Evolution of cooperation by phenotypic similarity

Abstract

The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much work in theoretical biology. Here, we study the evolution of cooperation in a model where individuals are characterized by phenotypic properties that are visible to others. The population is well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies can depend on distance in phenotype space. We study the interaction of cooperators and defectors. In our model, cooperators cooperate with those who are similar and defect otherwise. Defectors always defect. Individuals mutate to nearby phenotypes, which generates a random walk of the population in phenotype space. Our analysis brings together ideas from coalescence theory and evolutionary game dynamics. We obtain a precise condition for natural selection to favor cooperators over defectors. Cooperation is favored when the phenotypic mutation rate is large and the strategy mutation rate is small. In the optimal case for cooperators, in a one-dimensional phenotype space and for large population size, the critical benefit-to-cost ratio is given by b/c = 1 + 2/square root(3). We also derive the fundamental condition for any two-strategy symmetric game and consider high-dimensional phenotype spaces.

Fig. 1.

The basic geometry of evolution in phenotype space. There are two types of individuals (red and blue), which can refer to arbitrary traits or different strategies in an evolutionary game. Individuals inherit the strategy of their parent subject to a small mutation rate u. Moreover, each individual has a phenotype. Here, we consider a discrete one-dimensional phenotype space. An individual of phenotype i produces offspring of phenotype i − 1, i, or i + 1 with probabilities v, 1 − 2v, and v, respectively. The total population (of size N) performs a random walk in phenotype space with diffusion coefficient v. Sometimes the cluster breaks into two or more pieces, but typically only one of them survives. If evolutionary updating occurs according to a Wright–Fisher process then the distribution of individuals in phenotype space has a standard deviation of $\sqrt{2Nv}$. For the Moran process, the standard deviation is reduced to $\sqrt{Nv}$.

Fig. 2.

Random walks in phenotype space. Shown are two computer simulations of a Wright–Fisher process in a one-dimensional discrete phenotype space. The phenotypic mutation rate is v = 0.25. The colors, red and blue, refer to arbitrary traits, because no game is yet being played. All individuals have the same fitness. The population size is (Left) N = 10, and (Right) N = 100. The strategy mutation probability (between red and blue) is u = 0.004. Therefore, a given color dominates on average for 2/u = 500 generations (since new mutations arrive at rate Nu/2 and fixate with probability 1/N). The standard deviation of the distribution in phenotype space is $\sqrt{2Nv}$. Approximately 95% of all individuals are within 4 standard deviations. Often the population fragments into two or several pieces, but only one branch survives in the long run. We use the statistics of these neutral “phenotypic space walks” for calculating the fundamental conditions of evolutionary games in the limit of weak selection.

Fig. 3.

Excellent agreement between numerical simulations and analytic calculations. We show the critical benefit-to-cost ratio that is needed for cooperators to be more abundant than defectors in the stationary distribution. We have used a Wright–Fisher process with a phenotypic mutation rate v = 1/2 and a strategy mutation probability u = 1/(2N). The red line indicates the result of our analytic calculation. For these parameter values the asymptotic limit for large N is $b/c=(1+12\sqrt{2}/7)\approx 2.5672$. The red dots indicate the result of numerical simulations. The gray line illustrates the critical b/c-ratio for u → 0 with the asymptotic limit $b/c=1+2/\sqrt{3}\approx 2.1547$.