Strategy selection in structured populations

Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix . We study a mutation and selection process. For weak selection strategy A is favored over B if and only if σa+b>c+σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ for various examples including the well-mixed population, games on graphs, games in phenotype space and games on sets. We give a proof for the existence of such a σ, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between σ and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, σ, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.     

Mutation-selection equilibrium in games with multiple strategies

In evolutionary games the fitness of individuals is not constant but depends on the relative abundance of the various strategies in the population. Here we study general games among n strategies in populations of large but finite size. We explore stochastic evolutionary dynamics under weak selection, but for any mutation rate. We analyze the frequency dependent Moran process in well-mixed populations, but almost identical results are found for the Wright-Fisher and Pairwise Comparison processes. Surprisingly simple conditions specify whether a strategy is more abundant on average than 1/n, or than another strategy, in the mutation-selection equilibrium. We find one condition that holds for low mutation rate and another condition that holds for high mutation rate. A linear combination of these two conditions holds for any mutation rate. Our results allow a complete characterization of n×n games in the limit of weak selection.     

The fastest evolutionary trajectory

Given two mutants, A and B, separated by n mutational steps, what is the evolutionary trajectory which allows a homogeneous population of A to reach B in the shortest time? We show that the optimum evolutionary trajectory (fitness landscape) has the property that the relative fitness increase between any two consecutive steps is constant. Hence, the optimum fitness landscape between A and B is given by an exponential function. Our result is precise for small mutation rates and excluding back mutations. We discuss deviations for large mutation rates and including back mutations. For very large mutation rates, the optimum fitness landscape is flat and has a single peak at type B.     

The rate of multi-step evolution in Moran and Wright-Fisher populations

Several groups have recently modeled evolutionary transitions from an ancestral allele to a beneficial allele separated by one or more intervening mutants. The beneficial allele can become fixed if a succession of intermediate mutants are fixed or alternatively if successive mutants arise while the previous intermediate mutant is still segregating. This latter process has been termed stochastic tunneling. Previous work has focused on the Moran model of population genetics. I use elementary methods of analyzing stochastic processes to derive the probability of tunneling in the limit of large population size for both Moran and Wright-Fisher populations. I also show how to efficiently obtain numerical results for finite populations. These results show that the probability of stochastic tunneling is twice as large under the Wright-Fisher model as it is under the Moran model.     

Crossing fitness valleys during the evolution of limpet homing behaviour

Evolution is often considered a gradual hill-climbing process, slowly increasing the fitness of organisms. Here I investigate evolution of homing behaviour in simulated intertidal limpets. While the simulation of homing is only a possible mechanism by which homing may have evolved, the process allows an investigation of how evolution may occur over different fitness landscapes. With some fitness landscapes, in order to evolve path integration as a homing mechanism, a temporary reduction in an organism’s fitness was required — since high developmental costs occurred before successful homing strategies evolved. Simple hill-climbing algorithms, therefore, only rarely resulted in the evolution of a functional homing behaviour. The inclusion of trail-following greatly increases the frequency of success of evolution of a path integration strategy. Initially an emergent homing behaviour is formed combining path integration with trail-following. This also demonstrates evolution through exaptation, since in the simulation, the original role of trail-following is likely to be unrelated to homing. Analysis of the fitness landscapes of homing in the presence of trail-following behaviour shows a high variability of fitness, which results in the formation of ‘stepping-stones’ of high fitness across fitness valleys. By using these stepping-stones, simple hill-climbing algorithms can reach the global maximum fitness value.     

Evolutionary stability and quasi-stationary strategy in stochastic evolutionary game dynamics

Stochastic evolutionary game dynamics for finite populations has recently been widely explored in the study of evolutionary game theory. It is known from the work of Traulsen et al. [2005. Phys. Rev. Lett. 95, 238701] that the stochastic evolutionary dynamics approaches the deterministic replicator dynamics in the limit of large population size. However, sometimes the limiting behavior predicted by the stochastic evolutionary dynamics is not quite in agreement with the steady-state behavior of the replicator dynamics. This paradox inspired us to give reasonable explanations of the traditional concept of evolutionarily stable strategy (ESS) in the context of finite populations. A quasi-stationary analysis of the stochastic evolutionary game dynamics is put forward in this study and we present a new concept of quasi-stationary strategy (QSS) for large but finite populations. It is shown that the consistency between the QSS and the ESS implies that the long-term behavior of the replicator dynamics can be predicted by the quasi-stationary behavior of the stochastic dynamics. We relate the paradox to the time scales and find that the contradiction occurs only when the fixation time scale is much longer than the quasi-stationary time scale. Our work may shed light on understanding the relationship between the deterministic and stochastic methods of modeling evolutionary game dynamics.     

Darwinian fitness

The term Darwinian fitness refers to the capacity of a variant type to invade and displace the resident population in competition for available resources. Classical models of this dynamical process claim that competitive outcome is a deterministic event which is regulated by the population growth rate, called the Malthusian parameter. Recent analytic studies of the dynamics of competition in terms of diffusion processes show that growth rate predicts invasion success only in populations of infinite size. In populations of finite size, competitive outcome is a stochastic process--contingent on resource constraints--which is determined by the rate at which a population returns to its steady state condition after a random perturbation in the individual birth and death rates. This return rate, a measure of robustness or population stability, is analytically characterized by the demographic parameter, evolutionary entropy, a measure of the uncertainty in the age of the mother of a randomly chosen newborn. This article appeals to computational and numerical methods to contrast the predictive power of the Malthusian and the entropic principles. The computational analysis rejects the Malthusian model and is consistent with of the entropic principle. These studies thus provide support for the general claim that entropy is the appropriate measure of Darwinian fitness and constitutes an evolutionary parameter with broad predictive and explanatory powers.     

Evolutionary game dynamics in a Wright-Fisher process

Evolutionary game dynamics in finite populations can be described by a frequency dependent, stochastic Wright-Fisher process. We consider a symmetric game between two strategies, A and B. There are discrete generations. In each generation, individuals produce offspring proportional to their payoff. The next generation is sampled randomly from this pool of offspring. The total population size is constant. The resulting Markov process has two absorbing states corresponding to homogeneous populations of all A or all B. We quantify frequency dependent selection by comparing the absorption probabilities to the corresponding probabilities under random drift. We derive conditions for selection to favor one strategy or the other by using the concept of total positivity. In the limit of weak selection, we obtain the 1/3 law: if A and B are strict Nash equilibria then selection favors replacement of B by A, if the unstable equilibrium occurs at a frequency of A which is less than 1/3.     

Mutation-selection equilibrium in games with mixed strategies

We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an n×n payoff matrix. Players can use mixed strategies, which are given by the vector (p₁,…,p_n). Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S_n. We study evolutionary dynamics in a well-mixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutation-selection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the Hawk-Dove game. We prove general statements about the relationship between games with pure and with mixed strategies.     

The evolution of phenotypic traits in a predator-prey system subject to Allee effect

This paper considers the evolution of phenotypic traits in a community comprising the populations of predators and prey subject to Allee effect. The evolutionary model is constructed from a deterministic approximation of the stochastic process of mutation and selection. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that the strong Allee effect of prey facilitates the formation of continuously stable strategy in the case that prey population undergoes evolutionary branching if the Allee effect of prey is not strong enough. Secondly, we show that evolutionary suicide is impossible for prey population when the intraspecific competition of prey is symmetric about the origin. However, evolutionary suicide can occur deterministically on prey population if prey individuals undergo strong asymmetric competition and are subject to Allee effect. Thirdly, we show that the evolutionary model with symmetric interactions admits a stable limit cycle if the Allee effect of prey is weak. Evolutionary cycle is a likely outcome of the process, which depends on the strength of Allee effect and the mutation rates of predators and prey.     

Evolutionary jumping and breakthrough in tree masting evolution

Many long-lived plants such as trees show masting or intermittent and synchronized reproduction. In a coupled chaos system describing the dynamics of individual-plant resource budgets, masting occurs when the resource depletion coefficient k (ratio of the reproductive expenditure to the excess resource reserve) is large. Here, we mathematically studied the condition for masting evolution. In an infinitely large population, we obtained a deterministic dynamical system, to which we applied the pairwise invasibility plot and convergence stability of evolutionary singularity analyses. We prove that plants reproducing at the same rate every year are not evolutionarily stable. The resource depletion coefficient k increases, and the system oscillates with a period of 2 years (high and low reproduction) if k<1. Alternatively, k may evolve further and jump to a value >1, resulting in the sudden start of intermittent reproduction. We confirm that a high survivorship of young plants (seedlings) in the light-limited understory favors masting evolution, as previously suggested by computer simulations and field observations. The stochasticity caused by the finiteness of population size also promotes masting evolution.     

Stochastic payoff evaluation increases the temperature of selection

We study stochastic evolutionary game dynamics in populations of finite size. Moreover, each individual has a randomly distributed number of interactions with other individuals. Therefore, the payoff of two individuals using the same strategy can be different. The resulting "payoff stochasticity" reduces the intensity of selection and therefore increases the temperature of selection. A simple mean-field approximation is derived that captures the average effect of the payoff stochasticity. Correction terms to the mean-field theory are computed and discussed.     

The one-third law of evolutionary dynamics

Evolutionary game dynamics in finite populations provide a new framework for studying selection of traits with frequency-dependent fitness. Recently, a "one-third law" of evolutionary dynamics has been described, which states that strategy A fixates in a B-population with selective advantage if the fitness of A is greater than that of B when A has a frequency 13. This relationship holds for all evolutionary processes examined so far, from the Moran process to games on graphs. However, the origin of the "number"13 is not understood. In this paper we provide an intuitive explanation by studying the underlying stochastic processes. We find that in one invasion attempt, an individual interacts on average with B-players twice as often as with A-players, which yields the one-third law. We also show that the one-third law implies that the average Malthusian fitness of A is positive.     

A symmetry of fixation times in evoultionary dynamics

In this paper, we show that for evolutionary dynamics between two types that can be described by a Moran process, the conditional fixation time of either type is the same irrespective of the selective scenario. With frequency dependent selection between two strategies A and B of an evolutionary game, regardless of whether A dominates B, A and B are best replies to themselves, or A and B are best replies to each other, the conditional fixation times of a single A and a single B mutant are identical. This does not hold for Wright-Fisher models, nor when the mutants start from multiple copies.     

Diffusion of nuclear and mitochondrial genes across a zone of secondary contact in the maritime shrew, Sorex maritimensis: implications for the conservation of a Canadian endemic mammal

The maritime shrew, Sorex maritimensis, is a Canadian endemic species with a limited distribution in two provinces in eastern Canada. Phylogeographic analysis of mitochondrial DNA control region and cytochrome b sequences revealed two clades, one found in New Brunswick and the other primarily in Nova Scotia, Canada. We propose that these clades have come back into secondary contact following the Wisconsin glaciation via wetlands on the narrow Isthmus of Chigneto that connects these provinces. Despite evidence of an historic separation of maritime shrew subpopulations in Nova Scotia and New Brunswick, we conclude that shrews in these two regions should be considered a single evolutionary significant unit but separate, semi-isolated management units that should be recognized as such for conservation purposes. The susceptibility of this stenotopic species with limited dispersal capabilities raises concerns about its long-term persistence if climate-change induced habitat fragmentation increases. Maintenance of contiguous wetland habitats is needed to ensure connectivity and gene flow among populations of the maritime shrew.     

The origin of the tRNA molecule: Independent data favor a specific model of its evolution

The properties, historical and empirical observations of a model of the origin of the tRNA molecule are discussed. This model would predict that this molecule originated by means of the assembly of two hairpin-like structures of RNA. The conclusion is that the model possesses a relevant part of the truth on the origin of the tRNA molecule.     

The molecular evolution of pancreatic ribonuclease

Summary The primary structures of pancreatic ribonucleases from 26 species (18 artiodactyls, horse, whale, 5 rodents and turtle) are known. Several species contain identical ribonucleases (cow/bison; sheep/goat), other species show polymorphism (arabian camel) or the presence of two structural gene loci (guinea pig pancreas contains two ribonucleases that differ at 31 positions). 26 different sequences (including the ribonuclease from bovine seminal plasma which is paralogous to the pancreatic ribonucleases) were used to construct a most parsimonious tree. A second tree that most closely approximates current biological opinion requires 402 whereas the most parsimonious tree requires 389 nucleotide substitutions. The artiodactyl part of the most parsimonious tree conforms quite well with the biological one of this order, except for the position of the giraffe which is placed with the pronghorn. Other parts of the most parsimonious tree agree less with the biological tree, probably as a result of the occurrence of many parallel and back substitutions. Bovine seminal ribonuclease was found to be the result of a gene duplication which occurred before the divergence of the true ruminants, but after the divergence of this group from the cameloids.The evolutionary rate of ribonuclease was found to be 390, 3.0 and 11 nucleotide substitutions per 10⁹ yrs per ribonuclease gene, codon and covarion respectively. However, there is much variation in evolutionary rate in different taxa. Values ranging from about 100 (in the bovidae) to about 700 (in the rodents) nucleotide substitutions per 10⁹ yrs per gene were found.A method for counting parallel and back mutations is presented. The 389 nucleotide substitutions in the most parsimonious tree occur at 88 codon positions; 154 of them are the result of parallel and back mutations. Parallel evolution to a similar structure, including the presence of 2 sites with carbohydrate, was demonstrated in an extensive region at the surface of pig and guinea pig ribonuclease B. The presence of carbohydrate probably is important in a number of species. A correlation between the presence of heavily glycosidated ribonucleases and coecal digestion was observed. Hypothetical sequences of ancestral ungulate ribonucleases contain many recognition sites for carbohydrate attachment; this suggests that herbivores with coecal digestion might have preceded the true ruminants in mammalian evolution.