Chimera states in network-organized public goods games with destructive agents
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It is shown that a network-organized metapopulation of cooperators, defectors and destructive agents playing the public goods game with mutations, can collectively reach global synchronization or chimera states. Global synchronization is accompanied by a collective periodic burst of cooperation, whereas chimera states reflect the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation. Numerical simulations have shown that the system’s dynamics alternates between these two steady states through a first order transition. Depending on the parameters determining the dynamical and topological properties, chimera states with different numbers of coherent and incoherent clusters are observed. These results present (to my knowledge) the first systematic study of chimera states and their characterization in the context of evolutionary game theory. This provides a valuable insight into the details of their occurrence, extending the relevance of such states to natural and social systems. The evolution of the three strategies ( where This dynamical system has a Hopf bifurcation beyond which self-sustained oscillations emerge. They correspond to periodic bursting oscillations of cooperation. When the populations playing the public goods game (locally, in each node) are connected with a nonlocal ring networks as described by the equations then chimera states can emerge, reflecting the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation.
Further reading:
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: cooperators, 
: defectors, 
: destructive agents) is described by the replicator-mutator equations![\dot{x} =x (P_x -\bar{P}) + \mu (1 - 3 x)\,\\[5pt]\dot{y} =y (P_y -\bar{P}) + \mu (1 - 3 y) \,\\[5pt]\dot{z} = z (P_z -\bar{P})+ \mu (1 - 3 z)](http://s0.wp.com/latex.php?latex=%5Cdot%7Bx%7D+%3Dx+%28P_x+-%5Cbar%7BP%7D%29+%2B+%5Cmu+%281+-+3+x%29%5C%2C%5C%5C%5B5pt%5D%5Cdot%7By%7D+%3Dy+%28P_y+-%5Cbar%7BP%7D%29+%2B+%5Cmu+%281+-+3+y%29+%5C%2C%5C%5C%5B5pt%5D%5Cdot%7Bz%7D+%3D+z+%28P_z+-%5Cbar%7BP%7D%29%2B+%5Cmu+%281+-+3+z%29+&bg=ffffff&fg=000&s=0 "\dot{x} =x (P_x -\bar{P}) + \mu (1 - 3 x)\,\\[5pt]\dot{y} =y (P_y -\bar{P}) + \mu (1 - 3 y) \,\\[5pt]\dot{z} = z (P_z -\bar{P})+ \mu (1 - 3 z)")
is the average payoff and![P_x = r \frac{x}{1-z} \left[ 1 - \frac{1-z^n}{n(1-z)} \right] + \frac{r}{n} \frac{1-z^n}{1-z} - 1 - d \left( \frac{1-z^n}{1-z} - 1\right)\,,\\[5pt] P_y = P_x + 1 - \frac{r}{n} \frac{1-z^n}{1-z}\,,\\[5pt] P_z = 0](http://s0.wp.com/latex.php?latex=P_x+%3D+r+%5Cfrac%7Bx%7D%7B1-z%7D+%5Cleft%5B+1+-+%5Cfrac%7B1-z%5En%7D%7Bn%281-z%29%7D+%5Cright%5D+%2B+%5Cfrac%7Br%7D%7Bn%7D+%5Cfrac%7B1-z%5En%7D%7B1-z%7D+-+1+-+d+%5Cleft%28+%5Cfrac%7B1-z%5En%7D%7B1-z%7D+-+1%5Cright%29%5C%2C%2C%5C%5C%5B5pt%5D+P_y+%3D+P_x+%2B+1+-+%5Cfrac%7Br%7D%7Bn%7D+%5Cfrac%7B1-z%5En%7D%7B1-z%7D%5C%2C%2C%5C%5C%5B5pt%5D+P_z+%3D+0+&bg=ffffff&fg=000&s=0 "P_x = r \frac{x}{1-z} \left[ 1 - \frac{1-z^n}{n(1-z)} \right] + \frac{r}{n} \frac{1-z^n}{1-z} - 1 - d \left( \frac{1-z^n}{1-z} - 1\right)\,,\\[5pt] P_y = P_x + 1 - \frac{r}{n} \frac{1-z^n}{1-z}\,,\\[5pt] P_z = 0")
![\dot{x}_i =x_i (P_{x,i} -\bar{P}_i) + \mu (1 - 3 x_i)+ \frac{\sigma}{2R} \sum_{j=i-R}^{j=i+R}\! (x_j - x_i)\,,\\[5pt]\dot{y}_i = y_i (P_{y,i} -\bar{P}_i)+ \mu (1 - 3 y_i) + \frac{\sigma}{2R} \sum_{j=i-R}^{j=i+R}\! (y_j - y_i)\,,\\[5pt]z_i = 1 - x_i - y_i](http://s0.wp.com/latex.php?latex=%5Cdot%7Bx%7D_i+%3Dx_i+%28P_%7Bx%2Ci%7D+-%5Cbar%7BP%7D_i%29+%2B+%5Cmu+%281+-+3+x_i%29%2B+%5Cfrac%7B%5Csigma%7D%7B2R%7D+%5Csum_%7Bj%3Di-R%7D%5E%7Bj%3Di%2BR%7D%5C%21+%28x_j+-+x_i%29%5C%2C%2C%5C%5C%5B5pt%5D%5Cdot%7By%7D_i+%3D+y_i+%28P_%7By%2Ci%7D+-%5Cbar%7BP%7D_i%29%2B+%5Cmu+%281+-+3+y_i%29+%2B+%5Cfrac%7B%5Csigma%7D%7B2R%7D+%5Csum_%7Bj%3Di-R%7D%5E%7Bj%3Di%2BR%7D%5C%21+%28y_j+-+y_i%29%5C%2C%2C%5C%5C%5B5pt%5Dz_i+%3D+1+-+x_i+-+y_i+&bg=ffffff&fg=000&s=0 "\dot{x}_i =x_i (P_{x,i} -\bar{P}_i) + \mu (1 - 3 x_i)+ \frac{\sigma}{2R} \sum_{j=i-R}^{j=i+R}\! (x_j - x_i)\,,\\[5pt]\dot{y}_i = y_i (P_{y,i} -\bar{P}_i)+ \mu (1 - 3 y_i) + \frac{\sigma}{2R} \sum_{j=i-R}^{j=i+R}\! (y_j - y_i)\,,\\[5pt]z_i = 1 - x_i - y_i")