scholarly journals Dominant Cubic Coefficients of the ‘1/3-Rule’ Reduce Contest Domains

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 491 ◽  
Author(s):  
Paul F. Slade
Keyword(s):  
Population Size ◽  
Probability Function ◽  
Social Dilemma ◽  
Payoff Matrix ◽  
Fixation Probability ◽  
Random Genetic Drift ◽  
Selection Differential ◽  
Maclaurin Series ◽  
Third Order ◽  
Moran Model

Antagonistic exploitation in competition with a cooperative strategy defines a social dilemma, whereby eventually overall fitness of the population decreases. Frequency-dependent selection between two non-mutating strategies in a Moran model of random genetic drift yields an evolutionary rule of biological game theory. When a singleton fixation probability of co-operation exceeds the selectively neutral value being the reciprocal of population size, its relative frequency in the population equilibrates to less than 1/3. Maclaurin series of a singleton type fixation probability function calculated at third order enables the convergent domain of the payoff matrix to be identified. Asymptotically dominant third order coefficients of payoff matrix entries were derived. Quantitative analysis illustrates non-negligibility of the quadratic and cubic coefficients in Maclaurin series with selection being inversely proportional to population size. Novel corollaries identify the domain of payoff matrix entries that determines polarity of second order terms, with either non-harmful or harmful contests. Violation of this evolutionary rule observed with non-harmful contests depends on the normalized payoff matrix entries and selection differential. Significant violations of the evolutionary rule were not observed with harmful contests.

scholarly journals Stochasticity, Selection, and the Evolution of Cooperation in a Two-Level Moran Model of the Snowdrift Game

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Brian McLoone ◽  
Wai-Tong Louis Fan ◽  
Adam Pham ◽  
Rory Smead ◽  
Laurence Loewe
Keyword(s):  
Stochastic Dynamics ◽  
Cooperative Behavior ◽  
Social Dilemma ◽  
Threshold Effect ◽  
Payoff Matrix ◽  
Multilevel Selection ◽  
Evolution Of Cooperation ◽  
Interior Equilibrium ◽  
Moran Model ◽  
Snowdrift Game

The Snowdrift Game, also known as the Hawk-Dove Game, is a social dilemma in which an individual can participate (cooperate) or not (defect) in producing a public good. It is relevant to a number of collective action problems in biology. In a population of individuals playing this game, traditional evolutionary models, in which the dynamics are continuous and deterministic, predict a stable, interior equilibrium frequency of cooperators. Here, we examine how finite population size and multilevel selection affect the evolution of cooperation in this game using a two-level Moran process, which involves discrete, stochastic dynamics. Our analysis has two main results. First, we find that multilevel selection in this model can yield significantly higher levels of cooperation than one finds in traditional models. Second, we identify a threshold effect for the payoff matrix in the Snowdrift Game, such that below (above) a determinate cost-to-benefit ratio, cooperation will almost surely fix (go extinct) in the population. This second result calls into question the explanatory reach of traditional continuous models and suggests a possible alternative explanation for high levels of cooperative behavior in nature.

scholarly journals Looking Forwards and Backwards in the Multi-Allelic Neutral Cannings Population Model

2010 ◽  
Vol 47 (03) ◽  
pp. 713-731 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Markov Chain ◽  
Population Size ◽  
Side Effect ◽  
Population Model ◽  
Inversion Formula ◽  
Transition Matrix ◽  
Duality Relation ◽  
Moran Model ◽  
Fisher Model ◽  
Fixed Population

We look forwards and backwards in the multi-allelic neutral exchangeable Cannings model with fixed population size and nonoverlapping generations. The Markov chain X is studied which describes the allelic composition of the population forward in time. A duality relation (inversion formula) between the transition matrix of X and an appropriate backward matrix is discussed. The probabilities of the backward matrix are explicitly expressed in terms of the offspring distribution, complementing the work of Gladstien (1978). The results are applied to fundamental multi-allelic Cannings models, among them the Moran model, the Wright-Fisher model, the Kimura model, and the Karlin and McGregor model. As a side effect, number theoretical sieve formulae occur in these examples.

Analysis of genetic diversity in four Canadian swine breeds using pedigree data

2010 ◽  
Vol 90 (3) ◽  
pp. 331-340 ◽  
Author(s):  
M G Melka ◽  
F. Schenkel
Keyword(s):  
Genetic Diversity ◽  
Population Size ◽  
Effective Population Size ◽  
Genetic Drift ◽  
Relative Proportion ◽  
Random Genetic Drift ◽  
Pedigree Data ◽  
Effective Population ◽  
Animal Genetic Resources ◽  
Loss Of Genetic Diversity

Conservation of animal genetic resources entails judicious assessment of genetic diversity as a first step. The objective of this study was to analyze the trend of within-breed genetic diversity and identify major causes of loss of genetic diversity in four swine breeds based on pedigree data. Pedigree files from Duroc (DC), Hampshire (HP), Lacombe (LC) and Landrace (LR) containing 480 191, 114 871, 51 397 and 1 080 144 records, respectively, were analyzed. Pedigree completeness, quality and depth were determined. Several parameters derived from the in-depth pedigree analyses were used to measure trends and current levels of genetic diversity. Pedigree completeness indexes of the four breeds were 90.4, 52.7, 89.6 and 96.1%, respectively. The estimated percentage of genetic diversity lost within each breed over the last three decades was approximately 3, 22, 12 and 2%, respectively. The relative proportion of genetic diversity lost due to random genetic drift in DC, HP, LC and LR was 74.5, 63.6, 72.9 and 60.0%, respectively. The estimated current effective population size for DC, HP, LC and LR was 72, 14, 36 and 125, respectively. Therefore, HP and LC have been found to have lost considerable genetic diversity, demanding priority for conservation. Key words: Genetic drift, effective population size

scholarly journals Fixation probability of beneficial mutations in a fluctuating population

2009 ◽  
Vol 91 (1) ◽  
pp. 73-82 ◽  
Author(s):  
STEINAR ENGEN ◽  
RUSSELL LANDE ◽  
BERNT-ERIK SÆTHER
Keyword(s):  
Population Size ◽  
Logistic Model ◽  
Return Time ◽  
Fixation Probability ◽  
Population Fluctuations ◽  
Effective Population ◽  
Environmental Variance ◽  
Fixation Rate ◽  
Beneficial Mutations ◽  
Probability Of Fixation

SummaryWe compute an accurate approximation to the probability of fixation for a beneficial mutation in a population fluctuating with a stationary distribution of population size. The population dynamics are described by the theta-logistic model with environmental variance, assuming that the population size is large enough to ignore demographic variance. We show that stochastic fluctuations of population size reduce the probability of fixation. However, it is not the magnitude of the population fluctuationsper sethat creates this reduction. Only the environmental variance has a substantial effect on the probability of fixation. The strength of density dependence (or expected return time to equilibrium) and the functional form of density-regulation, given by the parameter θ in the theta-logistic model, have little effect on the fixation probability. Effective population size based on harmonic mean population size will therefore underestimate the expected fixation rate of beneficial mutations in fluctuating populations.

scholarly journals The probability of fixation of a favoured allele in a subdivided population

1993 ◽  
Vol 62 (2) ◽  
pp. 149-157 ◽  
Author(s):  
N. H. Barton
Keyword(s):  
Population Structure ◽  
Population Size ◽  
Discrete Model ◽  
Fixation Probability ◽  
Effective Population ◽  
Weak Selection ◽  
Extinction Rates ◽  
Subdivided Population ◽  
Probability Of Fixation ◽  
Neutral Markers

SummaryIn a stably subdivided population with symmetric migration, the chance that a favoured allele will be fixed is independent of population structure. However, random extinction introduces an extra component of sampling drift, and reduces the probability of fixation. In this paper, the fixation probability is calculated using the diffusion approximation; comparison with exact solution of the discrete model shows this to be accurate. The key parameters are the rates of selection, migration and extinction, scaled relative to population size (S = 4Ns, M = 4Nm, Λ = 4Nλ); results apply to a haploid model, or to diploids with additive selection. If new colonies derive from many demes, the fixation probability cannot be reduced by more than half. However, if colonies are initially homogeneous, fixation probability can be much reduced. In the limit of low migration and extinction rates (M, Λ ≪ 1), it is 2s/{1 + (Λ/MS)(1 −exp(−S))}, whilst in the opposite limit (S ≪ 1), it is 4sM/{Λ(Λ + M)}. In the limit of weak selection (M, Λ ≫ 1), it is 4sM/{Λ(Λ + M)}. These factors are not the same as the reduction in effective population size (Ne/N), showing that the effects of population structure on selected alleles cannot be understood from the behaviour of neutral markers.

scholarly journals Genetic differentiation of quantitative characters between populations or species: I. Mutation and random genetic drift

1982 ◽  
Vol 39 (3) ◽  
pp. 303-314 ◽  
Author(s):  
Ranajit Chakraborty ◽  
Masatoshi Nei
Keyword(s):  
Population Size ◽  
Effective Population Size ◽  
Genetic Drift ◽  
Genetic Variance ◽  
Genetic Model ◽  
Neutral Evolution ◽  
Random Genetic Drift ◽  
Effective Population ◽  
Quantitative Characters ◽  
Allelic State

SummaryIntroducing a new genetic model called the discrete allelic-state model, the evolutionary change of genetic variation of quantitative characters within and between populations is studied under the assumption of no selection. This model allows us to study the effects of mutation and random genetic drift in detail. It is shown that when the allelic effects on phenotype are additive, the rate of approach of the genetic variance within populations to the equilibrium value depends only on the effective population size. It is also shown that the distribution of genotypic value often deviates from normality particularly when the effective population size and the number of loci concerned are small. On the other hand, the interpopulational variance increases linearly with time, if the intrapopu-lational variance remains constant. Therefore, the ratio of interpopulational variance to intrapopulational variance can be used for testing the hypothesis of neutral evolution of quantitative characters.

scholarly journals Haldane’s formula in Cannings models: the case of moderately strong selection

2021 ◽  
Vol 83 (6-7) ◽  
Author(s):  
Florin Boenkost ◽  
Adrián González Casanova ◽  
Cornelia Pokalyuk ◽  
Anton Wakolbinger
Keyword(s):  
Population Size ◽  
Early Phase ◽  
Large Population ◽  
Selective Advantage ◽  
Fixation Probability ◽  
Strong Selection

AbstractFor a class of Cannings models we prove Haldane’s formula, $$\pi (s_N) \sim \frac{2s_N}{\rho ^2}$$ π ( s N ) ∼ 2 s N ρ 2 , for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for $$s_N \sim N^{-b}$$ s N ∼ N - b and $$0< b<1/2$$ 0 < b < 1 / 2 . Here, $$s_N$$ s N is the selective advantage of an individual carrying the beneficial type, and $$\rho ^2$$ ρ 2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.

scholarly journals Theoretical study of near neutrality. I. Heterozygosity and rate of mutant substitution.

Genetics ◽  
1990 ◽  
Vol 126 (1) ◽  
pp. 219-229 ◽  
Author(s):  
T Ohta ◽  
H Tachida
Keyword(s):  
Molecular Evolution ◽  
Population Size ◽  
Theoretical Study ◽  
Simulation Studies ◽  
Random Genetic Drift ◽  
Time Dependency ◽  
Weak Selection ◽  
Closely Related Species ◽  
Subdivided Populations ◽  
New Mutations

Abstract In order to clarify the nature of "near neutrality" in molecular evolution and polymorphism, extensive simulation studies were performed. Selection coefficients of new mutations are assumed to be small so that both random genetic drift and selection contribute to determining the behavior of mutants. The model also incorporates normally distributed spatial fluctuation of selection coefficients. If the system starts from "average neutrality," it will move to a better adapted state, and most new mutations will become "slightly deleterious." Monte Carlo simulations have indicated that such adaptation is attained, but that the rate of such "progress" is very low for weak selection. In general, the larger the population size, the more effective the selection becomes. Also, as selection becomes weaker, the behavior of the mutants approaches that of completely neutral genes. Thus, the weaker the selection, the smaller is the effect of population size on mutant dynamics. Increase of heterozygosity with population size is very pronounced for subdivided populations. The significance of these results is discussed in relation to various observed facts on molecular evolution and polymorphism, such as generation-time dependency and overdispersion of the molecular clock, or contrasting patterns of DNA and protein polymorphism among some closely related species.

scholarly journals Fixation of Advantageous Alleles in Partially Self-Fertilizing Populations: The Effect of Different Selection Modes

Genetics ◽  
2000 ◽  
Vol 154 (2) ◽  
pp. 813-821 ◽  
Author(s):  
Christian Damgaard
Keyword(s):  
Sexual Selection ◽  
Population Size ◽  
Diffusion Approximation ◽  
Selective Advantage ◽  
Fixation Probability ◽  
Selfing Rate ◽  
Male Fitness ◽  
Viability Selection ◽  
Fecundity Selection ◽  
Hermaphroditic Plant

Abstract The expected fixation probability of an advantageous allele was examined in a partially self-fertilizing hermaphroditic plant species using the diffusion approximation. The selective advantage of the advantageous allele was assumed to be increased viability, increased fecundity, or an increase in male fitness. The mode of selection, as well as the selfing rate, the population size, and the dominance of the advantageous allele, affect the fixation probability of the allele. In general it was found that increases in selfing rate decrease the fixation probability under male sexual selection, increase fixation probability under fecundity selection, and increase when recessive and decrease when dominant under viability selection. In some cases the highest fixation probability of advantageous alleles under fecundity or under male sexual selection occurred at an intermediary selfing rate. The expected mean fixation times of the advantageous allele were also examined using the diffusion approximation.

scholarly journals Probability of Identity by Descent in Metapopulations

Genetics ◽  
1999 ◽  
Vol 152 (3) ◽  
pp. 1217-1228
Author(s):  
Ingemar Kaj ◽  
Martin Lascoux
Keyword(s):  
Population Size ◽  
Colony Size ◽  
Continuous Time ◽  
Identity By Descent ◽  
Moran Model ◽  
Metapopulation Models ◽  
Size Model ◽  
Varying Population ◽  
F Statistics ◽  
Birth Death

Abstract Equilibrium probabilities of identity by descent (IBD), for pairs of genes within individuals, for genes between individuals within subpopulations, and for genes between subpopulations are calculated in metapopulation models with fixed or varying colony sizes. A continuous-time analog to the Moran model was used in either case. For fixed-colony size both propagule and migrant pool models were considered. The varying population size model is based on a birth-death-immigration (BDI) process, to which migration between colonies is added. Wright's F statistics are calculated and compared to previous results. Adding between-island migration to the BDI model can have an important effect on the equilibrium probabilities of IBD and on Wright's index.

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