Anisotropic invasion and its consequences in two-strategy evolutionary games on a square lattice

We have studied invasion processes in two-strategy evolutionary games on a square lattice for imitation rule when the players interact with their nearest neighbors. Monte Carlo simulations are performed for systems where the pair interactions are composed of a unit strength coordination game when varying the strengths of the self-dependent and cross-dependent components at a fixed noise level. The visualization of strategy distributions has clearly indicated that circular homogeneous domains evolve into squares with an orientation dependent on the composition. This phenomenon is related to the anisotropy of invasion velocities along the interfaces separating the two homogeneous regions. The quantified invasion velocities indicate the existence of a parameter region in which the invasions are opposite for the horizontal (or vertical) and the tilted interfaces. In this parameter region faceted islands of both strategies shrink and the system evolves from a random initial state into the homogeneous state that first percolated.

Figure 1

Evolution of the distribution of strategies 1 (black) and 2 (white) on the square lattice if the MC simulations are started from a state where the minority strategies form a circular domain (with a radius $r=100$) as illustrated in the upper two snapshots at $t=0$ MCS. The next two rows of snapshots show the distortion of domains (at $t=500$ MCS) that are shrinking continuously, as it is illustrated in the bottom snapshots at $t=4000$ MCS. These simulations are performed for $\delta =0.5, \varepsilon =-0.03, K=0.3$, and $L=400$.

Figure 2

The sufficiently large and initially circular white (strategy 2) and black (strategy 1) domains exhibit opposite behaviors during the evolutionary process at a high noise level ($K=3$) for the same payoff parameters we used in the simulations plotted in Fig. 1. The snapshots show the strategy distributions at times $t=0$ (upper), 1000 (middle), and 3000 MCS (lower).

Figure 3

Horizontal and tilted interfaces in the initial states used for quantifying the interfacial velocities.

Figure 4

Average vertical velocity of the horizontal (open boxes) and tilted (open diamonds) interfaces as a function of $\varepsilon$ for $\delta =0.5$ and $K=0.3$.

Figure 5

Boxes and diamonds indicate the values of $\delta$ and $\varepsilon$ where the velocity of the horizontal and tilted interfaces vanish for $K=0.3$. In the gray territory both homogeneous absorbing states can occur when varying the composition of the random initial strategy distribution.

Figure 6

Boxes and diamonds indicate the threshold values of $\delta$ and $\varepsilon$ where the velocity of the horizontal and tilted interfaces vanish. The filled and open symbols refer to interpolated data obtained for $K=0.1$ and 0.6, respectively. In order to illustrate the effect of noise, the boundary lines of the gray territory show the same results plotted in Fig. 5 at $K=0.3$.

Figure 7

Time-dependence of the portion of strategy 1 if the simulations are started from ${\rho }_1(t=0)=0.7$, 0.6, 0.55, 0.54, 0.53, and 0.5 (from top to bottom) for $\delta =0.5, \varepsilon =-0.03, K=0.3$, and $L=1000$. The plotted MC data are averaged over 100 runs.

Figure 8

Extinction probability of strategy 1 (solid lines) and 2 (dashed lines) vs. time for different system sizes (see the labels) for ${\rho }_1(t=0)=0.54, \delta =0.5, \varepsilon =-0.03$, and $K=0.3$. These curves are obtained by averaging over 1000 runs.