Fig. 3: Evolution of cooperation.
a Time average total frequency of cooperators, \({\rho }_{C}={\rho }_{C}^{1}+{\rho }_{C}^{2}\) in the r − cg plane is color plotted. The dynamics can settle in fixed point (FP) (small and large enhancement factors), or two different periodic orbits, red queen periodic orbit (RPO) where cooperation only in the costly institution evolve and black queen periodic orbit (BPO) where cooperation in both institutions evolve. b Time average difference between the probability that an individual in the costly institution is a cooperator from the probability that an individual in the free institution is a cooperator, \(\gamma ={\rho }_{C}^{1}/({\rho }_{C}^{1}+{\rho }_{D}^{1})-{\rho }_{C}^{2}/({\rho }_{C}^{2}+{\rho }_{D}^{2})\). Individuals are more likely to be cooperators in a costly institution. c The time average total frequency of cooperators in the r − cg plane under pure selection dynamic (ν = 0). Red queen and black queen periodic orbit can occur for, respectively, small and large enhancement factors. In other regions, the dynamics settle in a fixed point where either non-costly defectors (small enhancement factors), costly cooperators (inside the region marked with dashed black line), or non-costly cooperators survive. Parameter values: g = 5, and π0 = 2. In (a, b) ν = 10−3, and in (c) ν = 0. In (a, b) the replicator dynamic is solved for 8000 time steps, and the time average is taken over the last 2000 steps. In (c) the replicator dynamic is solved for 200,000 time steps, and the time average is taken over the last 150,000 time steps.
