The structure and dynamics of woodreed Calamagrostis canescens population: a modelling approach

A scale of ontogenetic states has been developed for woodreed Calamagrostis canescens, a perennial species dominating the grass layer of fell forest areas. The population structure is considered as a set of age-stage groups of individuals differing both in the ontogenetic stage and the chronological age measured in years. to describe the dynamics through years a special kind of matrix formalism has been proposed which is reducible neither to the classic Leslie matrix for an age-structured population, nor to the well-known Lefkovitch matrix for a stage-structured one, and which does not suffer from excessiveness of the "two-dimensional" representation for the structure implying the projection matrix of a block pattern. It has been shown however that the protection matrix corresponding to C. canescens life-history graph embodies the canonical features of matrix formalism for structured population dynamics, such as the exponential population growth or decline, the convergence to a stable equilibrium structure, the calculable indicator of growth/decline/equilibrium (i.e., a measure of the population reproductive potential) as well as possibility to determine the relative reproductive value of each group. On the other hand, "left-sidedness of the age spectrum", a property that is often observed in real populations and is inherent in Leslie models of growing populations, may fail in the age-stage-structured model. The aggregation of age-stage groups into the age classes is possible only under special strict relationship among the age-stage-specific vital rates of the population. The both circumstances serve a methodical indication that an additional dimension such as the stages, for example, ought to be introduced into the age structure of the model population.     

Demographic variance in heterogeneous populations: matrix models and sensitivity analysis

The demographic consequences of stochasticity in processes such as survival and reproduction are modulated by the heterogeneity within the population. Therefore, to study effects of stochasticity on population growth and extinction risk, it is critical to use structured population models in which the most important sources of heterogeneity (e.g. age, size, developmental stage) are incorporated as i‐state variables. Demographic stochasticity in heterogeneous populations has often been studied using one of two approaches: multitype branching processes and diffusion approximations. Here, we link these approaches, through the demographic stochasticity in age‐ or stage‐structured matrix population models. We derive the demographic variance, σ²_d, which measures the per capita contribution to the variance in population growth increment, and we show how it can be decomposed into contributions from transition probabilities and fertility across ages or stages. Furthermore, using matrix calculus we derive the sensitivity of σ²_d to age‐ or stage‐specific mortality and fertility. We apply the methods to an extensive set of data from age‐classified human populations (long‐term time‐series for Sweden, Japan and the Netherlands; two hunter–gatherer populations, and the high‐fertility Hutterites), and to a size‐classified population of the herbaceous plant Calathea ovandensis. For the human populations our analysis reveals substantial temporal changes in the demographic variance as well as its main components across age. These new methods provide a powerful approach for calculating the demographic variance for any structured model, and for analyzing its main components and sensitivities. This will make possible new analyses of demographic variance across different kinds of heterogeneity in different life cycles, which will in turn improve our understanding of mechanisms underpinning extinction risk and other important biological outcomes.     

A general formula for the sensitivity of population growth rate to changes in life history parameters.

This paper considers the sensitivity of population growth to small changes in birth, growth, survival, and migration probabilities for an arbitrary population classification (i.e., age, instar, size, developmental stage, age, and spatial location, etc.). The stage-specific life history parameters are expressed in a discrete-time system of linear difference equations, the dominant eigenvalue of which defines the population growth rate. The sensitivity of this eigenvalue to production of class i by class j individuals is shown to be proportional to the product of the reproductive value of stage i and the abundance of stage j in the stable stage distribution. This formula is readily computable, and several examples are presented. For the special case of age-structured populations, this formula reduces to those derived by Hamilton, Emlen, and Goodman.     

Aggregation of Leslie matrix models with application to ten diverse animal species

Some grouping is necessary when constructing a Leslie matrix model because it involves discretizing a continuous process of births and deaths. The level of grouping is determined by the number of age classes and frequency of sampling. It is largely unknown what is lost or gained by using fewer age classes, and I address this question using aggregation theory. I derive an aggregator for a Leslie matrix model using weighted least squares, determine what properties an aggregated matrix inherits from the original matrix, evaluate aggregation error, and measure the influence of aggregation on asymptotic and transient behaviors. To gauge transient dynamics, I employ reactivity of the standardized Leslie matrix. I apply the aggregator to 10 Leslie models developed for animal populations drawn from a diverse set of species. Several properties are inherited by the aggregated matrix: (a) it is a Leslie matrix; (b) it is irreducible whenever the original matrix is irreducible; (c) it is primitive whenever the original matrix is primitive; and (d) its stable population growth rate and stable age distribution are consistent with those of the original matrix if the least squares weights are equal to the original stable age distribution. In the application, depending on the population modeled, when the least squares weights do not follow the stable age distribution, the stable population growth rate of the aggregated matrix may or may not be approximately consistent with that of the original matrix. Transient behavior is lost with high aggregation.     

Cladoceran birth and death rates estimates: experimental comparisons of egg-ratio methods

I. Birth and death rates of natural cladoceran populations cannot be measured directly. Estimates of these population parameters must be calculated using methods that make assumptions about the form of population growth. These methods generally assume that the population has a stable age distribution.
2. To assess the effect of variable age distributions, we tested six egg ratio methods for estimating birth and death rates with data from thirty-seven laboratory populations of Daphnia pulicaria. The populations were grown under constant conditions, but the initial age distributions and egg ratios of the populations varied. Actual death rates were virtually zero, so the difference between the estimated and actual death rates measured the error in both birth and death rate estimates.
3. The results demonstrate that unstable population structures may produce large errors in the birth and death rates estimated by any of these methods. Among the methods tested, Taylor and Slatkin's formula and Paloheimo's formula were most reliable for the experimental data.
4. Further analyses of three of the methods were made using computer simulations of growth of age-structured populations with initially unstable age distributions. These analyses show that the time interval between sampling strongly influences the reliability of birth and death rate estimates. At a sampling interval of 2.5 days (equal to the duration of the egg stage), Paloheimo's formula was most accurate. At longer intervals (7.5–10 days), Taylor and Slatkin's formula which includes information on population structure was most accurate.     

Fixed point sensitivity analysis of interacting structured populations

Sensitivity analysis of structured populations is a useful tool in population ecology. Historically, methodological development of sensitivity analysis has focused on the sensitivity of eigenvalues in linear matrix models, and on single populations. More recently there have been extensions to the sensitivity of nonlinear models, and to communities of interacting populations. Here we derive a fully general mathematical expression for the sensitivity of equilibrium abundances in communities of interacting structured populations. Our method yields the response of an arbitrary function of the stage class abundances to perturbations of any model parameters. As a demonstration, we apply this sensitivity analysis to a two-species model of ontogenetic niche shift where each species has two stage classes, juveniles and adults. In the context of this model, we demonstrate that our theory is quite robust to violating two of its technical assumptions: the assumption that the community is at a point equilibrium and the assumption of infinitesimally small parameter perturbations. Our results on the sensitivity of a community are also interpreted in a niche theoretical context: we determine how the niche of a structured population is composed of the niches of the individual states, and how the sensitivity of the community depends on niche segregation.     

Demographic analysis of continuous-time life-history models

I present a computational approach to calculate the population growth rate, its sensitivity to life-history parameters and associated statistics like the stable population distribution and the reproductive value for exponentially growing populations, in which individual life history is described as a continuous development through time. The method is generally applicable to analyse population growth and performance for a wide range of individual life-history models, including cases in which the population consists of different types of individuals or in which the environment is fluctuating periodically. It complements comparable methods developed for discrete-time dynamics modelled with matrix or integral projection models. The basic idea behind the method is to use Lotka's integral equation for the population growth rate and compute the integral occurring in that equation by integrating an ordinary differential equation, analogous to recently derived methods to compute steady-states of physiologically structured population models. I illustrate application of the method using a number of published life-history models.     

When genes go to sleep: the population genetic consequences of seed dormancy and monocarpic perenniality

In many annual plant populations, seeds may be dormant for several seasons before they germinate. Here, we investigate the consequences of both conditional (dispersed seeds cannot enter a dormant stage) and unconditional seed dormancy on the amount and the distribution of neutral genetic diversity within and among populations. We present joint demographic and population genetics models for single and subdivided populations and derive the effective size and population differentiation at both local and metapopulation scales. We suggest that a Wahlund effect is unlikely to result from age structure alone. Furthermore, the differentiation among populations is decreased by the presence of seed banks. We also extend these models to describe monocarpic (semelparous) perennial life cycle, where the nonreproductive stages are vegetative rosettes instead of dormant seeds. The main difference between the models relies in the way the density-dependent regulation is acting. The effective size of monocarpic perennial species may be less than the census number of individuals, and among-population differentiation is always larger than in annual species. We discuss our results in the light of recent population genetics surveys of annual plants with seed banks.     

Demography when history matters: construction and analysis of second-order matrix population models

History matters when individual prior conditions contain important information about the fate of individuals. We present a general framework for demographic models which incorporates the effects of history on population dynamics. The framework incorporates prior condition into the i-state variable and includes an algorithm for constructing the population projection matrix from information on current state dynamics as a function of prior condition. Three biologically motivated classes of prior condition are included: prior stages, linear functions of current and prior stages, and equivalence classes of prior stages. Taking advantage of the matrix formulation of the model, we show how to calculate sensitivity and elasticity of any demographic outcome. Prior condition effects are a source of inter-individual variation in vital rates, i.e., individual heterogeneity. As an example, we construct and analyze a second-order model of Lathyrus vernus, a long-lived herb. We present population growth rate, the stable population distribution, the reproductive value vector, and the elasticity of λ to changes in the second-order transition rates. We quantify the contribution of prior conditions to the total heterogeneity in the stable population of Lathyrus using the entropy of the stable distribution.     

甘南高山林线岷江冷杉—杜鹃种群结构与动态

高山林线是一种典型的生态交错带,是对气候反映最敏感的地区之一。甘肃南部高山林线区域主要以原始岷江冷杉种群和杜鹃种群为优势种,通过对岷江冷杉和杜鹃种群建立静态生命表,绘制存活曲线描述其结构特征,利用种群数量动态预测时间序列分析定量研究未来的发展趋势。结果显示:(1)岷江冷杉种群幼苗比较丰富,能很好的维持种群个体的自疏死亡,存活曲线呈Deevey-Ⅲ型;杜鹃种群幼苗缺乏,存活曲线趋向于Deevey-Ⅰ型;死亡曲线和危险率曲线都随着龄级的增加而增加,杜鹃种群的死亡率在各个龄级一直大于岷江冷杉种群,危险率在Ⅱ龄级以后杜鹃种群也始终大于岷江冷杉种群。(2)岷江冷杉种群结构动态变化指数Vpi大于修正后的种群结构动态变化指数V′pi且大于0,而杜鹃种群结构动态变化指数Vpi小于修正后的种群结构动态变化指数V′pi且小于0,则岷江冷杉种群属于增长型,杜鹃种群属于衰退型,岷江冷杉、杜鹃随机干扰极大值分别为0.027、0.011,说明二者对外界干扰均比较敏感。(3)杜鹃种群时间序列预测为前期幼苗比较缺乏,中期稳定,后期衰退的动态特征,而岷江冷杉种群表现出各龄级时间变化较小,幼苗个体数较多,种群为稳定增长型,岷江冷杉更能适应甘肃南部高山林线区域当前环境。     

Lifetime reproductive output: individual stochasticity,variance, and sensitivity analysis

Lifetime reproductive output (LRO) determines per-generation growth rates, establishes criteria for population growth or decline, and is an important component of fitness. Empirical measurements of LRO reveal high variance among individuals. This variance may result from genuine heterogeneity in individual properties, or from individual stochasticity, the outcome of probabilistic demographic events during the life cycle. To evaluate the extent of individual stochasticity requires the calculation of the statistics of LRO from a demographic model. Mean LRO is routinely calculated (as the net reproductive rate), but the calculation of variances has only recently received attention. Here, we present a complete, exact, analytical, closed-form solution for all the moments of LRO, for age- and stage-classified populations. Previous studies have relied on simulation, iterative solutions, or closed-form analytical solutions that capture only part of the sources of variance. We also present the sensitivity and elasticity of all of the statistics of LRO to parameters defining survival, stage transitions, and (st)age-specific fertility. Selection can operate on variance in LRO only if the variance results from genetic heterogeneity. The potential opportunity for selection is quantified by Crow’s index \(\mathcal {I}\), the ratio of the variance to the square of the mean. But variance due to individual stochasticity is only an apparent opportunity for selection. In a comparison of a range of age-classified models for human populations, we find that proportional increases in mortality have very small effects on the mean and variance of LRO, but large positive effects on \(\mathcal {I}\). Proportional increases in fertility increase both the mean and variance of LRO, but reduce \(\mathcal {I}\). For a size-classified tree population, the elasticity of both mean and variance of LRO to stage-specific mortality are negative; the elasticities to stage-specific fertility are positive.     

Complexity in matrix population models: Polyvariant ontogeny and reproductive uncertainty

Linear matrix models of stage-structured population dynamics are widely used in plant and animal demography as a tool to evaluate the growth potential of a population in a given environment. The potential is identified with λ₁, the dominant eigenvalue of the projection matrix, which is compiled of stage-specific transition and fertility rates. Advanced botanical studies reveal polyvariant ontogeny in perennial plants, i.e., multiple different versions of individual development within a local population of a single species. This phenomenon complicates any standard, successive-stage, life cycle graph to a digraph defined on a 2D lattice in the age and stage dimensions, the pattern of projection matrix becoming more complex too. In a kind of experimental design, the transition rates can be calculated directly from the data for two successive time moments, but the age-stage-specific rates of reproduction still remain uncertain, adding more complexity to the calibration problem. Simple additional assumptions could technically eliminate the uncertainty, but they contravene the biology of a species in which polyvariant ontogeny is considered to be the major mechanism of adaptation. Given the data and expert constraints, the calibration can be reduced instead to a nonlinear maximization problem, yet with linear constraints. I prove that it has a unique solution to be attained at a vertex of the constraint polyhedral. To facilitate searching for the solution in practice, I use the net reproductive rate R₀, a well-known indicator for the principal property of λ₁ to be greater or less than 1. The method is exemplified with the calibration of a projection matrix in an age-stage-structured model (published elsewhere) for Calamagrostis canescens, a perennial herbaceous species with a complex (multivariant) life cycle that features unlimited growth when colonizing open areas.     

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

Estimating population birth rates of zooplankton when rates of egg deposition and hatching are periodic

Summary I present a general method of computing finite birth and death rates of natural zooplankton populations from changes in the age distribution of eggs and changes in population size. The method is applicable to cases in which eggs hatch periodically owing to variable rates of oviposition. When morphological criteria are used to determine the age distribution of eggs at the beginning and end of a sampling interval, egg mortality can be incorporated in estimates of population birth rate. I raised laboratory populations of Asplanchna priodonta, a common planktonic rotifer, in semicontinuous culture to evaluate my method of computing finite birth rate. The Asplanchna population became synchronized to a daily addition of food but grew by the same amount each day once steady state was achieved. The steady-state rate of growth, which can be computed from the volume-specific dilution rate of the culture, was consistent with the finite birth rate predicted from the population's egg ratio and egg age distribution.     

昆虫多状态发育的生存分析

针对昆虫种群变态发育过程,本文给出了一个多状态生存函数的模型。使用生存分析的方法对模型进行丁分析。本文还对有关的生存参数如各状态的死亡风险,发育风险,年龄特征死亡率,年龄特征发育率以及状态发育历期等进行了讨论并且给出了它们的极大似然估计值。关于马铃薯块茎蛾数值例子的分析表明所提出的摸型用来描述昆虫种群的发育过程是有效的。本文的结论可以做为组建描述昆虫种群多状发育的年龄一状态特征生命表的理论基础。     

Prediction vs. reality: Can a PVA model predict population persistence 13 years later?

The challenge of conservation biology is to make models that predict population dynamics and have a high probability of accurately tracking population change (increase, decrease, constancy). In this study we tested whether the transition model is accurate enough to predict population persistence and size 13 years down and whether after 13 years populations had achieved a stable stage distribution. We modeled 6 small populations of an epiphytic orchid using a Lefkovitch type analysis to predict population growth pattern based on monthly surveys for approximately 1.5 years. In addition, sensitivity and elasticity analyses were used to identify life stages with high sensitivity or elasticity that have the largest influence on population growth rate. We re-censused the populations 13 years after the first study and compared the structure of the populations to predictions based on the earlier census data. Effective population growth rates were similar to those expected except for one where the population went extinct. The prediction slightly (but not significantly) overestimated the actual population growth rates of some populations. Elasticity analysis revealed that the adult stage is critical in the life cycle. The observed stage distributions of the populations were not stable at the beginning of the survey and neither were they after 13 years. We suggest that this might be caused by external perturbations that result in unequal mortality between life stages and stochastic recruitment events. The ability of the matrices to predict population size approximately eight generations in the future is encouraging and warrants the continued use of these approaches for PVA.     

年龄-龄期两性生命表及其在种群生态学与害虫综合治理中的应用

生命表是种群生态学与害虫治理的重要工具,由于传统雌性生命表无法正确描述昆虫的变态且忽略雄性个体,近年来国内外学者普遍采用年龄-龄期两性生命表。本文首先从昆虫种群的龄期分化、性比对种群增长的影响、总产卵前期与成虫产卵前期的差异、产卵期与产卵日数的差异4个方面概述了年龄-龄期两性生命表(age-stage, two-sex life table)的基本原理,进而阐明了基于bootstrap技术的生命表分析技术及其主要优点,然后介绍了年龄-龄期两性生命表各软件(TWOSEX-MSChart, CONSUME-MSChart, TIMING-MSChart)的主要用途,即预测种群的增长与防治适期、正确分析天敌的捕食率与害虫的取食量、预测天敌的种群增长与捕食潜能以及指导天敌的大量繁育。昆虫生命表作为一种强有力的分析技术,不仅在研究种群生态学和害虫治理方面已有广泛的应用,展望未来,这项技术还可以用于昆虫生理、抗药性、亚致死剂量、共生菌等方面的研究。     

Using statistics to design and estimate vital rates in matrix population models for a perennial herb

Matrix population models are widely used to assess population status and to inform management decisions. Despite existing theories for building such models, model construction is often partially based on expert opinion. So far, model structure has received relatively little attention, although it may affect estimates of population dynamics. Here, we assessed the consequences of two published matrix structures (a 4 × 4 matrix based on expert opinion and a 10 × 10 matrix based on statistical modeling) for estimates of vital rates and stochastic population dynamics of the long-lived herb Astragalus scaphoides. We explored the ways in which choice of model structure alters the accuracy (i.e., mean) and precision (i.e., variance) of predicted population dynamics. We found that model structure had a negligible effect on the accuracy and precision of vital rates and stochastic stage distribution. However, the 10 × 10 matrix produced lower estimates of stochastic population growth rates than the 4 × 4 matrix, and more accurately predicted the observed trends in population abundance for three out of four study populations. Moreover, estimates of realized variation in population growth rate due to fluctuations in population stage structure over time were occasionally sensitive to matrix structure, suggesting differential roles of transient dynamics. Our study indicates that statistical modeling for choosing categories in matrix models might be preferable over expert opinion to accurately predict population trends and can provide a more objective way for model construction when the biological knowledge of the species is limited.     

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.     

Demography of the twospotted spider mite,Tetranychus urticae Koch

Summary A simple life table model was constructed for Tetranychus urticae in which daily survivorship of eggs and motil stages, fecundity, and development time was altered to assess the impact of each parameter on the intrinsic rate of increase. r. Interpretation of the trade-offs focused on management considerations.A second aspect of the study concerned age and stage structure in mite populations including the time path of convergence to a stable age distribution and the effect of changes in birth and death rates on the age profile. The stable stage distributions of 7 tetranychid mite species were computed using 25 separate life tables. In spite of the wide range of r-values induced by different experimental conditions, all of the stage distributions were quite similar averaging roughly 66% eggs, 26% immatures, and 8% adults. Several population studies were cited which reported stage distributions of growing mite populations. The empirical evidence suggested that natural mite populations are often quite near this stable distribution.A practical problem involving the extent to which hormoligosis (insecticide stimulation) affects mite population growth rate was addressed using the life table model and laboratory data from controlled studies. The findings suggested that mite populations treated with insecticide may attain a 1.4- to a 4.2-fold difference in population size relative to an untreated population after 2 generations and over a 1,300-fold potential difference after 10 generations.