Fixation probabilities of evolutionary coordination games on two coupled populations

Evolutionary forces resulted from competitions between different populations are common, which change the evolutionary behavior of a single population. In an isolated population of coordination games of two strategies (e.g., $s_1$ and $s_2$), the previous studies focused on determining the fixation probability that the system is occupied by only one strategy ($s_1$) and their expectation times, given an initial mixture of two strategies. In this work, we propose a model of two interdependent populations, disclosing the effects of the interaction strength on fixation probabilities. In the well-mixing limit, a detailed linear stability analysis is performed, which allows us to find and to classify the different equilibria, yielding a clear picture of the bifurcation patterns in phase space. We demonstrate that the interactions between populations crucially alter the dynamic behavior. More specifically, if the coupling strength is larger than some threshold value, the critical initial density of one strategy ($s_1$) that corresponds to fixation is significantly delayed. Instead, the two populations evolve to the opposite state of all ($s_2$) strategy, which are in favor of the red queen hypothesis. We delineate the extinction time of strategy ($s_1$) explicitly, which is an exponential form. These results are validated by systematic numerical simulations.

Figure 1

Illustrations of the bifurcation diagrams in $({\lambda }_P,{\lambda }_Q)$. Note that $(0,0)$ and $(1,1)$ always exist and are stable. (a) Fixed point $(0,1)$ is stable in the area of ${\lambda }_P<{\lambda }_P^{c_1}$ and ${\lambda }_Q<{\lambda }_Q^{c_1}$. The existence areas of fixed points (saddles) $(x^L\left(1\right),1)$ and $(1,y^L\left(1\right))$ are filled with diagonal continuous lines as indicated by legends, respectively. (b) Fixed point $(1,0)$ is stable in the area of ${\lambda }_P<{\lambda }_P^{c_2}$ and ${\lambda }_Q<{\lambda }_Q^{c_2}$. The existence areas of fixed points (saddles) $(x^L\left(0\right),0)$ and $(0,y^L\left(0\right))$ are filled with diagonal continuous lines as indicated by legends, respectively.

Figure 2

Schematic illustrations of the existence conditions [Eqs. (28) and (29)] of the unstable fixed point $(x^{♯},y^{♯})$ in the parameter space of $({\varepsilon }_1,{\varepsilon }_2)$, where ${\lambda }^{c_1},{\lambda }^{c_2}$ are defined by Eq. (30). (a) ${\lambda }_P{\lambda }_Q<1$ and (b) ${\lambda }_P{\lambda }_Q>1$.

Figure 3

Phase space depending on different coupling strength between two groups $P$ and $Q$. Dashed lines are nullclines $x^L\left(y\right)$ and $y^L\left(x\right)$. The velocity field is indicated by the arrows, showing the direction of evolution. Stable fixed points are denoted by filled dots, while unstable ones are represented by open circles. The relationship between $\mathrm{\Psi }$ and each stable state is the following: $\mathrm{\Psi }=0$ means that the system is attracted to ${\mathrm{\Phi }}_{00}$ with probability 1 (dark green); $\mathrm{\Psi }=1$ means that the system is attracted to ${\mathrm{\Phi }}_{10}$ with probability 1 (green); $\mathrm{\Psi }=2$ means that the system is attracted to ${\mathrm{\Phi }}_{01}$ with probability 1 (light green); and $\mathrm{\Psi }=3$ means that the system is absorbed into ${\mathrm{\Phi }}_{11}$ with probability 1 (yellow).

Figure 4

Fixation probabilities $p^Q$ to pure $s_1$ players (selection strength $\beta =100$). (a) Effects of coupling strength ${\lambda }_Q$ on $p^Q$. Legends are for various ${\lambda }_Q$ values. The critical values of the unstable point $y_{{\lambda }_Q}^L(x_0=0.6,{\lambda }_Q=0.8)\approx 0.47,y_{{\lambda }_Q}^L(x_0=0.6,{\lambda }_Q=1.0)\approx 0.45$ (when $P$ and $Q$ are coupled) are highlighted by two vertical dashed lines. Each dot is an average over $1\times {10}^5$ random realizations. (b) Fixation probabilities for an isolated $Q$ population, where an explosive transition is observed when $y_0>y^{*}\approx 0.53$.

Figure 5

Fixation probabilities $p^Q$ versus coupling strength ${\lambda }_Q$ in $Q$ population. Parameter settings are the same as Fig. 4. Four cases of initial number of $s_2$ players in $Q$ are denoted by $n=1,2,5,10$. The vertical dashed line corresponds to the critical coupling ${\lambda }_Q^{c_1}=0.9$. Continuous lines are from the corresponding theoretical predictions [Eq. (38)].

Figure 6

Effects of selection strength $\beta$ on the estimation of fixation probabilities $p^Q$ [Eq. (38)]. The parameters are the same as described in the caption of Fig. 5, besides ${\lambda }_Q=1.2$. (a) $p^Q$ vs. $\beta$ for various initial number of $s_2$ players in $Q$ population. (b) Estimation errors.

Figure 7

Fixation probabilities $p^Q$ to pure $s_1$ players in $Q$ population for various coupling strength ${\lambda }_Q$ as indicated by legends. (a) The initial density of $s_1$ players in $P$ population is $x_0=0.6$. The critical values of the unstable point $y_{{\lambda }_Q}^L(x_0=0.6,{\lambda }_Q=0.8)\approx 0.47,y_{{\lambda }_Q}^L(x_0=0.6,{\lambda }_Q=1.0)\approx 0.45$ are highlighted by two vertical dashed lines. Each dot is an average over $1\times {10}^5$ random realizations. (b) The same as (a) but the initial density of $s_1$ players in $P$ population is $x_0=0.4$.