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Wrote the paper: CGN CET MAN. Conducted the research: CET, CGN, MAN.

The authors have declared that no competing interests exist.

Evolution is shaping the world around us. At the core of every evolutionary process is a population of reproducing individuals. The outcome of an evolutionary process depends on population structure. Here we provide a general formula for calculating evolutionary dynamics in a wide class of structured populations. This class includes the recently introduced “games in phenotype space” and “evolutionary set theory.” There can be local interactions for determining the relative fitness of individuals, but we require global updating, which means all individuals compete uniformly for reproduction. We study the competition of two strategies in the context of an evolutionary game and determine which strategy is favored in the limit of weak selection. We derive an intuitive formula for the structure coefficient, σ, and provide a method for efficient numerical calculation.

At the center of any evolutionary process is a population of reproducing individuals. The structure of this population can greatly affect the outcome of evolution. If the fitness of an individual is determined by its interactions with others, then we are in the world of evolutionary game theory. The population structure specifies who interacts with whom. We derive a simple formula that holds for a wide class of such evolutionary processes. This formula provides an efficient computational method for studying evolutionary dynamics in structured populations.

Constant selection implies that the fitness of individuals does not depend on the composition of the population. In general, however, the success of individuals is affected by what others are doing. Then we are in the realm of game theory

The classical approach to evolutionary game dynamics is based on deterministic differential equations describing infinitely large, well-mixed populations

Evolutionary graph theory is an extension of spatial games, which are normally studied on regular lattices, to general graphs

‘Games in phenotype space’

‘Evolutionary set theory’ represents another type of spatial model

In all three frameworks – evolutionary graph theory, games in phenotype space and evolutionary set theory – the fitness of individuals is a consequence of local interactions. In evolutionary graph theory there is also a local update rule: individuals learn from their neighbors on the graph or compete with nearby individuals for placing offspring. For evolutionary set theory, however,

Consider a game between two strategies,

Based on these interactions, individuals derive a cumulative payoff,

Reproduction is proportional to fitness but subject to mutation. With probability

A state of the population contains all information that can affect the payoffs of players. It assigns to each player a strategy (

For our proof we assume a finite state space and we study the Markov process defined by gameplay together with the update rule on this state space. The Markov process has a unique stationary distribution defined over all states.

It is shown in

For a large well-mixed population we obtain

Here we derive a formula for

These assumptions are fulfilled, for example, by games in phenotype space

For each state of the system, let

Suppose there is a ‘spatial’ process which has two mixed states. These two states must have the same frequency in the stationary distribution at neutrality, because the process cannot introduce asymmetries between

This formula suggests a simple numerical algorithm for calculating the

The rigorous proof of eq (3) is given in Appendix A; here we provide an intuition for it. For symmetry reasons, at neutrality, we have the following identities

A standard replicator equation for deterministic evolutionary game dynamics of two strategies in a well-mixed population can be written as

As a particular game we can study the evolution of cooperation. Consider the simplified Prisoner's Dilemma payoff matrix:

As shown in

From eqs (3) and (7) we can write

Our new formula for

In this section we use the simple numerical algorithm suggested by our formula (3) to find

We test our simulation procedure against the analytic results of the set model of

An individual can be in 1, 2, or 3 sets; when he mutates set membership, the number of sets he joins is drawn with uniform probability. Parameter values are

The original set-structured model describes a population of

To obtain exact analytical calculations, it is assumed that each individual belongs to exactly

In

It has been shown that evolutionary dynamics in a structured population can be described by a single parameter,

Here we provide a general formula for the

Our main result, eq (3), provides both an intuitive description of what the

Here we give the proof of equation (3). It is based on the following three claims which we prove in the next subsection:

First, we show that for structures and update rules with either constant death rate or constant birth rate the condition

We show that for global updating, condition (11) is equivalent to

Finally we claim that, in the limit of weak selection, for structures satisfying global updating and constant death or birth, the difference between the birth rate and death rate of an individual

Combining the three claims, we conclude that condition (10) is equivalent to

By assumption, either birth or death has a fixed rate; assume without loss of generality that death is constant with rate

As in

Again, we assume without loss of generality that the death rate is constant, equal to

We would like to thank Tibor Antal and Dave Rand for useful discussions.