simplex is $(x,y,z)=(0,0,1)$, the bottom right corner point is $(1,0,0)$, and the bottom left corner point is $(0,1,0)$. Black outlined points indicate points to which at least some trajectories converge.

Figure 3

Phase plane view of the simplex for (a) $r=0.05$, (b) $r=0.15$, (c) $r=0.3$, and (d) $r=0.45$, with $p=0.2$ and $s=0$. Red points indicate the corner points of the simplex or the coexistence point $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. Black outlined points indicate points to which at least some trajectories converge.

Figure 4

Phase plane view of the simplex for (a) $p=0$, (b) $p=0.1$, (c) $p=0.2$, and (d) $p=0.4$, with $r=p=s$. Red points indicate the corner points of the simplex or the coexistence point $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. Black outlined points indicate points to which at least some trajectories converge. Note that the coexistence point is a spiral sink and the only steady-state point for $p>0$.

Figure 5

System survival time as a function of system size for various zealot fraction sizes $p$ for paper, with $r=s=0$.

Figure 6

System survival time vs $p$ for various system sizes $N$, with $r=s=0$. F.C. is an abbreviation for fully connected. The (sim) plots are obtained by simulating the system dynamics until extinction.

Figure 7

System survival time as a function of system size for various zealot fraction sizes $p$ for paper and $r$ for rock.

Figure 8

System survival time vs $p$ for various system sizes $N$ and zealot fraction $r=0.2,s=0$ or $r=0,s=0.2$.

Figure 9

Locations of initial conditions within the RPS simplex having various properties for (a) $p=0$, (b) $p=0.075$, (c) $p=0.15$, and (d) $p=0.225$. Corner refers to the simplex corner points, center refers to the point $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$, and coexist indicates the theoretical coexistence point found in Sec. 2c, namely, $(\frac{1-2p}{3},\frac{p+1}{3},\frac{p+1}{3})$. The other points are described in the text.

Figure 10

Density estimates showing distribution of natural log of survival time for $N=100$ for (a) $p=0$, (b) $p=0.075$, (c) $p=0.15$, and (d) $p=0.225$, based on simulations of 2000 runs, starting from the min-mean initial condition. Three open circles indicate the $0.1,0.5$, and $0.9$ quantiles.

Figure 11

Density estimates showing distribution of natural log of survival time for $N=100$ for (a) $p=0$, (b) $p=0.075$, (c) $p=0.15$, and (d) $p=0.225$, based on simulations of 2000 runs, starting from the max-mean initial condition. Three open circles indicate the $0.1,0.5$, and $0.9$ quantiles. Distribution for $p=0.15$ is maximally shifted towards larger survival times, compared to other values of $p$.

Figure 12

System survival time as a function of system size for various zealot fraction sizes $p$, with $r=s=0$, for a random geometric and Erdős Renyi graphs, compared with fully connected network (denoted by FC).

Figure 13

System survival time as a function of system size for various zealot fraction sizes $p$, with $r=s=0$, for lattices ($10\times 10$ and $13\times 13$ grid), compared with theoretical survival time for the corresponding fully connected topology (denoted by FC) of the same sizes.

Figure 14

Average minimum population size vs $p$ for various system sizes $N$, with $r=s=0$, for a $30\times 30$ lattice.