Fig 1.
Phenotypic diversity and contingent cooperation.
We consider variation in the capacity of expressing different phenotypes. G1 possesses only one expressible phenotype, say Red. G4 possesses four expressible phenotypes, say Red, Green, Blue and Yellow. Each individual just expresses one phenotype. G2 can express either Red or Blue, while G3 can express Red, Blue or Green. When G2 and G3 express the same phenotype, there will be an interaction (solid line) between them. When they express different phenotypes, there will be no interaction between them. The interaction outcome is dependent on their strategic behaviors. When both are cooperators, they each get the benefit b − c. When both are defectors, they get zero payoff each. When a cooperator meets a defector, the former gets the payoff −c, while the later reaps the payoff b.
Fig 2.
Transition rate means the probability that the population moves from an invaded state to an invading state. The capital letter, C or D, along the Y-axis, denotes the mutant’s behavioral strategy, while the one along the X-axis denotes the residents’ behavioral strategy. The coordinate value denotes the number of potentially expressible phenotypes that individuals are endowed with. Parameters: N = 20, b = 1, c = 0.3, β = 0.1, θ = 0.1.
Fig 3.
Stationary distribution for 2M competing strains.
Fraction of these 2M strains in the long run. The bars are obtained by solving the eigenvector of the 2M by 2M transition matrix. Empty circles and empty triangles are obtained by simulations. Blue denotes cooperator, and red defector. The abscissa value represents how many potentially expressible phenotypes individuals can switch to. The evolutionary process is fully characterized in the main text. Parameters: N = 20, b = 1, c = 0.3, β = 0.1, μ = 0.002. From A to F, θ is 0, 0.05, 0.1, 0.3, 0.5, 1, correspondingly, and the overall cooperation level is 0.91, 0.88, 0.84, 0.63, 0.49, and 0.29, respectively.
Fig 4.
Time evolution of the competition between cooperative strains and defective strains.
When the population is occupied by defective strain, it is most likely that the defective strain possesses a very small number of potentially expressible phenotypes. It is either followed by the invasion of defective strain with similar numbers of phenotypes, or by cooperative strain with a moderate number of phenotypes. In the former case, the evolutionary process advances just as it starts. In the later case, it gets very hard for the cooperative strain to be invaded, since it possesses the strongest resistance power against invasion of other strains. In the average sense, cooperative strain endowed with a moderate number of phenotypes prevails most of the time. Parameters: N = 20, b = 1, c = 0.3, β = 0.1, μ = 0.002, and θ = 0.1.
Fig 5.
Cooperation level, the optimal phenotypic diversity of cooperative strain as a function of θ and β, respectively.
The overall cooperation level decreases with θ. So does the optimal diversity. Even when cooperation is disfavored, the optimal diversity of all cooperative strains still exists. Quite differently, there exists an optimal selection intensity at which the overall cooperation level arrives at the highest and, correspondingly the optimum of diversity level is maximized. Parameters: N = 20, b = 1, c = 0.3. In panels A and B β = 0.1. In panels C and D θ = 0.1.