There has been much interest in studying evolutionary games in structured populations, often modelled as graphs. However, most analytical results so far have only been obtained for two-player or additive games, while the study of more complex multiplayer games has been usually tackled by computer simulations. Here we investigate evolutionary multiplayer games in regular graphs updated with a Moran process. Using a combination of pair approximation and diffusion approximation, we obtain an analytical condition for cooperation to be favored by natural selection, given in terms of the payoffs of the game and a set of structure coefficients. We show that, for a large class of cooperative dilemmas, graph-structured populations are stronger promoters of cooperation than populations lacking spatial structure. Computer simulations validate our results, showing that the complexity arising from many-person social interactions and spatial structure can be often captured by analytical methods.
The mutation rate of a well adapted population is prone to reduction so as to have lower mutational load. The aim here is to understand the role of epistatic interactions in this process. Using a multitype branching process, the probability of fixation of a rare nonmutator in an asexual mutator population undergoing deleterious mutations at constant, but much higher rate than that of the nonmutator is analytically calculated here. We find that antagonistic epistasis lowers chances of mutation rate reduction, while synergistic epistasis enhances it. Below a critical value of epistasis, it can be seen that the fixation probability behaves nonmonotonically with variation in mutation rate of the background population for constant selection. Also, the variation of this critical value of epistasis parameter with the strength of the mutator is discussed. For synergistic epistasis, fixation probability shows a nonmonotonic trend with respect to selection when mutation rate is held constant.
On the stochastic evolution of finite populations
Fabio A. C. C. Chalub, Max O. Souza
This work is concerned with Markov chain models used in population genetics. Namely, we start from two celebrated models, the Moran and the Wright-Fisher processes, and study them from a very general viewpoint. Our aim is three fold: to identify the algebraic structures associated to time-homogeneous processes; to study the monotonicity properties of the fixation probability, with respect to the initial condition— in particular we show that there are situations, that are by no means exceptional, where an increase in the initial presence of a type can lead to a decrease in the fixation probability of this type; to understand time-inhomogeneous processes in a more systematic way. In addition, we also discuss the traditional identification of frequency dependent fitnesses and pay-offs, extensively used in evolutionary game theory, the role of weak selection when the population is finite, and the relations between jumps in evolutionary processes and frequency dependent fitnesses.