Variant evolutionary trees under phenotypic variance

Evolutionary branching, which is a coevolutionary phenomenon of the development of two or more distinctive traits from a single trait in a population, is the issue of recent studies on adaptive dynamics. In previous studies, it was revealed that trait variance is a minimum requirement for evolutionary branching, and that it does not play an important role in the formation of an evolutionary pattern of branching. Here we demonstrate that the trait evolution exhibits various evolutionary branching paths starting from an identical initial trait to different evolutional terminus traits as determined by only changing the assumption of trait variance. The key feature of this phenomenon is the topological configuration of equilibria and the initial point in the manifold of dimorphism from which dimorphic branches develop. This suggests that the existing monomorphic or polymorphic set in a population is not an unique inevitable consequence of an identical initial phenotype.     

Trait Substitution Sequence process and Canonical Equation for age-structured populations

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin–McCamy equation in Demography. Another particularity is that they involve an establishment probability, describing the probability of invasion of the resident population by the mutant one, that cannot always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.      

Phylogenomic assessment of the role of hybridization and introgression in trait evolution

Trait evolution among a set of species—a central theme in evolutionary biology—has long been understood and analyzed with respect to a species tree. However, the field of phylogenomics, which has been propelled by advances in sequencing technologies, has ushered in the era of species/gene tree incongruence and, consequently, a more nuanced understanding of trait evolution. For a trait whose states are incongruent with the branching patterns in the species tree, the same state could have arisen independently in different species (homoplasy) or followed the branching patterns of gene trees, incongruent with the species tree (hemiplasy). Another evolutionary process whose extent and significance are better revealed by phylogenomic studies is gene flow between different species. In this work, we present a phylogenomic method for assessing the role of hybridization and introgression in the evolution of polymorphic or monomorphic binary traits. We apply the method to simulated evolutionary scenarios to demonstrate the interplay between the parameters of the evolutionary history and the role of introgression in a binary trait’s evolution (which we call xenoplasy). Very importantly, we demonstrate, including on a biological data set, that inferring a species tree and using it for trait evolution analysis in the presence of gene flow could lead to misleading hypotheses about trait evolution.     

A compound poisson process for relaxing the molecular clock

The molecular clock hypothesis remains an important conceptual and analytical tool in evolutionary biology despite the repeated observation that the clock hypothesis does not perfectly explain observed DNA sequence variation. We introduce a parametric model that relaxes the molecular clock by allowing rates to vary across lineages according to a compound Poisson process. Events of substitution rate change are placed onto a phylogenetic tree according to a Poisson process. When an event of substitution rate change occurs, the current rate of substitution is modified by a gamma-distributed random variable. Parameters of the model can be estimated using Bayesian inference. We use Markov chain Monte Carlo integration to evaluate the posterior probability distribution because the posterior probability involves high dimensional integrals and summations. Specifically, we use the Metropolis-Hastings-Green algorithm with 11 different move types to evaluate the posterior distribution. We demonstrate the method by analyzing a complete mtDNA sequence data set from 23 mammals. The model presented here has several potential advantages over other models that have been proposed to relax the clock because it is parametric and does not assume that rates change only at speciation events. This model should prove useful for estimating divergence times when substitution rates vary across lineages.     

A new bayesian method for fitting evolutionary models to comparative data with intraspecific variation

Phylogenetic comparative methods that incorporate intraspecific variability are relatively new and, so far, not especially widely used in empirical studies. In the present short article we will describe a new Bayesian method for fitting evolutionary models to comparative data that incorporates intraspecific variability. This method differs from an existing likelihood-based approach in that it requires no a priori inference about species means and variances; rather it takes phenotypic values from individuals and a phylogenetic tree as input, and then samples species means and variances, along with the parameters of the evolutionary model, from their joint posterior probability distribution. One of the most novel and intriguing attributes of this approach is that jointly sampling the species means with the evolutionary model parameters means that the model and tree can influence our estimates of species mean trait values, not just the reverse. In the present implementation, we first apply this method to the most widely used evolutionary model for continuously valued phenotypic trait data (Brownian motion). However, the general approach has broad applicability, which we illustrate by also fitting the λ model, another simple model for quantitative trait evolution on a phylogeny. We test our approach via simulation and by analyzing two empirical datasets obtained from the literature. Finally, we have implemented the methods described herein in a new function for the R statistical computing environment, and this function will be distributed as part of the 'phytools' R library.     

Lévy flights in evolutionary ecology

We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law and the corresponding diffusion admits jumps. This could be seen as an alternative to Gould and Eldredge's model of evolutionary punctuated equilibria. We investigate the large-population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of non-local fractional reaction-diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such non-trivial fractional objects; their uniqueness is also proved.     

Ecological interactions drive evolutionary loss of traits

Loss of traits can dramatically alter the fate of species. Evidence is rapidly accumulating that the prevalence of trait loss is grossly underestimated. New findings demonstrate that traits can be lost without affecting the external phenotype, provided the lost function is compensated for by species interactions. This is important because trait loss can tighten the ecological relationship between partners, affecting the maintenance of species interactions. Here, we develop a new perspective on so-called `compensated trait loss' and how this type of trait loss may affect the evolutionary dynamics between interacting organisms. We argue that: (1) the frequency of compensated trait loss is currently underestimated because it can go unnoticed as long as ecological interactions are maintained; (2) by analysing known cases of trait loss, specific factors promoting compensated trait loss can be identified and (3) genomic sequencing is a key way forwards in detecting compensated trait loss. We present a comprehensive literature survey showing that compensated trait loss is taxonomically widespread, can involve essential traits, and often occurs as replicated evolutionary events. Despite its hidden nature, compensated trait loss is important in directing evolutionary dynamics of ecological relationships and has the potential to change facultative ecological interactions into obligatory ones.     

Reconstructing pedigrees: some identifiability questions for a recombination-mutation model

Pedigrees are directed acyclic graphs that represent ancestral relationships between individuals in a population. Based on a schematic recombination process, we describe two simple Markov models for sequences evolving on pedigrees—Model R (recombinations without mutations) and Model RM (recombinations with mutations). For these models, we ask an identifiability question: is it possible to construct a pedigree from the joint probability distribution of extant sequences? We present partial identifiability results for general pedigrees: we show that when the crossover probabilities are sufficiently small, certain spanning subgraph sequences can be counted from the joint distribution of extant sequences. We demonstrate how pedigrees that earlier seemed difficult to distinguish are distinguished by counting their spanning subgraph sequences.     

A new method to uncover signatures of divergent and stabilizing selection in quantitative traits

While it is well understood that the pace of evolution depends on the interplay between natural selection, random genetic drift, mutation, and gene flow, it is not always easy to disentangle the relative roles of these factors with data from natural populations. One popular approach to infer whether the observed degree of population differentiation has been influenced by local adaptation is the comparison of neutral marker gene differentiation (as reflected in FST) and quantitative trait divergence (as reflected in QST). However, this method may lead to compromised statistical power, because FST and QST are summary statistics which neglect information on specific pairs of populations, and because current multivariate tests of neutrality involve an averaging procedure over the traits. Further, most FST-QST comparisons actually replace QST by its expectation over the evolutionary process and are thus theoretically flawed. To overcome these caveats, we derived the statistical distribution of population means generated by random genetic drift and used the probability density of this distribution to test whether the observed pattern could be generated by drift alone. We show that our method can differentiate between genetic drift and selection as a cause of population differentiation even in cases with FST=QST and demonstrate with simulated data that it disentangles drift from selection more accurately than conventional FST-QST tests especially when data sets are small.     

Semi-Markov graph dynamics

In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.     

Ecological fitting by phenotypically flexible genotypes: implications for species associations, community assembly and evolution

Ecological fitting is the process whereby organisms colonize and persist in novel environments, use novel resources or form novel associations with other species as a result of the suites of traits that they carry at the time they encounter the novel condition. This paper has four major aims. First, we review the original concept of ecological fitting and relate it to the concept of exaptation and current ideas on the positive role of phenotypic plasticity in evolution. Second, we propose phenotypic plasticity, correlated trait evolution and phylogenetic conservatism as specific mechanisms behind ecological fitting. Third, we attempt to operationalize the concept of ecological fitting by providing explicit definitions for terms. From these definitions, we propose a simple conceptual model of ecological fitting. Using this model, we demonstrate the differences and similarities between ecological fitting and ecological resource tracking and illustrate the process in the context of species colonizing new areas and forming novel associations with other species. Finally, we discuss how ecological fitting can be both a precursor to evolutionary diversity or maintainer of evolutionary stasis, depending on conditions. We conclude that ecological fitting is an important concept for understanding topics ranging from the assembly of ecological communities and species associations, to biological invasions, to the evolution of biodiversity.     

Measures of success in a class of evolutionary models with fixed population size and structure

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.     

Delayed maturation in temporally structured populations with non-equilibrium dynamics

In this paper we study the evolutionary dynamics of delayed maturation in semelparous individuals. We model this in a two-stage clonally reproducing population subject to density-dependent fertility. The population dynamical model allows multiple — cyclic and/or chaotic — attractors, thus allowing us to illustrate how (i) evolutionary stability is primarily a property of a population dynamical system as a whole, and (ii) that the evolutionary stability of a demographic strategy by necessity derives from the evolutionary stability of the stationary population dynamical systems it can engender, i.e., its associated population dynamical attractors. Our approach is based on numerically estimating invasion exponents or “mutant fitnesses”. The invasion exponent is defined as the theoretical long-term average relative growth rate of a population of mutants in the stationary environment defined by a resident population system. For some combinations of resident and mutant trait values, we have to consider multi-valued invasion exponents, which makes the evolutionary argument more complicated (and more interesting) than is usually envisaged. Multi-valuedness occurs (i) when more than one attractor is associated with the values of the residents' demographic parameters, or (ii) when the setting of the mutant parameters makes the descendants of a single mutant reproduce exclusively either in even or in odd years, so that a mutant population is affected by either subsequence of the fluctuating resident densities only. Non-equilibrium population dynamics or random environmental noise selects for strategists with a non-zero probability to delay maturation. When there is an evolutionarily attracting pair of such a strategy and a population dynamical attractor engendered by it, this delaying probability is a Continuously Stable Strategy, that is an Evolutionarily Unbeatable Strategy which is also Stable in a long term evolutionary sense. Population dynamical coexistence of delaying and non-delaying strategists is possible with non-equilibrium dynamics, but adding random environmental noise to the model destroys this coexistence. Adding random noise also shifts the CSS towards a higher probability of delaying maturation.     

How quickly do brains catch up with bodies? A comparative method for detecting evolutionary lag.

A trait may be at odds with theoretical expectation because it is still in the process of responding to a recent selective force. Such a situation can be termed evolutionary lag. Although many cases of evolutionary lag have been suggested, almost all of the arguments have focused on trait fitness. An alternative approach is to examine the prediction that trait expression is a function of the time over which the trait could evolve. Here we present a phylogenetic comparative method for using this 'time' approach and we apply the method to a long-standing lag hypothesis: evolutionary changes in brain size lag behind evolutionary changes in body size. We tested the prediction in primates that brain mass contrast residuals, calculated from a regression of pairwise brain mass contrasts on positive pairwise body mass contrasts, are correlated with the time since the paired species diverged. Contrary to the brain size lag hypothesis, time since divergence was not significantly correlated with brain mass contrast residuals. We found the same result when we accounted for socioecology, used alternative body mass estimates and used male rather than female values. These tests do not support the brain size lag hypothesis. Therefore, body mass need not be viewed as a suspect variable in comparative neuroanatomical studies and relative brain size should not be used to infer recent evolutionary changes in body size.     

Evolutionary Branching in a Finite Population: Deterministic Branching vs. Stochastic Branching

Adaptive dynamics formalism demonstrates that, in a constant environment, a continuous trait may first converge to a singular point followed by spontaneous transition from a unimodal trait distribution into a bimodal one, which is called “evolutionary branching.” Most previous analyses of evolutionary branching have been conducted in an infinitely large population. Here, we study the effect of stochasticity caused by the finiteness of the population size on evolutionary branching. By analyzing the dynamics of trait variance, we obtain the condition for evolutionary branching as the one under which trait variance explodes. Genetic drift reduces the trait variance and causes stochastic fluctuation. In a very small population, evolutionary branching does not occur. In larger populations, evolutionary branching may occur, but it occurs in two different manners: in deterministic branching, branching occurs quickly when the population reaches the singular point, while in stochastic branching, the population stays at singularity for a period before branching out. The conditions for these cases and the mean branching-out times are calculated in terms of population size, mutational effects, and selection intensity and are confirmed by direct computer simulations of the individual-based model.     

Third-party mutualists have contrasting effects on host invasion under the enemy-release and biotic-resistance hypotheses

Plants engage in complex multipartite interactions with mutualists and antagonists, but these interactions are rarely included in studies that explore plant invasiveness. When considered in isolation, we know that beneficial microbes can enhance an exotic plant’s invasive ability and that herbivorous insects often decrease an exotic plant’s likeliness of success. However, the effect of these partners on plant fitness has not been well characterized when all three species coevolve. We use computational evolutionary modeling of a trait-based system to test how microbes and herbivores simultaneously coevolving with an invading plant affect the invaders’ probability of becoming established. Specifically, we designed a model that explores how a beneficial microbe would influence the outcome of an interaction between a plant and herbivore. To model novel interactions, we included a phenotypic trait shared by each species. Making this trait continuous and selectable allows us to explore how trait similarities between coevolving plants, herbivores and microbes affect fitness. Using this model, we answer the following questions: (1) Can a beneficial plant-microbe interaction influence the evolutionary outcome of antagonistic interactions between plants and herbivores? (2) How does the initial trait similarity between interacting organisms affect the likelihood of plant survival in novel locations? (3) Does the effect of tripartite interactions on the invasion success of a plant depend on whether organisms interact through trait similarity [Enemy Release Hypothesis (ERH)] or dissimilarity (Biotic Resistance Hypothesis)? We found that it was much more difficult for plants to invade under the ERH but that beneficial microbes increase the probability of plant survival in a novel range under both hypotheses. To our knowledge, this model is the first to use tripartite interactions to explore novel species introductions. It represents a step towards gaining a better understanding of the factors influencing establishment of exotic species to prevent future invasions.     

THE CONTRIBUTION OF MATERNAL EFFECTS TO SELECTION RESPONSE: AN EMPIRICAL TEST OF COMPETING MODELS

Maternal effects can dramatically influence the evolutionary process, in some cases facilitating and in others hindering adaptive evolution. Maternal effects have been incorporated into quantitative genetic models using two theoretical frameworks: the variance‐components approach, which partitions variance into direct and maternal components, and the trait‐based approach, which assumes that maternal effects are mediated by specific maternal traits. Here, we demonstrate parallels between these models and test their ability to predict evolutionary change. First, we show that the two approaches predict equivalent responses to selection in the absence of maternal effects mediated by traits that are themselves maternally influenced. We also introduce a correction factor that may be applied when such cascading maternal effects are present. Second, we use several maternal effect models, as well as the standard breeder's equation, to predict evolution in response to artificial selection on flowering time in American bellflower, Campanulastrum americanum. Models that included maternal effects made much more accurate predictions of selection response than the breeder's equation. Maternal effect models differed somewhat in their fit, with a version of the trait‐based model providing the best fit. We recommend fitting such trait‐based models when possible and appropriate to make the most accurate evolutionary predictions.     

Most Undirected Random Graphs Are Amplifiers of Selection for Birth-Death Dynamics,but Suppressors of Selection for Death-Birth Dynamics

We analyze evolutionary dynamics on graphs, where the nodes represent individuals of a population. The links of a node describe which other individuals can be displaced by the offspring of the individual on that node. Amplifiers of selection are graphs for which the fixation probability is increased for advantageous mutants and decreased for disadvantageous mutants. A few examples of such amplifiers have been developed, but so far it is unclear how many such structures exist and how to construct them. Here, we show that almost any undirected random graph is an amplifier of selection for Birth-death updating, where an individual is selected to reproduce with probability proportional to its fitness and one of its neighbors is replaced by that offspring at random. If we instead focus on death-Birth updating, in which a random individual is removed and its neighbors compete for the empty spot, then the same ensemble of graphs consists of almost only suppressors of selection for which the fixation probability is decreased for advantageous mutants and increased for disadvantageous mutants. Thus, the impact of population structure on evolutionary dynamics is a subtle issue that will depend on seemingly minor details of the underlying evolutionary process.     

The capture of heritable variation for genetic quality through social competition

In theory, females of many species choose mates based on traits that are indicators of male genetic quality. A fundamental question in evolutionary biology is why genetic variation for such indicator traits persists despite strong persistent selection imposed by female preference, which is known as the lek paradox. One potential solution to the lek paradox suggests that the traits that are targets of mate choice should evolve condition-dependent expression and that condition should have a large genetic variance. Condition is expected to exhibit high genetic variance because it is affected by a large number of physiological processes and hence, condition-dependent traits should 'capture' variation contributed by a large number of loci. We suggest that a potentially important cause of variation in condition is competition for limited resources. Here, we discuss a pair of models to analyze the evolutionary genetics of traits affected by success in social competition for resources. We show that competition can contribute to genetic variation of 'competition-dependent' traits that have fundamentally different evolutionary properties than other sources of variation. Competition dependence can make traits honest indicators of genetic quality by revealing the relative competitive ability of males, can provide a component of heritable variation that does not contribute to trait evolution, and can help maintain heritable variation under directional selection. Here we provide a general introduction to the concept of competition dependence and briefly introduce two models to demonstrate the potential evolutionary consequences of competition-dependent trait expression.