Coevolution of nodes and links: Diversity-driven coexistence in cyclic competition of three species

When three species compete cyclically in a well-mixed, stochastic system of $N$ individuals, extinction is known to typically occur at times scaling as the system size $N$. This happens, for example, in rock-paper-scissors games or conserved Lotka-Volterra models in which every pair of individuals can interact on a complete graph. Here we show that if the competing individuals also have a “social temperament” to be either introverted or extroverted, leading them to cut or add links, respectively, then long-living states in which all species coexist can occur. These nonequilibrium quasisteady states only occur when both introverts and extroverts are present, thus showing that diversity can lead to stability in complex systems. In this case, it enables a subtle balance between species competition and network dynamics to be maintained.

Figure 1

Schematic diagram showing the results of a node update in a $A_E{B}_E{C}_I$ QSS, showing the coevolutionary nature of the dynamics. Particles in the square boxes represent individuals of each node type, while those in circular urns represent links of the appropriate category. The dark urns (purple-online) represent E-E links, while the medium (green-online) and light colored (yellow-online) urns represent I-E and I-I links, respectively. If a node changes in, say, an $A+B\to 2A$ update, then a particle moves from box $B$ to box $A$, represented by the long thin dark gray (red-online) arrow. All the links that the changed node has also change, thus particles moves from urns $BA, BB$, and $BC$ to urns $AA, AB$, and $AC$, respectively, represented by the short thick dark gray (red-online) arrows. The medium gray (blue-online) and light gray (yellow-online) arrows indicate the particle moves resulting from $B+C\to 2B$ and $C+A\to 2C$ updates, respectively. By contrast, during a link update, at most, a single particle will be added to or removed from the appropriate urn.

Figure 2

Results from a typical $5\times {10}^4$ section of a very long (${10}^9$ MCS) run with $N=1000$ and $r=4/N$. (a) Population numbers of A (dark gray, blue online), B (medium gray, red online), and C (light gray, orange online). (b) Total self-links AA (dark gray, blue online), BB(medium gray, red online), and CC (light gray, orange online). (c) Total cross-links AB (dark gray, blue online), AC (medium gray, red online), and BC (light gray, orange online). (d) Same data for the cross-links but shown as a fraction of the maximum number possible. Note that the results in (d) are remarkably in phase.

Figure 3

Solution of the mean-field equation and fixed-point stability analysis, as a function of $\rho$. (a) Fixed point values of $\eta$, from top to bottom: C (green online), B (blue online), A (red online). (b) Fixed point values of $\lambda$. Dashed line is for all three types of cross-links. For the self-links, from top to bottom: BB (blue online), AA (red online), CC (green online). (c) Real part of the slowest eigenvalue, $\mu$.

Figure 4

Contour plot of the frequencies of the occurrence of $(A,B,C)$ during a very long (${10}^9$ MCS) run, representing (a section of) the quasistationary distribution, PQSS. The units in the legend are arbitrary; total number of occurrences being the length of the run: ${10}^9$. Also shown is a typical trajectory of length 5000 MCS (jagged line, gray, red online), which started near the center at t=90,022,001.