Abstract:
The stage of evolution is the population of reproducing individuals. The structure of the population is know to affect the dynamics and outcome of evolutionary processes, but analytical results for generic random structures have been lacking. The most general result so far, the isothermal theorem, assumes the propensity for change in each position is exactly the same, but realistic biological structures are always subject to variation and noise. We consider a population of finite size $n$ under constant selection whose structure is given by a wide variety of weighted, directed, random graphs; vertices represent individuals and edges interactions between individuals. By establishing a robustness result for the isothermal theorem and using large deviation estimates to understand the typical structure of random graphs, we prove that for a generalization of the Erd\H{o}s-R\'{e}nyi model the fixation probability of an invading mutant is approximately the same as that of a mutant of equal fitness in a well-mixed population with high probability. Simulations of perturbed lattices, small-world networks, and scale-free networks behave similarly. We conjecture that the fixation probability in a well-mixed population, $(1-r^{-1})/(1-r^{-n})$, is universal: for many random graph models, the fixation probability approaches the above function uniformly as the graphs become large.

Abstract:
Stable mixtures of cooperators and defectors are often seen in nature. This fact is at odds with predictions based on linear public goods games under weak selection. That model implies fixation either of cooperators or of defectors, and the former scenario requires a level of group relatedness larger than the cost/benefit ratio, being therefore expected only if there is either kin recognition or a very low cost/benefit ratio, or else under stringent conditions with low gene flow. This motivated us to study here social evolution in a large class of group structured populations, with arbitrary multi-individual interactions among group members and random migration among groups. Under the assumption of weak selection, we analyze the equilibria and their stability. For some significant models of social evolution with non-linear fitness functions, including contingent behavior in iterated public goods games and threshold models, we show that three regimes occur, depending on the migration rate among groups. For sufficiently high migration rates, a rare cooperative allele A cannot invade a monomorphic population of asocial alleles N. For sufficiently low values of the migration rate, allele A can invade the population, when rare, and then fixate, eliminating N. For intermediate values of the migration rate, allele A can invade the population, when rare, producing a polymorphic equilibrium, in which it coexists with N. The equilibria and their stability do not depend on the details of the population structure. The levels of migration (gene flow) and group relatedness that allow for invasion of the cooperative allele leading to polymorphic equilibria with the non-cooperative allele are common in nature.

Abstract:
Natural selection and random drift are competing phenomena for explaining the evolution of populations. Combining a highly fit mutant with a population structure that improves the odds that the mutant spreads through the whole population tips the balance in favor of natural selection. The probability that the spread occurs, known as the fixation probability, depends heavily on how the population is structured. Certain topologies, albeit highly artificially contrived, have been shown to exist that favor fixation. We introduce a randomized mechanism for network growth that is loosely inspired in some of these topologies' key properties and demonstrate, through simulations, that it is capable of giving rise to structured populations for which the fixation probability significantly surpasses that of an unstructured population. This discovery provides important support to the notion that natural selection can be enhanced over random drift in naturally occurring population structures.

Abstract:
Analysis of HIV-1 gene sequences sampled longitudinally from infected individuals can reveal the evolutionary dynamics that underlie associations between disease outcome and viral genetic diversity and divergence. Here we extend a statistical framework to estimate rates of viral molecular adaptation by considering sampling error when computing nucleotide site-frequencies. This is particularly beneficial when analyzing viral sequences from within-host viral infections if the number of sequences per time point is limited. To demonstrate the utility of this approach, we apply our method to a cohort of 24 patients infected with HIV-1 at birth. Our approach finds that viral adaptation arising from recurrent positive natural selection is associated with the rate of HIV-1 disease progression, in contrast to previous analyses of these data that found no significant association. Most surprisingly, we discover a strong negative correlation between viral population size and the rate of viral adaptation, the opposite of that predicted by standard molecular evolutionary theory. We argue that this observation is most likely due to the existence of a confounding third variable, namely variation in selective pressure among hosts. A conceptual non-linear model of virus adaptation that incorporates the two opposing effects of host immunity on the virus population can explain this counterintuitive result.

Abstract:
Extortion strategies can dominate any opponent in an iterated prisoner's dilemma game. But if players are able to adopt the strategies performing better, extortion becomes widespread and evolutionary unstable. It may sometimes act as a catalyst for the evolution of cooperation, and it can also emerge in interactions between two populations, yet it is not the evolutionary stable outcome. Here we revisit these results in the realm of spatial games. We find that pairwise imitation and birth-death dynamics return known evolutionary outcomes. Myopic best response strategy updating, on the other hand, reveals new counterintuitive solutions. Defectors and extortioners coarsen spontaneously, which allows cooperators to prevail even at prohibitively high temptations to defect. Here extortion strategies play the role of a Trojan horse. They may emerge among defectors by chance, and once they do, cooperators become viable as well. These results are independent of the interaction topology, and they highlight the importance of coarsening, checkerboard ordering, and best response updating in evolutionary games.

Abstract:
The emergence and maintenance of cooperative behavior is a fascinating topic in evolutionary biology and social science. The public goods game (PGG) is a paradigm for exploring cooperative behavior. In PGG, the total resulting payoff is divided equally among all participants. This feature still leads to the dominance of defection without substantially magnifying the public good by a multiplying factor. Much effort has been made to explain the evolution of cooperative strategies, including a recent model in which only a portion of the total benefit is shared by all the players through introducing a new strategy named persistent cooperation. A persistent cooperator is a contributor who is willing to pay a second cost to retrieve the remaining portion of the payoff contributed by themselves. In a previous study, this model was analyzed in the framework of well-mixed populations. This paper focuses on discussing the persistent cooperation in lattice-structured populations. The evolutionary dynamics of the structured populations consisting of three types of competing players (pure cooperators, defectors and persistent cooperators) are revealed by theoretical analysis and numerical simulations. In particular, the approximate expressions of fixation probabilities for strategies are derived on one-dimensional lattices. The phase diagrams of stationary states, the evolution of frequencies and spatial patterns for strategies are illustrated on both one-dimensional and square lattices by simulations. Our results are consistent with the general observation that, at least in most situations, a structured population facilitates the evolution of cooperation. Specifically, here we find that the existence of persistent cooperators greatly suppresses the spreading of defectors under more relaxed conditions in structured populations compared to that obtained in well-mixed population.

Abstract:
A stochastic evolutionary dynamics of two strategies given by 2 x 2 matrix games is studied in finite populations. We focus on stochastic properties of fixation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with arbitrary hopping rates. The time of fixation is found to be identical for both strategies in any particular game. The asymptotic behavior of the fixation time and fixation probabilities in the large population size limit is also discussed. We show that fixation is fast when there is at least one pure evolutionary stable strategy (ESS) in the infinite population size limit, while fixation is slow when the ESS is the coexistence of the two strategies.

Abstract:
In subdivided populations, migration acts together with selection and genetic drift and determines their evolution. Building up on a recently proposed method, which hinges on the emergence of a time scale separation between local and global dynamics, we study the fixation properties of subdivided populations in the presence of balancing selection. The approximation implied by the method is accurate when the effective selection strength is small and the number of subpopulations is large. In particular, it predicts a phase transition between species coexistence and biodiversity loss in the infinite-size limit and, in finite populations, a nonmonotonic dependence of the mean fixation time on the migration rate. In order to investigate the fixation properties of the subdivided population for stronger selection, we introduce an effective coarser description of the dynamics in terms of a voter model with intermediate states, which highlights the basic mechanisms driving the evolutionary process.

Abstract:
We propose a model for the frequency of an altruistic defense trait. More precisely, we consider Lotka-Volterra-type models involving a host/prey population consisting of two types and a parasite/predator population where one type of host individuals (modeling carriers of a defense trait) is more effective in defending against the parasite but has a weak reproductive disadvantage. Under certain assumptions we prove that the relative frequency of these altruistic individuals in the total host population converges to spatially structured Wright-Fisher diffusions with frequency-dependent migration rates. For the many-demes limit (mean-field approximation) hereof, we show that the defense trait goes to fixation/extinction if and only if the selective disadvantage is smaller/larger than an explicit function of the ecological model parameters.

Abstract:
Population structure induced by both spatial embedding and more general networks of interaction, such as model social networks, have been shown to have a fundamental effect on the dynamics and outcome of evolutionary games. These effects have, however, proved to be sensitive to the details of the underlying topology and dynamics. Here we introduce a minimal population structure that is described by two distinct hierarchical levels of interaction. We believe this model is able to identify effects of spatial structure that do not depend on the details of the topology. We derive the dynamics governing the evolution of a system starting from fundamental individual level stochastic processes through two successive meanfield approximations. In our model of population structure the topology of interactions is described by only two parameters: the effective population size at the local scale and the relative strength of local dynamics to global mixing. We demonstrate, for example, the existence of a continuous transition leading to the dominance of cooperation in populations with hierarchical levels of unstructured mixing as the benefit to cost ratio becomes smaller then the local population size. Applying our model of spatial structure to the repeated prisoner's dilemma we uncover a novel and counterintuitive mechanism by which the constant influx of defectors sustains cooperation. Further exploring the phase space of the repeated prisoner's dilemma and also of the "rock-paper-scissor" game we find indications of rich structure and are able to reproduce several effects observed in other models with explicit spatial embedding, such as the maintenance of biodiversity and the emergence of global oscillations.