Antagonistic selection factors induce a continuous population divergence in a polymorphism

Understanding the relative importance of selection and stochastic factors in population divergence of adaptive traits is a classical topic in evolutionary biology. However, it is difficult to separate these factors and detect the effects of selection when two or more contrasting selective factors are simultaneously acting on a single locus. In the damselfly Ischnura senegalensis, females exhibit color dimorphism and morph frequencies change geographically. We here evaluated the role of selection and stochastic factors in population divergence of morph frequencies by comparing the divergences in color locus and neutral loci. Comparisons between population pairwise F_ST for neutral loci and for the color locus did not detect any stochastic factors affecting color locus. Although comparison between population divergence in color and neutral loci using all populations detected only divergent selection, we detected two antagonistic selective factors acting on the color locus, that is, balancing and divergent selection, when considering geographical distance between populations. Our results suggest that a combination of two antagonistic selective factors, rather than stochastic factors, establishes the geographic cline in morph frequency in this system.     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

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.     

Pools versus Queues: The Variable Dynamics of Stochastic “Steady States”

Mathematical models in ecology and epidemiology often consider populations “at equilibrium”, where in-flows, such as births, equal out-flows, such as death. For stochastic models, what is meant by equilibrium is less clear – should the population size be fixed or growing and shrinking with equal probability? Two different mechanisms to implement a stochastic steady state are considered. Under these mechanisms, both a predator-prey model and an epidemic model have vastly different outcomes, including the median population values for both predators and prey and the median levels of infection within a hospital (P < 0.001 for all comparisons). These results suggest that the question of how a stochastic steady state is modeled, and what it implies for the dynamics of the system, should be carefully considered.     

Unmasking Chaotic Attributes in Time Series of Living Cell Populations

Background

Long-range oscillations of the mammalian cell proliferation rate are commonly observed both in vivo and in vitro. Such complicated dynamics are generally the result of a combination of stochastic events and deterministic regulation. Assessing the role, if any, of chaotic regulation is difficult. However, unmasking chaotic dynamics is essential for analysis of cellular processes related to proliferation rate, including metabolic activity, telomere homeostasis, gene expression, and tumor growth.

Methodology/Principal Findings

Using a simple, original, nonlinear method based on return maps, we previously found a geometrical deterministic structure coordinating such fluctuations in populations of various cell types. However, nonlinearity and determinism are only necessary conditions for chaos; they do not by themselves constitute a proof of chaotic dynamics. Therefore, we used the same analytical method to analyze the oscillations of four well-known, low-dimensional, chaotic oscillators, originally designed in diverse settings and all possibly well-adapted to model the fluctuations of cell populations: the Lorenz, Rössler, Verhulst and Duffing oscillators. All four systems also display this geometrical structure, coordinating the oscillations of one or two variables of the oscillator. No such structure could be observed in periodic or stochastic fluctuations.

Conclusion/Significance

Theoretical models predict various cell population dynamics, from stable through periodically oscillating to a chaotic regime. Periodic and stochastic fluctuations were first described long ago in various mammalian cells, but by contrast, chaotic regulation had not previously been evidenced. The findings with our nonlinear geometrical approach are entirely consistent with the notion that fluctuations of cell populations can be chaotically controlled.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Moment closure approximations for stochastic kinetic models with rational rate laws

Stochastic models are often used when modelling chemical species that have low numbers of molecules. However, as these models become large, it can become computationally expensive to simulate even a single realisation of the system since even efficient simulation techniques have a high computational cost. One possible technique to approximate the stochastic system is moment closure. The moment closure approximation is used to provide analytic approximations to non-linear stochastic models. Until now, this approximation has only been applied to models with polynomial rate laws. In this paper we extend the moment closure method to cover models with rational rate laws.     

Spatial variation in phenotypic selection on floral characteristics in an epiphytic orchid

Spatial variation in twelve floral characters was examined in an epiphytic orchidLepanthes rupestris to evaluate the strength and direction of phenotypic selection in seven riparian populations along two river basins in the Caribbean National Forest “El Yunque” for a range of 18–34 months. We evaluated selection on floral characters based on male (pollinaria removal) and female fitness (fruit set). Simple linear and quadratic regressions were used to evaluate the strength of directional, disruptive and stabilizing selections. Univariate and multivariate analyses were used to estimate the total strength of the selection acting on a character. Phenotypic selection was inconsistent among characters and populations. Few of the characters appeared to be under selection and none of them was found to be consistent throughout all populations. Inconsistency in selection coefficients among populations could suggest that selection is spatially variable. We only noted one character (column length) which had some consistency in differential selection coefficients among populations. Previous studies have shown that effective population sizes inL. rupestris are small and the observed “fitness differences” among populations could as easily be explained as stochastic events at play. We argue that the observed “fitness differences” in most characters and inconsistency among populations are likely from stochastic noise and not phenotypic selection. Consequently, we propose that random selection on character state support the hypothesis of genetic drift in small orchid populations.     

Density-dependent demographic variation determines extinction rate of experimental populations

Understanding population extinctions is a chief goal of ecological theory. While stochastic theories of population growth are commonly used to forecast extinction, models used for prediction have not been adequately tested with experimental data. In a previously published experiment, variation in available food was experimentally manipulated in 281 laboratory populations of Daphnia magna to test hypothesized effects of environmental variation on population persistence. Here, half of those data were used to select and fit a stochastic model of population growth to predict extinctions of populations in the other half. When density-dependent demographic stochasticity was detected and incorporated in simple stochastic models, rates of population extinction were accurately predicted or only slightly biased. However, when density-dependent demographic stochasticity was not accounted for, as is usual when forecasting extinction of threatened and endangered species, predicted extinction rates were severely biased. Thus, an experimental demonstration shows that reliable estimates of extinction risk may be obtained for populations in variable environments if high-quality data are available for model selection and if density-dependent demographic stochasticity is accounted for. These results suggest that further consideration of density-dependent demographic stochasticity is required if predicted extinction rates are to be relied upon for conservation planning.     

Systematization of a set of closure techniques

Approximations in population dynamics are gaining popularity since stochastic models in large populations are time consuming even on a computer. Stochastic modeling causes an infinite set of ordinary differential equations for the moments. Closure models are useful since they recast this infinite set into a finite set of ordinary differential equations. This paper systematizes a set of closure approximations. We develop a system, which we call a power p closure of n moments, where 0≤p≤n. [Keeling, 2000a] and [Keeling, 2000b] approximation with third order moments is shown to be an instantiation of this system which we call a power 3 closure of 3 moments. We present an epidemiological example and evaluate the system for third and fourth moments compared with Monte Carlo simulations.     

Mixed mating in androdioecious Mercurialis annua inferred using progeny arrays and diploid-acting microsatellite loci in a hexaploid background

Background and Aims

The frequency at which males can be maintained with hermaphrodites in androdioecious populations is predicted to depend on the selfing rate, because self-fertilization by hermaphrodites reduces prospective siring opportunities for males. In particular, high selfing rates by hermaphrodites are expected to exclude males from a population. Here, the first estimates are provided of the mating system from two wild hexaploid populations of the androdioecious European wind-pollinated plant M. annua with contrasting male frequencies.

Methods

Four diploid microsatellite loci were used to genotype 19–20 progeny arrays from two populations of M. annua, one with males and one without. Mating-system parameters were estimated using the program MLTR.

Key Results

Both populations had similar, intermediate outcrossing rates (t_m = 0·64 and 0·52 for the population with and without males, respectively). The population without males showed a lower level of correlated paternity and biparental inbreeding and higher allelic richness and gene diversity than the population with males.

Conclusions

The results demonstrate the utility of new diploid microsatellite loci for mating system analysis in a hexaploid plant. It would appear that androdioecious M. annua has a mixed-mating system in the wild, an uncommon finding for wind-pollinated species. This study sets a foundation for future research to assess the relative importance of the sexual system, plant-density variation and stochastic processes for the regulation of male frequencies in M. annua over space and time.     

Genome-wide population genetic analysis identifies evolutionary forces establishing continuous population divergence

Elucidating the mechanism shaping the spatial variations of traits has long been a central concern of evolutionary biologists. Geographic clines of allele/morph frequencies along environmental gradients are suggested to be established and maintained by the balancing of two opposing evolutionary forces, namely selection that generates spatial differentiation in morph frequencies, and selection and/or stochastic factors that lead to the coexistence of multiple morphs within a population. Thus, testing for both selection and stochastic factors is necessary for a comprehensive understanding of the mechanism underlying clinal variation in morph/allele frequency in natural populations. Here, I identified the evolutionary forces responsible for clinal variation of color morph frequency in Ischnura senegalensis by comparing the population divergence of putatively neutral loci generated by high-throughput next-generation sequencing (F _STn) with that of the putative color locus (F _STc). No strong correlation was observed between F _STn and F _STc, suggesting that stochastic factors contribute less to color-locus population divergence. F _STc was less than F _STn between populations exposed to similar environmental conditions, but greater than F _STn between populations exposed to different environmental conditions, suggesting that both balancing selection and divergent selection act on the color locus. Therefore, two antagonistic selection factors rather than stochastic and historical factors contribute to establishing the clinal variation of morph frequency in I. senegalensis.     

Predator-prey interactions in a nonequilibrium context: the metapopulation approach to modeling "hide-and-seek" dynamics in a spatially explicit tri-trophic system

Predators and prey are usually heterogeneously distributed in space so that the ability of the predators to respond to the distribution of their prey may have a profound influence on the stability and persistence of a predator‐prey system. A special type of dynamics is “hide‐and‐seek” characterized by a high turnover rate of local populations of prey and predators, because once the predators have found a patch of prey they quickly overexploit it, whereupon the starving predators either should move to better places or die. Continued persistence of prey and predators thus hinges on a long‐term balance between local extinctions and founding of new subpopulations. The colonization rate depends on the rate of emigration from occupied patches and the likelihood of successfully arriving at a suitable new patch, while extinction rate depends on the local population dynamics. Since extinctions and colonizations are both discrete probabilistic events, these phenomena are most adequately modeled by means of a stochastic model. In order to demonstrate the qualitative differences between a deterministic and stochastic approach to population dynamics, a spatially explicit tritrophic predator‐prey model is developed in a deterministic and a stochastic version. The model is parameterized using data for the two‐spotted spider mite (Tetranychus urticae) and the phytoseiid mite predator Phytoseiulus persimilis inhabiting greenhouse cucumbers.
Simulations show that the deterministic and stochastic approaches yield different results. The deterministic version predicts that the populations will exhibit violent fluctuations, implying that the system is fundamentally unstable. In contrast, the stochastic version predicts that the two species will be able to coexist in spite of frequent local extinctions of both species, provided the system consists of a sufficiently large number of local populations. This finding is in agreement with experimental results. It is therefore concluded that demographic stochasticity in combination with dispersal is capable of producing and maintaining sufficient asynchrony between local populations to ensure long‐term regional (metapopulation) persistence.     

The effects of large and small-scale random events on the synchrony of metapopulation dynamics: a theoretical analysis

We present an analysis of the conditions under which migration and global random factors may determine large scale synchrony in the dynamics of spatially structured populations. We derive an analytic approximation which describes how the desynchronizing influence of local environmental stochasticity combines with the synchronizing influences of larger scale environmental stochastic variation and migration to determine population cross correlation coefficients. Despite the simplifications made by this analysis, computer simulations show that the behaviour of more complicated models is well described by our approximation over considerable regions of parameter space. We conclude that population synchrony is largely determined by the coefficients of variation (CVs) of the local and larger scale stochastic processes, and that migration alone is only likely to maintain population synchrony when the CV of the local stochastic process is very small.     

Stochastic modeling of the transmission of respiratory syncytial virus (RSV) in the region of Valencia,Spain

In this paper, we study the dynamics of the transmission of respiratory syncytial virus (RSV) in the population using stochastic models. The stochastic models are developed introducing stochastic perturbations on the demographic parameter as well as on the transmission rate of the RSV. Numerical simulations of the deterministic and stochastic models are performed in order to understand the effect of fluctuating birth rate and transmission rate of the RSV on the population dynamics. The numerical solutions of stochastic models are calculated using Euler-Maruyama and Milstein schemes, and confidence intervals for stochastic solutions are given using Monte-Carlo method. Analysis of the numerical results reveals that perturbations on the transmission rate are more decisive in the dynamics of RSV than perturbations on demographic parameters. In addition, the stochastic models show the advantage of reproducing more effectively the noisy RSV hospitalization data. It is concluded that these stochastic models are a viable option to provide a realistic modeling of the RSV dynamics on the population.     

Proposing a two-level stochastic model for epileptic seizure genesis

By assuming the brain as a multi-stable system, different scenarios have been introduced for transition from normal to epileptic state. But, the path through which this transition occurs is under debate. In this paper a stochastic model for seizure genesis is presented that is consistent with all scenarios: a two-level spontaneous seizure generation model is proposed in which, in its first level the behavior of physiological parameters is modeled with a stochastic process. The focus is on some physiological parameters that are essential in simulating different activities of ElectroEncephaloGram (EEG), i.e., excitatory and inhibitory synaptic gains of neuronal populations. There are many depth-EEG models in which excitatory and inhibitory synaptic gains are the adjustable parameters. Using one of these models at the second level, our proposed seizure generator is complete. The suggested stochastic model of first level is a hidden Markov process whose transition matrices are obtained through analyzing the real parameter sequences of a seizure onset area. These real parameter sequences are estimated from real depth-EEG signals via applying a parameter identification algorithm. In this paper both short-term and long-term validations of the proposed model are done. The long-term synthetic depth-EEG signals simulated by this model can be taken as a suitable tool for comparing different seizure prediction algorithms.     

Invasion genetics of <Emphasis Type="Italic">Senecio vulgaris</Emphasis>: loss of genetic diversity characterizes the invasion of a selfing annual,despite multiple introductions

Genetic variation in invasive populations is affected by a variety of processes including stochastic forces, multiple introductions, population dynamics and mating system. Here, we compare genetic diversity between native and invasive populations of the selfing, annual plant Senecio vulgaris to infer the relative importance of genetic bottlenecks, multiple introductions, post-introduction genetic drift and gene flow to genetic diversity in invasive populations. We scored multilocus genotypes at eight microsatellite loci from nine native European and 19 Chinese introduced populations and compared heterozygosity and number of alleles between continents. We inferred possible source populations for introduced populations by performing assignment analyses and evaluated the relative contributions of gene flow and genetic drift to genetic diversity based on correlations of pairwise genetic and geographic distance. Genetic diversity within Chinese populations was significantly reduced compared to European populations indicating genetic bottlenecks accompanying invasion. Assignment tests provided support for multiple introductions with populations from Central China and southwestern China descended from genotypes matching those from Switzerland and the UK, respectively. Genetic differentiation among populations in China and Europe was not correlated with geographic distance. However, European populations exhibited less variation in the relation between G _ST and geographical distance than populations in China. These results suggest that gene flow probably plays a more significant role in structuring genetic diversity in native populations, whereas genetic drift appears to predominate in introduced populations. High rates of selfing in Chinese populations may restrict opportunities for pollen-mediated gene flow. Repeated colonization-extinction cycles associated with ongoing invasion is likely to maintain low genetic diversity in Chinese populations.     

Informativity of some of the molecular markers most frequently used in population genetics of salmonids

The genetic parameters of donor and artificial populations of Siberian salmon (Oncorhynchus keta) have been analyzed. The parameters were obtained with the use of different molecular markers as well as mathematic modeling based on experimental data. The difference between results is explained by the different resolving power of individual markers and/or stochastic effects. It is assumed that these factors together with temporal differentiation of spawning salmonid populations and the stable interpopulation gene flow constrain resolution of methods based on the use of molecular markers and designated for the analysis of differentiation of salmonid populations.