Generation cycles

Generation cycles are population cycles with a period of roughly one generation. They have been observed in tropical pest populations and in laboratory populations. Theory suggests that they can arise both from intraspecific competition and from the influence of natural enemies, and ecological studies of populations of insects showing these cycles are starting to shed light on the mechanisms that maintain them.     

Association mapping in structured populations

The use, in association studies, of the forthcoming dense genomewide collection of single-nucleotide polymorphisms (SNPs) has been heralded as a potential breakthrough in the study of the genetic basis of common complex disorders. A serious problem with association mapping is that population structure can lead to spurious associations between a candidate marker and a phenotype. One common solution has been to abandon case-control studies in favor of family-based tests of association, such as the transmission/disequilibrium test (TDT), but this comes at a considerable cost in the need to collect DNA from close relatives of affected individuals. In this article we describe a novel, statistically valid, method for case-control association studies in structured populations. Our method uses a set of unlinked genetic markers to infer details of population structure, and to estimate the ancestry of sampled individuals, before using this information to test for associations within subpopulations. It provides power comparable with the TDT in many settings and may substantially outperform it if there are conflicting associations in different subpopulations.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

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.     

Self-organized dynamics in spatially structured populations

Self-organization and pattern formation represent the emergence of order in temporal and spatial processes. Self-organization in population ecology is gaining attention due to the recent advances concerning temporal fluctuations in the population size of dispersal-linked subunits. We shall report that spatially structured models of population renewal promote the emergence of a complex power law order in spatial population dynamics. We analyse a variety of population models showing that self-organization can be identified as a temporal match in population dynamics among local units, and how the synchrony changes in time. Our theoretical results are concordant with analyses of population data on the Canada lynx.     

Unbiased relatedness estimation in structured populations

Knowledge of the genetic relatedness between individuals is important in many research areas in quantitative genetics, conservation genetics, forensics, evolution, and ecology. In the absence of pedigree records, relatedness can be estimated from genetic marker data using a number of estimators. These estimators, however, make the critical assumption of a large random mating population without genetic structures. The assumption is frequently violated in the real world where geographic/social structures or nonrandom mating usually lead to genetic structures. In this study, I investigated two approaches to the estimation of relatedness between a pair of individuals from a subpopulation due to recent common ancestors (i.e., relatedness is defined and measured with the current focal subpopulation as reference). The indirect approach uses the allele frequencies of the entire population with and without accounting for the population structure, and the direct approach uses the allele frequencies of the current focal subpopulation. I found by simulations that currently widely applied relatedness estimators are upwardly biased under the indirect approach, but can be modified to become unbiased and more accurate by using Wright's F(st) to account for population structures. However, the modified unbiased estimators under the indirect approach are clearly inferior to the unmodified original estimators under the direct approach, even when small samples are used in estimating both allele frequencies and relatedness.     

Selection for recombination in structured populations

In finite populations, linkage disequilibria generated by the interaction of drift and directional selection (Hill-Robertson effect) can select for sex and recombination, even in the absence of epistasis. Previous models of this process predict very little advantage to recombination in large panmictic populations. In this article we demonstrate that substantial levels of linkage disequilibria can accumulate by drift in the presence of selection in populations of any size, provided that the population is subdivided. We quantify (i) the linkage disequilibrium produced by the interaction of drift and selection during the selective sweep of beneficial alleles at two loci in a subdivided population and (ii) the selection for recombination generated by these disequilibria. We show that, in a population subdivided into n demes of large size N, both the disequilibrium and the selection for recombination are equivalent to that expected in a single population of a size intermediate between the size of each deme (N) and the total size (nN), depending on the rate of migration among demes, m. We also show by simulations that, with small demes, the selection for recombination is stronger than both that expected in an unstructured population (m = 1 - 1/n) and that expected in a set of isolated demes (m = 0). Indeed, migration maintains polymorphisms that would otherwise be lost rapidly from small demes, while population structure maintains enough local stochasticity to generate linkage disequilibria. These effects are also strong enough to overcome the twofold cost of sex under strong selection when sex is initially rare. Overall, our results show that the stochastic theories of the evolution of sex apply to a much broader range of conditions than previously expected.     

Sex ratios in geographically structured populations

In his book on sexual selection (1), Darwin documented evidence that the primary sex ratio (the proportion of males at conception) is about 1/2 in a wide variety of species. Otherwise, he explained, a newly conceived member of the rare sex will, on average, have more offspring than one of the common sex, since each offspring has one mother and one father; thus there is frequency-dependent selection in favour of parents producing the rare sex. Darwin formulated this explanation in the first edition (1871) for monogamous species, but he failed to extend it to polygamous species, and in the second edition (1874) he retracted it completely. It was left to Fisher (2) to develop the theory in the more general form that there should be equal parental expenditure on the two sexes, allowing for the possibility that one sex may cost more to produce than the other. Despite the wide applicability of Fisher's principle, recent work on sex ratio evolution has focused on situations where it breaks down (3). Hamilton (4) first pointed out that Fisher's argument assumes population-wide competition for mates, whereas most natural populations have a geographical population structure in which limited dispersal imposes constraints on mating patterns. What are the consequences for the sex ratio?     

Survival in continuous structured populations models

The extended McKendrick-von Foerster structured population model is employed to derive a nonautonomous ordinary differential equation model of a population. The derivation assumes that the individual life history can be delineated into several physiological stages. We study the persistence of the population when the model is autonomous and base the nonautonomous survival analysis on the autonomous case and a comparison principle. A brief excursion into alternate life history strategies is presented.This work was supported in part by the U.S. Environmental Protection Agency under cooperative agreement CR 813353010     

Optimal sex ratios in structured populations

In this paper, we develop a general method to determine evolutionary equilibrium sex ratios and to check evolutionary stability, continuous stability and invadability in exact genetic models with or without dominance. This method is then applied to three kinds of models for structured populations: the first one concerns Hamilton's LMC model, except that only a fraction beta of female offspring mate with male offspring born in the same colonies, while a fraction 1-beta mate with male offspring chosen at random within the whole population; in the second model, it is assumed that partial dispersal of inseminated females occurs after mating; in the third model, partial dispersal of male and female offspring occurs before mating. In the first model, the effect of population regulation is studied while, in the other models, two kinds of dispersal are considered: proportional and uniform.     

Epidemic outbreaks on structured populations

Our chances to halt epidemic outbreaks rely on how accurately we represent the population structure underlying the disease spread. When analysing global epidemics this force us to consider metapopulation models taking into account intra- and inter-community interactions. Here I introduce and analyze a metapopulation model which accounts for several features observed in real outbreaks. First, I demonstrate that depending on the intra-community expected outbreak size and the fraction of social bridges the epidemic outbreaks die out or there is a finite probability to observe a global epidemics. Second, I show that the global scenario is characterized by resurgent epidemics, their number increasing with increasing the intra-community average distance between individuals. Finally, I present empirical data for the AIDS epidemics supporting the model predictions.     

Equilibria in systems of interacting structured populations

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k–1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.     

Gene genealogies in geographically structured populations.

Population genetics theory has dealt only with the spatial or geographic pattern of degrees of relatedness or genetic similarity separately for each point in time. However, a frequent goal of experimental studies is to infer migration patterns that occurred in the past or over extended periods of time. To fully understand how a present geographic pattern of genetic variation reflects one in the past, it is necessary to build genealogy models that directly relate the two. For the first time, space-time probabilities of identity by descent and coalescence probabilities are formulated and characterized in this article. Formulations for general migration processes are developed and applied to specific types of systems. The results can be used to determine the level of certainty that genes found in present populations are descended from ancient genes in the same population or nearby populations vs. geographically distant populations. Some parameter combinations result in past populations that are quite distant geographically being essentially as likely to contain ancestors of genes at a given population as the past population located at the same place. This has implications for the geographic point of origin of ancestral, "Eve," genes. The results also form the first model for emerging "space-time" molecular genetic data.     

Persistence of structured populations in random environments

Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and density-dependent interactions influence population dynamics, we provide a general theory for persistence for density-dependent matrix models in random environments. For populations with compensating density dependence, exhibiting “bounded” dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating density-dependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stage-structured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space.     

The strong-migration limit in geographically structured populations

Summary Some strong-migration limits are established for geographically structured populations. A diploid monoecious population is subdivided into a finite number of colonies, which exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus. In all the limiting results, an effective population number N _e ( N _T ) appears instead of the actual total population number N _T . 1. If there is no selection, every allele mutates at rate u to types not preexisting in the population, and the (finite) subpopulation numbers N _i are very large, then the ultimate rate and pattern of convergence of the probabilities of allelic identity are approximately the same as for panmixia. If, in addition, the N _i are proportional to 1/u, as N _T 8, the equilibrium probabilities of identity converge to the panmictic value. 2. With a finite number of alleles, any mutation pattern, an arbitrary selection scheme for each colony, and the mutation rates and selection coefficients proportional to 1/N _T , let P _j be the frequency of the allele A _j in the entire population, averaged with respect to the stationary distribution of the backward migration matrix M. As N _T 8, the deviations of the allelic frequencies in each of the subpopulations from P _j converge to zero; the usual panmictic mutation-selection diffusion is obtained for P _j , with the selection intensities averaged with respect to the stationary distribution of M. In both models, N _e = N _T and all effects of population subdivision disappear in the limit if, and only if, migration does not alter the subpopulation numbers.Supported by the National Science Foundation (Grant No. DEB77-21494)     

On indirect genetic effects in structured populations

Indirect genetic effects (IGEs) occur when the phenotype of an individual, and possibly its fitness, depends, at least in part, on the genes of its social partners. The effective result is that environmental sources of phenotypic variance can themselves evolve. Simple models have shown that IGEs can alter the rate and direction of evolution for traits involved in interactions. Here we expand the applicability of the theory of IGEs to evolution in metapopulations by including nonlinear interactions between individuals and population genetic structure. Although population subdivision alone generates some dramatic and nonintuitive evolutionary dynamics for interacting phenotypes, the combination of nonlinear interactions with subdivision reveals an even greater importance of IGEs. The presence of genetic structure links the evolution of interacting phenotypes and the traits that influence their expression ("effector traits") even in the absence of genetic correlations. When nonlinear social effects occur in subdivided populations, evolutionary response is altered and can even oppose the direction expected due to direct selection. Because population genetic structure allows for multilevel selection, we also investigate the role of IGEs in determining the response to individual and group selection. We find that nonlinear social effects can cause interference between levels of selection even when they act in the same direction. In some cases, interference can be so extreme that the actual evolutionary response to multilevel selection is opposite in direction to that predicted by summing selection at each level. This theoretical result confirms empirical data that show higher levels of selection cannot be ignored even when selection acts in the same direction at all levels.     

Generation separation in simple structured life cycles: models and 48 years of field data on a tea tortrix moth

Population cycles have fascinated ecologists since the early nineteenth century, and the dynamics of insect populations have been central to understanding the intrinsic and extrinsic biological processes responsible for these cycles. We analyzed an extraordinary long-term data set (every 5 days for 48 years) of a tea tortrix moth (Adoxophyes honmai) that exhibits two dominant cycles: an annual cycle with a conspicuous pattern of four or five single-generation cycles superimposed on it. General theory offers several candidate mechanisms for generation cycles. To evaluate these, we construct and parameterize a series of temperature-dependent, stage-structured models that include intraspecific competition, parasitism, mate-finding Allee effects, and adult senescence, all in the context of a seasonal environment. By comparing the observed dynamics with predictions from the models, we find that even weak larval competition in the presence of seasonal temperature forcing predicts the two cycles accurately. None of the other mechanisms predicts the dynamics. Detailed dissection of the results shows that a short reproductive life span and differential winter mortality among stages are the additional life-cycle characteristics that permit the sustained cycles. Our general modeling approach is applicable to a wide range of organisms with temperature-dependent life histories and is likely to prove particularly useful in temperate systems where insect pest outbreaks are both density and temperature dependent.     

Coexistence of competing juvenile-adult structured populations

The Leslie-Gower model is a discrete time analog of the competition Lotka-Volterra model and is known to possess the same dynamic scenarios of that famous model. The Leslie-Gower model played a historically significant role in the history of competition theory in its application to classic laboratory experiments of two competing species of flour beetles (carried out by Park in the 1940s-1960s). While these experiments generally supported what became the Competitive Exclusion Principle, Park observed an anomalous coexistence case. Recent literature has discussed Park's 'coexistence case' by means of non-Lotka-Volterra, non-equilibrium dynamics that occur in a high dimensional model with life cycle stages. We study this dynamic possibility in the lowest possible dimension, that is to say, by means of a model involving only two species each with two life cycle stages. We do this by extending the Leslie-Gower model so as to describe the competitive interaction of two species with juvenile and adult classes. We give a complete account of the global dynamics of the resulting model and show that it allows for non-equilibrium competitive coexistence as competition coefficients are increased. We also show that this phenomenon occurs in a general class of models for competing populations structured by juvenile and adult life cycle stages.