Identifying the density-dependent structure underlying ecological time series

A central problem in ecology is explaining the causes of population fluctuations, and an important step in the solution is determining the structure of the negative feedback (density dependent) process regulating population dynamics. The conventional way to determine the dimension or order of density dependence in a time series is to calculate the partial autorcorrelation function (PACF). We maintain, however, that PACF is not designed with biological populations in mind and has the wrong null model for detecting the structure of density dependence. We suggest an alternative diagnostic, the partial rate correlation function (PRCF), which is specifically designed for biological populations and has an appropriate null model for detecting their density dependent structures. Tests with simulated data show PRCF to be superior to PACF in detecting the underlying density dependent structure of two simple mathematical models.     

Extinction risks in metapopulations of a beetle inhabiting hollow trees predicted from time series

Ancient and dead trees are declining habitats harbouring many threatened species. These habitats are naturally patchy, and inhabiting species might exhibit metapopulation dynamics at a small spatial scale. In this study, the demography and metapopulation dynamics was analysed for Osmoderma eremita , which is an endangered beetle species associated with tree hollows in Europe. Extinction risks of O. eremita populations were predicted using Monte Carlo simulations based on time series of population assessments. Predicted occurrence patterns were consistent with field observations from an area with many small stands in which the populations are believed to have been more or less isolated from each other during the last 150–200 yr. Population growth was found to be density dependent. Carrying capacity was proportional to the volume of wood mould (i.e. loose material of dead wood in the tree hollows), which varied widely between hollow trees. This generates large differences in local extinction risks between hollow trees. The predicted metapopulation extinction risk was much higher if the habitat dynamics (formation, gradual increase and deterioration of tree hollows) were taken into consideration than in predictions yielded by models in which the amount of wood mould was assumed to be constant over time. Thus, this system has features from both mainland-island metapopulations and habitat-tracking metapopulations, and is rather far from a classic metapopulation. For the long-term persistence of the species in hollow trees, the habitat dynamics seem to be more important than demographic processes. Since the formation and deterioration of suitable tree are partly stochastic processes, there is a considerable extinction risk for many O. eremita populations, because they mainly rely on only one or a few trees with large amounts of wood mould.     

利用全基因组连锁不平衡估计中国荷斯坦牛有效群体大小

有效群体大小是群体遗传学研究的一个重要内容,有助于我们更清楚地了解群体的遗传变异、进化和复杂性状的遗传机制等。随着高密度SNP标记的出现,越来越多的研究利用SNP标记间连锁不平衡估计有效群体大小。文章采集北京地区中国荷斯坦牛2 093个样本,并利用牛SNP芯片(Illumina BovineSNP50,含5 4001 SNPs)进行基因型测定,估计不同世代中国荷斯坦牛的有效群体大小。质量控制标准设定为SNP检出率0.95,最小等位基因频率>0.05,样本检出率0.95,哈代温伯格平衡检验显著性水平P<0.0001。经过质量控制,共1 968个样本和38 796个SNPs用于连锁不平衡分析。文章选取SNP间距0.1、0.2、0.5、1、2、5、10、15(Mb),估计中国荷斯坦牛在4世代之前有效群体大小。结果表明,中国荷斯坦牛的有效群体呈逐代下降趋势,至4世代前,中国荷斯坦牛平均有效群体为45头左右。     

Extinction can be estimated from moderately sized molecular phylogenies

Hundreds of studies have been dedicated to estimating speciation and extinction from phylogenies of extant species. Although it has long been known that estimates of extinction rates using trees of extant organisms are often uncertain, an influential paper by Rabosky (2010) suggested that when birth rates vary continuously across the tree, estimates of the extinction fraction (i.e., extinction rate/speciation rate) will appear strongly bimodal, with a peak suggesting no extinction and a peak implying speciation and extinction rates are approaching equality. On the basis of these results, and the realistic nature of this form of rate variation, it is now generally assumed by many practitioners that extinction cannot be understood from molecular phylogenies alone. Here, we reevaluated and extended the analyses of Rabosky (2010) and come to the opposite conclusion—namely, that it is possible to estimate extinction from molecular phylogenies, even with model violations due to heritable variation in diversification rate. Note that while it may be tempting to interpret our study as advocating the application of simple birth–death models, our goal here is to show how a particular model violation does not necessitate the abandonment of an entire field: use prudent caution, but do not abandon all hope.     

Extinction time and age of an allele in a large finite population

For a single locus with two alleles we study the expected extinction and fixation times of the alleles under the influence of selection and genetic drift. Using a diffusion model we derive asymptotic approximations for these expected exit times for large populations. We consider the case where the fitness of the heterozygote is in between the fitnesses of the homozygotes (incomplete dominance). The asymptotic analysis not only yields new quantitative results but also reveals interesting features that remain hidden in the exact solution. Some of the outcomes are extensions of results known in the literature. The asymptotic approximations also apply to the expected first arrival time of an allele at a specified frequency and to the expected age of an allele.     

Is tuna critically endangered? Extinction risk of a large and overexploited population

In 1994, (the World Conservation Union (IUCN) made new quantitative criteria for Red List Categories. Among these, criterion A is based on the reduction rate of population size and is not linked to absolute population size. This is because for most unexploited threatened species absolute population size is completely uncertain. Criterion E is directly concerned with the extinction probability within a specified period. Criterion A is applied to marine fish species, including tuna, which are exploited by commercial fishing. However, for many fish that are commercially exploited, the absolute number of mature individuals and the variance in its yearly reduction rate are often known. In addition, extinction probability depends on absolute population size. There is an inconsistency between criterion A and E when population size is large enough. This is the case for tuna. In this paper, we propose a new criterion for threatened species based on the average reduction rate and the current population size. Criterion A is consistent with criterion E as long as the population size is very small. We also propose a method for estimating the extinction risk of tuna based on the variance of the reduction rate. We investigated the sensitivity in the uncertain parameters involved in the models and concluded that tuna is unlikely to be listed as critically endangered but that southern bluefin tuna may be listed as vulnerable.     

Local stability of a population with density-dependent fertility

The local stability of an equilibrium population configuration is investigated. The population is governed by a density-dependent, age-structured, nonlinear renewal equation. Local stability is shown to depend essentially on the rate of change of the net reproduction rate as a function of the population size. Examples illustrating the results and the characteristics promoting stability are discussed.     

Weak population regulation in ecological time series

How strongly natural populations are regulated has a long history of debate in ecology. Here, we discuss concepts of population regulation appropriate for stochastic population dynamics. We then analyse two large collections of data sets with autoregressive-moving average (ARMA) models, using model selection techniques to find best-fitting models. We estimated two metrics of population regulation: the characteristic return rate of populations to stationarity and the variability of the stationary distribution (the long-term distribution of population abundance). Empirically, longer time series were more likely to show weakly regulated population dynamics. For data sets of length ≥ 20, more than 35% had characteristic return times > 6 years, and more than 29% had stationary distributions whose coefficients of variation were more than two times greater than would be the case if they were maximally regulated. These results suggest that many natural populations are weakly regulated.
Ecology Letters (2010) 13: 21–31     

Fractal dimension of birds population sizes time series

Information about fractal dimension is collected so that it can be applied to time series interpreting Hurst coefficient. The population size of a species is modelled as a dynamic system. The Hurst coefficient is calculated for these times series. A computer programme has been elaborated to compute the Hurst exponent of time series using the algorithms of range increment, second order moment increment and local second order moment increment. It has been applied to time series of birds' populations.     

Interspecific competition among macroparasites in a density-dependent host population

 We analyze the dynamics of a community of macroparasite species that share the same host. Our work extends an earlier framework for a host species that would grow exponentially in the absence of parasitism, to one where an uninfected host population is regulated by factors other than parasites. The model consists of one differential equation for each parasite species and a single density-dependent nonlinear equation for the host. We assume that each parasite species has a negative binomial distribution within the host and there is zero covariance between the species (exploitation competition). New threshold conditions on model parameters for the coexistence and competitive exclusion of parasite species are derived via invadibility and stability analysis of corresponding equilibria. The main finding is that the community of parasite species coexisting at the stable equilibrium is obtained by ranking the species according t! o th e minimum host density H ^* above which a parasite species can grow when rare: the lower H ^* , the higher the competitive ability. We also show that ranking according to the basic reproduction number Q ₀ does not in general coincide with ranking according to H ^* . The second result is that the type of interaction between host and parasites is crucial in determining the competitive success of a parasite species, because frequency-dependent transmission of free-living stages enhances the invading ability of a parasite species while density-dependent transmission makes a parasite very sensitive to other competing species. Finally, we show that density dependence in the host population entails a simplification of the portrait of possible outcomes with respect to previous studies, because all the cases resulting in the exponential growth of host and parasite populations are eliminated.. Received: 24 June 1996 / Revised version: 28 April 1998     

Linking individual phenotype to density-dependent population growth: the influence of body size on the population dynamics of malaria vectors

Understanding the endogenous factors that drive the population dynamics of malaria mosquitoes will facilitate more accurate predictions about vector control effectiveness and our ability to destabilize the growth of either low- or high-density insect populations. We assessed whether variation in phenotypic traits predict the dynamics of Anopheles gambiae sensu lato mosquitoes, the most important vectors of human malaria. Anopheles gambiae dynamics were monitored over a six-month period of seasonal growth and decline. The population exhibited density-dependent feedback, with the carrying capacity being modified by rainfall (97% wAIC(c) support). The individual phenotypic expression of the maternal (p = 0.0001) and current (p = 0.040) body size positively influenced population growth. Our field-based evidence uniquely demonstrates that individual fitness can have population-level impacts and, furthermore, can mitigate the impact of exogenous drivers (e.g. rainfall) in species whose reproduction depends upon it. Once frontline interventions have suppressed mosquito densities, attempts to eliminate malaria with supplementary vector control tools may be attenuated by increased population growth and individual fitness.     

Estimating density dependence from population time series using demographic theory and life-history data

For populations with a density-dependent life history reproducing at discrete annual intervals, we analyze small or moderate fluctuations in population size around a stable equilibrium, which is applicable to many vertebrate populations. Using a life history having age at maturity alpha, with stochasticity and density dependence in adult recruitment and mortality, we derive a linearized autoregressive equation with time lags from 1 to alpha yr. Contrary to current interpretations, the coefficients corresponding to different time lags in the autoregressive dynamics are not simply measures of delayed density dependence but also depend on life-history parameters. The theory indicates that the total density dependence in a life history, D, should be defined as the negative elasticity of population growth rate per generation with respect to change in population size, [Formula: see text], where lambda is the asymptotic multiplicative growth rate per year, T is the generation time, and N is adult population size. The total density dependence in the life history, D, can be estimated from the sum of the autoregression coefficients. We estimate D in populations of seven vertebrate species for which life-history studies and unusually long time series of complete population censuses are available. Estimates of D were statistically significant and large, on the order of 1 or higher, indicating strong density dependence in five of the seven species. We also show that life history can explain the qualitative features of population autocorrelation functions and power spectra and observations of increasing empirical variance in population size with increasing length of time series.     

Intraspecific genetic variation of a Fagus sylvatica population in a temperate forest derived from airborne imaging spectroscopy time series

1. The growing pace of environmental change has increased the need for large‐scale monitoring of biodiversity. Declining intraspecific genetic variation is likely a critical factor in biodiversity loss, but is especially difficult to monitor: assessments of genetic variation are commonly based on measuring allele pools, which requires sampling of individuals and extensive sample processing, limiting spatial coverage. Alternatively, imaging spectroscopy data from remote platforms may hold the potential to reveal genetic structure of populations. In this study, we investigated how differences detected in an airborne imaging spectroscopy time series correspond to genetic variation within a population of Fagus sylvatica under natural conditions.
2. We used multi‐annual APEX (Airborne Prism Experiment) imaging spectrometer data from a temperate forest located in the Swiss midlands (Laegern, 47°28'N, 8°21'E), along with microsatellite data from F. sylvatica individuals collected at the site. We identified variation in foliar reflectance independent of annual and seasonal changes which we hypothesize is more likely to correspond to stable genetic differences. We established a direct connection between the spectroscopy and genetics data by using partial least squares (PLS) regression to predict the probability of belonging to a genetic cluster from spectral data.
3. We achieved the best genetic structure prediction by using derivatives of reflectance and a subset of wavebands rather than full‐analyzed spectra. Our model indicates that spectral regions related to leaf water content, phenols, pigments, and wax composition contribute most to the ability of this approach to predict genetic structure of F. sylvatica population in natural conditions.
4. This study advances the use of airborne imaging spectroscopy to assess tree genetic diversity at canopy level under natural conditions, which could overcome current spatiotemporal limitations on monitoring, understanding, and preventing genetic biodiversity loss imposed by requirements for extensive in situ sampling.

     

Effective population size of a population with stochastically varying size

For a Wright–Fisher model with mutation whose population size fluctuates stochastically from generation to generation, a heterozygosity effective population size is defined by means of the equilibrium average heterozygosity of the population. It is shown that this effective population size is equal to the harmonic mean of population size if and only if the stochastic changes of population size are uncorrelated. The effective population size is larger (resp. smaller) than the harmonic mean when the stochastic changes of population size are positively (resp. negatively) autocorrelated. These results and those obtained so far for other stochastic models with fluctuating population size suggest that the property that effective population sizes are always larger than the harmonic mean under the fluctuation of population size holds only for continuous time models such as diffusion and coalescent models, whereas effective population sizes can be equal to or smaller than the harmonic mean for discrete time models.     

Extinction risk of a meta-population: aggregation approach

Aggregation of variables of a complex mathematical model with realistic structure gives a simplified model which is more suitable than the original one when the amount of data for parameter estimation is limited. Here we explore use of a formula derived for a single unstructured population (canonical model) in predicting the extinction time for a population living in multiple habitats. In particular we focus multiple populations each following logistic growth with demographic and environmental stochasticities, and examine how the mean extinction time depends on the migration and environmental correlation. When migration rate and/or environmental correlation are very large or very small, we may express the mean extinction time exactly using the formula with properly modified parameters. When parameters are of intermediate magnitude, we generate a Monte Carlo time series of the population size for the realistic structured model, estimate the "effective parameters" by fitting the time series to the canonical model, and then calculate the mean extinction time using the formula for a single population. The mean extinction time predicted by the formula was close to those obtained from direct computer simulation of structured models. We conclude that the formula for an unstructured single-population model has good approximation capability and can be applicable in estimating the extinction risk of the structured meta-population model for a limited data set.     

Causality from palaeontological time series

As custodians of deep time, palaeontologists have an obligation to seek the causes and consequences of long‐term evolutionary trajectories and the processes of ecosystem assembly and collapse. Building explicit process models on the relevant scales can be fraught with difficulties, and causal inference is typically limited to patterns of association. In this review, we discuss some of the ways in which causal connections can be extracted from palaeontological time series and provide an overview of three recently developed analytical frameworks that have been applied to palaeontological questions, namely linear stochastic differential equations, convergent cross mapping and transfer entropy. We outline how these methods differ conceptually, and in practice, and point to available software and worked examples. We end by discussing why a paradigm of dynamical causality is needed to decipher the messages encrypted in palaeontological patterns.     

The evolution of density-dependent dispersal in a noisy spatial population model

It is well-known that dispersal is advantageous in many different ecological situations, e.g. to survive local catastrophes where populations live in spatially and temporally heterogeneous habitats. However, the key question, what kind of dispersal strategy is optimal in a particular situation, has remained unanswered. We studied the evolution of density-dependent dispersal in a coupled map lattice model, where the population dynamics are perturbed by external environmental noise. We used a very flexible dispersal function to enable evolution to select from practically all possible types of monotonous density-dependent dispersal functions. We treated the parameters of the dispersal function as continuously changing phenotypic traits. The evolutionary stable dispersal strategies were investigated by numerical simulations. We pointed out that irrespective of the cost of dispersal and the strength of environmental noise, this strategy leads to a very weak dispersal below a threshold density, and dispersal rate increases in an accelerating manner above this threshold. Decreasing the cost of dispersal increases the skewness of the population density distribution, while increasing the environmental noise causes more pronounced bimodality in this distribution. In case of positive temporal autocorrelation of the environmental noise, there is no dispersal below the threshold, and only low dispersal below it, on the other hand with negative autocorrelation practically all individual disperses above the threshold. We found our results to be in good concordance with empirical observations.     

Time constant for water transport in loblolly pine trees estimated from time series of evaporative demand and stem sapflow

 The use of stem sap flow data to estimate diurnal whole-tree transpiration and canopy stomatal conductance depends critically upon knowledge of the time lag between transpiration and water flux through the stem. In this study, the time constant for water movement in stems of 12-year-old Pinus taeda L. individuals was estimated from analysis of time series data of stem water flux and canopy transpiration computed from mean daytime canopy conductance, and diurnal vapor pressure deficit and solar radiation measurements. Water uptake through stems was measured using a constant-heat sapflow probe. Canopy transpiration was correlated to stem uptake using a resistance-capacitance equation that incorporates a time constant parameter. A least-squares auto-regression determined the parameters of the resistance-capacitance equation. The time constants for ten loblolly pine trees averaged 48.0 (SE = 2.0) min and the time lag for the diurnal frequency averaged 47.0 (SE = 2.0) min. A direct-cross correlation analysis between canopy transpiration and sap flow time series showed maximum correlation at an approximately 30 min lag. Residuals (model-predicted minus actual stem flow data) increased with increasing soil moisture depletion. While the time constants did not vary significantly within the range of tree sizes studied, hydraulic resistance and capacitance terms were individually dependent on stem cross-sectional area: capacitance increased and resistance decreased with stem volume. This result may indicate an inverse adjustment of resistance and capacitance to maintain a similar time constant over the range of tree sizes studied.     

Inference of population genetic parameters from an irregular time series of seasonal influenza virus sequences

Basic summary statistics that quantify the population genetic structure of influenza virus are important for understanding and inferring the evolutionary and epidemiological processes. However, the sampling dates of global virus sequences in the last several decades are scattered nonuniformly throughout the calendar. Such temporal structure of samples and the small effective size of viral population hampers the use of conventional methods to calculate summary statistics. Here, we define statistics that overcome this problem by correcting for the sampling-time difference in quantifying a pairwise sequence difference. A simple linear regression method jointly estimates the mutation rate and the level of sequence polymorphism, thus providing an estimate of the effective population size. It also leads to the definition of Wright’s F_ST for arbitrary time-series data. Furthermore, as an alternative to Tajima’s D statistic or the site-frequency spectrum, a mismatch distribution corrected for sampling-time differences can be obtained and compared between actual and simulated data. Application of these methods to seasonal influenza A/H3N2 viruses sampled between 1980 and 2017 and sequences simulated under the model of recurrent positive selection with metapopulation dynamics allowed us to estimate the synonymous mutation rate and find parameter values for selection and demographic structure that fit the observation. We found that the mutation rates of HA and PB1 segments before 2007 were particularly high and that including recurrent positive selection in our model was essential for the genealogical structure of the HA segment. Methods developed here can be generally applied to population genetic inferences using serially sampled genetic data.