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.     

Integral projection models perform better for small demographic data sets than matrix population models: a case study of two perennial herbs

1. Matrix population models are widely used to describe population dynamics, conduct population viability analyses and derive management recommendations for plant populations. For endangered or invasive species, management decisions are often based on small demographic data sets. Hence, there is a need for population models which accurately assess population performance from such small data sets.
2. We used demographic data on two perennial herbs with different life histories to compare the accuracy and precision of the traditional matrix population model and the recently developed integral projection model (IPM) in relation to the amount of data.
3. For large data sets both matrix models and IPMs produced identical estimates of population growth rate (λ). However, for small data sets containing fewer than 300 individuals, IPMs often produced smaller bias and variance for λ than matrix models despite different matrix structures and sampling techniques used to construct the matrix population models.
4. Synthesis and applications . Our results suggest that the smaller bias and variance of λ estimates make IPMs preferable to matrix population models for small demographic data sets with a few hundred individuals. These results are likely to be applicable to a wide range of herbaceous, perennial plant species where demographic fate can be modelled as a function of a continuous state variable such as size. We recommend the use of IPMs to assess population performance and management strategies particularly for endangered or invasive perennial herbs where little demographic data are available.     

Derivation of stochastic partial differential equations for size- and age-structured populations

Stochastic partial differential equations (SPDEs) for size-structured and age- and size-structured populations are derived from basic principles, i.e. from the changes that occur in a small time interval. Discrete stochastic models of size-structured and age-structured populations are constructed, carefully taking into account the inherent randomness in births, deaths, and size changes. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the size and age intervals decrease, SPDEs are derived for size-structured and age- and size-structured populations. Comparisons between numerical solutions of the SPDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations.     

滇牡丹自然种群数量动态

运用种群静态生命表、存活曲线、生殖力表和Leslie矩阵模型研究了中国西南特有濒危植物滇牡丹(Paeonia delavayi)种群数量动态过程。静态生命表和种群存活曲线表明: 滇牡丹在株龄3-6 a之间经历了较强的环境筛, 其单株生理寿命为15 a左右, 平均周期为8 a, 种群的净增殖率(R₀ = 0.985 7)、内禀增长率(r_m = -0.001 7)和周限增长率(λ = 0.998 3)表明其为衰退型种群; 滇牡丹种群存活表现为台阶型曲线(B1型), 分别在6 a和12 a阶段种群消亡率(K_x)较高。Leslie矩阵模型的模拟结果表明, 在未来30 a内种群数量呈现出下降趋势, 大约下降了50%, 其种群数量靠自身根系的萌蘖和种子繁殖共同维持。野生滇牡丹种群数量下降与其自身繁殖特性有关, 但主要原因是人为采挖和生态环境的破坏。     

Demography_Lab,an educational application to evaluate population growth: Unstructured and matrix models

Training in Population Ecology asks for scalable applications capable of embarking students on a trip from basic concepts to the projection of populations under the various effects of density dependence and stochasticity. Demography_Lab is an educational tool for teaching Population Ecology aspiring to cover such a wide range of objectives. The application uses stochastic models to evaluate the future of populations. Demography_Lab may accommodate a wide range of life cycles and can construct models for populations with and without an age or stage structure. Difference equations are used for unstructured populations and matrix models for structured populations. Both types of models operate in discrete time. Models can be very simple, constructed with very limited demographic information or parameter‐rich, with a complex density‐dependence structure and detailed effects of the different sources of stochasticity. Demography_Lab allows for deterministic projections, asymptotic analysis, the extraction of confidence intervals for demographic parameters, and stochastic projections. Stochastic population growth is evaluated using up to three sources of stochasticity: environmental and demographic stochasticity and sampling error in obtaining the projection matrix. The user has full control on the effect of stochasticity on vital rates. The effect of the three sources of stochasticity may be evaluated independently for each vital rate. The user has also full control on density dependence. It may be included as a ceiling population size controlling the number of individuals in the population or it may be evaluated independently for each vital rate. Sensitivity analysis can be done for the asymptotic population growth rate or for the probability of extinction. Elasticity of the probability of extinction may be evaluated in response to changes in vital rates, and in response to changes in the intensity of density dependence and environmental stochasticity.     

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.     

The contributions of age and sex to variation in common tern population growth rate

1. The decomposition of population growth rate into contributions from different demographic rates has many applications, ranging from evolutionary biology to conservation and management. Demographic rates with low variance may be pivotal for population persistence, but variable rates can have a dramatic influence on population growth rate. 2. In this study, the mean and variance in population growth rate (lambda) is decomposed into contributions from different ages and demographic rates using prospective and retrospective matrix analyses for male and female components of an increasing common tern (Sterna hirundo) population. 3. Three main results emerged: (1) subadult return was highly influential in prospective and retrospective analyses; (2) different age-classes made different contributions to variation in lambda: older age classes consistently produced offspring whereas young adults performed well only in high quality years; and (3) demographic rate covariation explained a significant proportion of variation in both sexes. A large contribution to lambda did not imply a large contribution to its variation. 4. This decomposition strengthens the argument that the relationship between variation in demographic rates and variation in lambda is complex. Understanding this relationship and its consequences for population persistence and evolutionary change demands closer examination of the lives, and deaths, of the individuals within populations within species.     

Plant self-incompatibility in natural populations: a critical assessment of recent theoretical and empirical advances

Self-incompatibility systems in plants are genetic systems that prevent self-fertilization in hermaphrodites through recognition and rejection of pollen expressing the same allelic specificity as that expressed in the pistils. The evolutionary properties of these self-recognition systems have been revealed through a fascinating interplay between empirical advances and theoretical developments. In 1939, Wright suggested that the main evolutionary force driving the genetic and molecular properties of these systems was strong negative frequency-dependent selection acting on pollination success. The empirical observation of high allelic diversity at the self-incompatibility locus in several species, followed by the discovery of very high molecular divergence among alleles in all plant families where the locus has been identified, supported Wright's initial theoretical predictions as well as many of its later developments. In the last decade, however, advances in the molecular characterization of the incompatibility reaction and in the analysis of allelic frequencies and allelic divergence from natural populations have stimulated new theoretical investigations that challenged some important assumptions of Wright's model of gametophytic self-incompatibility. We here review some of these recent empirical and theoretical advances that investigated: (i) the hypothesis that S-alleles are selectively equivalent, and the evolutionary consequences of genetic interactions between alleles; (ii) the occurrence of frequency-dependent selection in female fertility; (iii) the evolutionary genetics of self-incompatibility systems in subdivided populations; (iv) the evolutionary implications of the self-incompatibility locus's genetic architecture; and (v) of its interactions with the genomic environment.     

Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations

In nonlinear matrix models, strong Allee effects typically arise when the fundamental bifurcation of positive equilibria from the extinction equilibrium at r=1 (or R₀=1) is backward. This occurs when positive feedback (component Allee) effects are dominant at low densities and negative feedback effects are dominant at high densities. This scenario allows population survival when r (or equivalently R₀) is less than 1, provided population densities are sufficiently high. For r>1 (or equivalently R₀>1) the extinction equilibrium is unstable and a strong Allee effect cannot occur. We give criteria sufficient for a strong Allee effect to occur in a general nonlinear matrix model. A juvenile–adult example model illustrates the criteria as well as some other possible phenomena concerning strong Allee effects (such as positive cycles instead of equilibria).     

Linear analysis solves two puzzles in population dynamics: the route to extinction and extinction in coloured environments

In this paper, we give simple explanations to two unsolved puzzles that have emerged in recent theoretical studies in population dynamics. First, the tendency of some model populations to go extinct from high population densities, and second, the positive effect of autocorrelated environments on extinction risks for some model populations. Both phenomena are given general explanations by simple, linear, sto-chastic models. We emphasize the predictive and explanatory power of such models.     

Naturally occurring populations of Bemisia tabaci, biotype B and associated natural enemies in agro-ecosystem in northern China

The sweet potato whitefly, Bemisia tabaci (Gennadius) biotype B was first found in southern China in 1994 and has since then become a pest of increasing importance throughout the nation. The population dynamics of this invasive species and the effect of natural enemies on its population dynamics were studied in commercial cotton fields during the summers of 2003 through 2005 in Langfang, Hebei (northern China). The abundance of Bemisia tabaci was found to increase rapidly in late July, and reached its first peak in mid-August, then followed by the second peak in early September. Whitefly populations subsequently declined slowly to attain a plateau after mid-September. A total of 13 natural enemy species were discovered in association with B. tabaci in cotton fields, including dominant species such as Chrysopa sinica Tjeder, Leis axyridis Pallas, Propylaea japonica Thunberg, Orius similis Zheng, and Eretmocerus sp. Caged experiments on velvetleaf, Abutilon theophrasti Medic, one of the preferred host plants of the insect, indicated higher B. tabaci population densities in small-mesh cages with exclusion of all natural enemies versus those in medium- and large-mesh cages, and the uncaged plants retained the lowest population densities of B. tabaci. While no natural enemies were observed in the small-mesh cages, natural enemies like Misumenopos tricuspiaata, Eretmocerus sp., Encarsia formosa Orius similis, Deraeocoris punctulatus and Scolothrips takahashii were found in medium- and/or large-mesh cages. Findings from this study suggest that under open-field conditions, various indigenous natural enemies of different sizes may act in concert in the suppression of B. tabaci populations.     

Rapid and unpredictable changes of the G‐matrix in a natural bird population over 25 years

Knowledge of the genetic variances and covariances of traits (the G ‐matrix) is fundamental for the understanding of evolutionary dynamics of populations. Despite its essential importance in evolutionary studies, empirical tests of the temporal stability of the G ‐matrix in natural populations are few. We used a 25‐year‐long individual‐based field study on almost 7000 breeding attempts of the collared flycatcher (Ficedula albicollis) to estimate the stability of the G‐matrix over time. Using animal models to estimate G for several time periods, we show that the structure of the time‐specific G‐matrices changed significantly over time. The temporal changes in the G‐matrix were unpredictable, and the structure at one time period was not indicative of the structure at the next time period. Moreover, we show that the changes in the time‐specific G‐matrices were not related to changes in mean trait values or due to genetic drift. Selection, differences in acquisition/allocation patterns or environment‐dependent allelic effects are therefore likely explanations for the patterns observed, probably in combination. Our result cautions against assuming constancy of the G ‐matrix and indicates that even short‐term evolutionary predictions in natural populations can be very challenging.     

Comparative Population Dynamics of an Invading Species in its Native and Novel Ranges

Although the ecology of many exotic invaders has been intensively examined in the novel range, few studies have comparatively explored how population dynamics differ in native and novel parts of an invading plants’ range. The population dynamics of mile-a-minute weed, Polygonum perfoliatum L., was explored in both the native (Japan) and novel (northeastern USA) portions of its range and evaluated using periodic matrix models. Projected per capita population growth rate (λ) varied within and between native and novel range populations. Surprisingly, five of the six populations in the novel range were projected to fail to replace themselves (λ<1) while only two of the four native range populations were projected to decline, although these projections had wider confidence intervals than in the novel habitat. While changes in germination, survivorship, fecundity and seed banking would have equivalent effects on population growth in the invasive habitat, small increases in plant survivorship would greatly increase λ in native populations. The differences between native and novel population growth rates were driven by lower adult survival in the native range caused by annual flooding and higher fecundity. Simulation analyses indicated that a 50% reduction in plant survival would be required to control growing populations in the novel range. Further comparative studies of other invading species in both their native and novel ranges are needed to examine whether the high per capita population growth and strong regulatory effects of adult survival in the native habitat are generally predictive of invasive behavior in novel habitats. Sachiko Araki: (Deceased)     

Demographic models to simulate the stable ratio between ecologically similar gametophytes and tetrasporophytes in populations of the Gigartinaceae (Rhodophyta)

In populations of the Gigartinaceae (Rhodophyta), gametophytes often predominate numerically over tetrasporophytes. Several hypotheses have been proposed to explain this dominance, based on the usually implicit assumption that the stable ratio between gametophytes and tetrasporophytes (G:T ratio) should be 1 if both reproductive phases are ecologically similar. We developed demographic models to test this assumption, for which we considered that both phases are ecologically similar. Defining ecologic similarity for most demographic rates is relatively straightforward, except for rates of spore output. The first set of models considered the same spore output per thallus of both phases as representing ecologic similarity. Model iterations led to stable G:T ratios of 1 for triennial and for perennial thalli, regardless of the initial G:T ratio, but not for annual thalli with initial G:T ratios different from 1. However, equal spore output may not represent ecologic similarity, due to size differences between carpospores and tetraspores. The second set of models considered the lowest possible spore output for each phase, according to the life history of this family: only one carposporangium, with one carpospore, is produced from every two gametophytes and only one tetrasporangium, with four tetraspores, is produced by every tetrasporophyte. Model iterations led to stable G:T ratios of 2.8 for most cases, a ratio of 1 being obtained only every 2 years for annual thalli with an initial G:T ratio of 1. Increasing absolute spore output, without altering the relative output between phases and incorporating density-independent mortality through a matrix model, given the same mortality rate for both phases, did not modify results. We suggest that the combination of both modeling and field research may uncover more rapidly than otherwise the most relevant ecologic differences between phases, if any, that underlie the G:T ratio observed for a given population.     

Stochastic matrix models for conservation and management: a comparative review of methods

Stochastic matrix models are frequently used by conservation biologists to measure the viability of species and to explore various management actions. Models are typically parameterized using two or more sets of estimated transition rates between age/size/stage classes. While standard methods exist for analyzing a single set of transition rates, a variety of methods have been employed to analyze multiple sets of transition rates. We review applications of stochastic matrix models to problems in conservation and use simulation studies to compare the performance of different analytic methods currently in use. We find that model conclusions are likely to be robust to the choice of parametric distribution used to model vital rate fluctuations over time. However, conclusions can be highly sensitive to the within-year correlation structure among vital rates, and therefore we suggest using analytical methods that provide a means of conducting a sensitivity analysis with respect to correlation parameters. Our simulation results also suggest that the precision of population viability estimates can be improved by using matrix models that incorporate environmental covariates in conjunction with experiments to estimate transition rates under a range of environmental conditions.     

Effects of climate change on an emperor penguin population: analysis of coupled demographic and climate models

Sea ice conditions in the Antarctic affect the life cycle of the emperor penguin (Aptenodytes forsteri). We present a population projection for the emperor penguin population of Terre Adélie, Antarctica, by linking demographic models (stage‐structured, seasonal, nonlinear, two‐sex matrix population models) to sea ice forecasts from an ensemble of IPCC climate models. Based on maximum likelihood capture‐mark‐recapture analysis, we find that seasonal sea ice concentration anomalies (SIC_a) affect adult survival and breeding success. Demographic models show that both deterministic and stochastic population growth rates are maximized at intermediate values of annual SIC_a, because neither the complete absence of sea ice, nor heavy and persistent sea ice, would provide satisfactory conditions for the emperor penguin. We show that under some conditions the stochastic growth rate is positively affected by the variance in SIC_a. We identify an ensemble of five general circulation climate models whose output closely matches the historical record of sea ice concentration in Terre Adélie. The output of this ensemble is used to produce stochastic forecasts of SIC_a, which in turn drive the population model. Uncertainty is included by incorporating multiple climate models and by a parametric bootstrap procedure that includes parameter uncertainty due to both model selection and estimation error. The median of these simulations predicts a decline of the Terre Adélie emperor penguin population of 81% by the year 2100. We find a 43% chance of an even greater decline, of 90% or more. The uncertainty in population projections reflects large differences among climate models in their forecasts of future sea ice conditions. One such model predicts population increases over much of the century, but overall, the ensemble of models predicts that population declines are far more likely than population increases. We conclude that climate change is a significant risk for the emperor penguin. Our analytical approach, in which demographic models are linked to IPCC climate models, is powerful and generally applicable to other species and systems.