Fig 1.
Time series of the fraction of cooperators for different values of σ when r equals to 5.
The value of σ is equal to 0.0,0.2,0.5,0.8,1.0 respectively. While σ = 0, which is exactly the traditional case, the investment of all the cooperators is 1. With the increment of σ, the asymptotic fraction of cooperators will increase.
Fig 2.
Relationship between asymptotic fraction of cooperators ρC and enhancement factor r corresponding to σ = 0, 0.2, 0.5, 0.8, 1.0 respectively.
The curves in the figure show that the larger the value of σ is, the bigger the asymptotic fraction of cooperators ρC will be. In addition, larger value of σ can decrease the values of rC and rD effectively.
Fig 3.
Typical snapshots of strategy distributions on the square lattice when σ = 0.5 and r = 5.
Cooperators and defectors are colored red and blue, and the MCS of (a)-(d) is 1, 10, 100, 50000 respectively. The figure shows that the fraction of cooperators decreases at the beginning of the evolution, but as the evolution proceeds, the cooperators form into clusters to restrain the invasion of the defectors, and spread to the defectors reversely. At the end of the evolution all the players in the population hold the cooperation strategy.
Fig 4.
Relationship between asymptotic fraction of cooperators ρC and enhancement factor r corresponding to σ = 0, 0.2, 0.5, 0.8, 1.0 respectively when the coevolution of strategy and investment is taken into account.
The curves show that if σ>0, the thresholds rC and rD decrease compared to the case of σ = 0, which is exactly the traditional situation. This implies that cooperation can be promoted by introducing the coevolution of strategy and investment.
Fig 5.
The steady distribution of investment corresponding to strategy evolution only (Left panel) and that corresponding to coevolution of strategy and investment (Right panel), when σ = 0.8, r = 5.2.
From the figure, it can be found that when only strategy is updated, the distribution of the investment in steady state is still a uniform distribution [0.2,1.8], however, when the coevolution of strategy and investment is taken into account, the previous uniform distribution is severely distorted, and the steady investment satisfies a discrete distribution valued [1.786,1.790,1.792,1.793,1.795,1.797,1.798,1.799]. In conclusion, compared to the former case, as the heterogeneity of investment is weakened, the asymptotic fraction of cooperators ρC decreases in the case of coevolution.