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Stochastic game dynamics under demographic fluctuations

Significance This contribution breaks with the tradition to restrict stochastic evolutionary game dynamics to populations of constant size and introduces a theoretical framework to investigate relevant and natural changes arising in populations that vary in size according to fitness—a feature common to many real biological systems. Explicitly including ecological variation can result in significant effects on the stochastic evolutionary trajectories while providing a transparent link to the established, deterministic Lotka–Volterra systems. Frequency-dependent selection and demographic fluctuations play important roles in evolutionary and ecological processes. Under frequency-dependent selection, the average fitness of the population may increase or decrease based on interactions between individuals within the population. This should be reflected in fluctuations of the population size even in constant environments. Here, we propose a stochastic model that naturally combines these two evolutionary ingredients by assuming frequency-dependent competition between different types in an individual-based model. In contrast to previous game theoretic models, the carrying capacity of the population, and thus the population size, is determined by pairwise competition of individuals mediated by evolutionary games and demographic stochasticity. In the limit of infinite population size, the averaged stochastic dynamics is captured by deterministic competitive Lotka–Volterra equations. In small populations, demographic stochasticity may instead lead to the extinction of the entire population. Because the population size is driven by fitness in evolutionary games, a population of cooperators is less prone to go extinct than a population of defectors, whereas in the usual systems of fixed size the population would thrive regardless of its average payoff.