Evolutionary dynamics in structured populations

Abstract

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called 'spatial selection': cooperators prevail against defectors by clustering in physical or other spaces.

Figure 1.

In evolutionary graph theory, the individuals of a population occupy the vertices of a graph. The edges denote who interacts with whom—both for accumulating payoff and for reproductive competition. Here, we consider two strategies, A (blue) and B (red). Evolutionary dynamics on graphs depend on the update rule. In this paper, we use death–birth updating: a random individual dies; the neighbours compete for the empty site proportional to fitness.

Figure 2.

We study games in a one-dimensional, discrete phenotype space. The phenotype of an individual is given by an integer i. The offspring of this individual has phenotype i − 1, i, i + 1 with probabilities v, 1 − 2v, v, where v is the phenotypic mutation rate. Offspring also inherit the strategy of their parent (red or blue) with a certain mutation rate. Each individual interacts with others who have the same phenotype and thereby derives a payoff. The population drifts through phenotype space. Strategies tend to cluster. For evolution of cooperation, this model represents a very simple scenario of tag-based cooperation (or ‘Green beard’ effects).

Figure 3.

In evolutionary set theory, the individuals of a population are distributed over sets. Individuals interact with others who are in the same set. If two individuals share several sets, they interact several times. The interactions lead to payoff in terms of an evolutionary game. Strategies and set memberships of successful individuals are imitated. There is a strategy mutation rate and a set mutation rate. The population structure becomes effectively well mixed if the set mutation rate is too low or too high. There is an intermediate set mutation rate which maximizes the clustering of individuals according to strategies. Evolutionary set theory is a dynamical graph theory. The population structure changes as a consequence of evolutionary updating.