Incorporating arbitrary stage duration distributions in stage‐structured matrix population models

BackgroundUnderstanding how biotic and abiotic factors affect insect mortality is crucial for both fundamental knowledge of population ecology and for successful pest management. However, because these factors are difficult to quantify and interpret, patterns and dynamics of insect mortality remain unclear, especially comparative mortality across climate zones. Life table analysis provides robust information for quantifying population mortality and population parameters.MethodsIn this study, we estimated cause-of-death probabilities and irreplaceable mortality (the portion of mortality that cannot be replaced by another cause or combination of causes) using a Multiple Decrement Life Table (MDLT) analysis of 268 insect life tables from 107 peer-reviewed journal articles. In particular, we analyzed insect mortality between temperate and tropical climate zones.ResultsSurprisingly, our results suggest that non-natural enemy factors (abiotic) were the major source of insect mortality in both temperate and tropical zones. In addition, we observed that irreplaceable mortality from predators in tropical zones was 3.7-fold greater than in temperate zones. In contrast, irreplaceable mortality from parasitoids and pathogens was low and not different between temperate and tropical zones. Surprisingly, we did not observe differences in natural enemy and non-natural enemy factors based on whether the insect species was native or non-native. We suggest that characterizing predation should be a high priority in tropical conditions. Furthermore, because mortality from parasitoids was low in both tropical and temperate zones, this mortality needs to be better understood, especially as it relates to biological control and integrated pest management.

A matrix population model is a convenient tool for summarizing per capita survival and reproduction rates (collectively vital rates) of a population and can be used for calculating an asymptotic finite population growth rate (λ) and generation time. These two pieces of information can be used for determining the status of a threatened species. The use of stage-structured population models has increased in recent years, and the vital rates in such models are often estimated using a life table analysis. However, potential bias introduced when converting age-structured vital rates estimated from a life table into parameters for a stage-structured population model has not been assessed comprehensively. The objective of this study was to investigate the performance of methods for such conversions using simulated life histories of organisms. The underlying models incorporate various types of life history and true population growth rates of varying levels. The performance was measured by comparing differences in λ and the generation time calculated using the Euler-Lotka equation, age-structured population matrices, and several stage-structured population matrices that were obtained by applying different conversion methods. The results show that the discretization of age introduces only small bias in λ or generation time. Similarly, assuming a fixed age of maturation at the mean age of maturation does not introduce much bias. However, aggregating age-specific survival rates into a stage-specific survival rate and estimating a stage-transition rate can introduce substantial bias depending on the organism’s life history type and the true values of λ. In order to aggregate survival rates, the use of the weighted arithmetic mean was the most robust method for estimating λ. Here, the weights are given by survivorship curve after discounting with λ. To estimate a stage-transition rate, matching the proportion of individuals transitioning, with λ used for discounting the rate, was the best approach. However, stage-structured models performed poorly in estimating generation time, regardless of the methods used for constructing the models. Based on the results, we recommend using an age-structured matrix population model or the Euler-Lotka equation for calculating λ and generation time when life table data are available. Then, these age-structured vital rates can be converted into a stage-structured model for further analyses.

Stage‐based demographic methods, such as matrix population models (MPMs), are powerful tools used to address a broad range of fundamental questions in ecology, evolutionary biology and conservation science. Accordingly, MPMs now exist for over 3000 species worldwide. These data are being digitised as an ongoing process and periodically released into two large open‐access online repositories: the COMPADRE Plant Matrix Database and the COMADRE Animal Matrix Database. During the last decade, data archiving and curation of COMPADRE and COMADRE, and subsequent comparative research, have revealed pronounced variation in how MPMs are parameterized and reported. Here, we summarise current issues related to the parameterisation and reporting of MPMs that arise most frequently and outline how they affect MPM construction, analysis, and interpretation. To quantify variation in how MPMs are reported, we present results from a survey identifying key aspects of MPMs that are frequently unreported in manuscripts. We then screen COMPADRE and COMADRE to quantify how often key pieces of information are omitted from manuscripts using MPMs. Over 80% of surveyed researchers (n = 60) state a clear benefit to adopting more standardised methodologies for reporting MPMs. Furthermore, over 85% of the 300 MPMs assessed from COMPADRE and COMADRE omitted one or more elements that are key to their accurate interpretation. Based on these insights, we identify fundamental issues that can arise from MPM construction and communication and provide suggestions to improve clarity, reproducibility and future research utilising MPMs and their required metadata. To fortify reproducibility and empower researchers to take full advantage of their demographic data, we introduce a standardised protocol to present MPMs in publications. This standard is linked to www.compadre‐db.org, so that authors wishing to archive their MPMs can do so prior to submission of publications, following examples from other open‐access repositories such as DRYAD, Figshare and Zenodo. Combining and standardising MPMs parameterized from populations around the globe and across the tree of life opens up powerful research opportunities in evolutionary biology, ecology and conservation research. However, this potential can only be fully realised by adopting standardised methods to ensure reproducibility.

Introduction to Stage-Structured Population Models

Summary 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.

W.O. Kermack and A.G. McKendrick introduced in their fundamental paper, A Contribution to the Mathematical Theory of Epidemics, published in 1927, a deterministic model that captured the qualitative dynamic behavior of single infectious disease outbreaks. A Kermack–McKendrick discrete-time general framework, motivated by the emergence of a multitude of models used to forecast the dynamics of epidemics, is introduced in this manuscript. Results that allow us to measure quantitatively the role of classical and general distributions on disease dynamics are presented. The case of the geometric distribution is used to evaluate the impact of waiting-time distributions on epidemiological processes or public health interventions. In short, the geometric distribution is used to set up the baseline or null epidemiological model used to test the relevance of realistic stage-period distribution on the dynamics of single epidemic outbreaks. A final size relationship involving the control reproduction number, a function of transmission parameters and the means of distributions used to model disease or intervention control measures, is computed. Model results and simulations highlight the inconsistencies in forecasting that emerge from the use of specific parametric distributions. Examples, using the geometric, Poisson and binomial distributions, are used to highlight the impact of the choices made in quantifying the risk posed by single outbreaks and the relative importance of various control measures.

Landscape structure, habitat fragmentation, and the ecology of insects

Matrix population models are widely used to study the dynamics of stage‐structured populations. A census in these models is an event monitoring the number of individuals in each stage and occurs at discrete time intervals. The two most common methods used in building matrix population models are the prebreeding census and postbreeding census. Models using the prebreeding and postbreeding censuses assume that breeding occurs immediately before or immediately after the censuses, respectively. In some models such as age‐structured models, the results are identical regardless of the method used, rendering the choice of method a matter of preference. However, in stage‐structured models, where the duration of the first stage of life varies among newborns, a choice between the prebreeding and postbreeding censuses may result in different conclusions. This is attributed to the different first‐stage duration distributions assumed by the two methods. This study investigated the difference emerging in the structures of these models and its consequence on conclusions of eigenvalue and elasticity analyses using two‐stage models. Considerations required in choosing a modeling method are also discussed.

Matrix population models (MPMs) are powerful tools for translating demographic and life history information into a form that can be used to address a wide range of research topics, such as projecting population dynamics, evaluating stressor impacts on populations, and studying life history evolution. However, the reliability of such studies depends on the MPM being constructed in a way that accurately reflects the species’ life history. We highlight three errors commonly encountered in published MPMs: (1) failing to include survival in the fertility coefficient; (2) introducing a one-year delay in age at first reproduction; and (3) incorrectly calculating the growth rate out of a stage class. We review the sources of such errors and provide new analyses revealing the impact of such errors on model predictions, using lionfish and American alligator models as examples. To quantify the prevalence of such errors we examined and scored the original publications underlying the models in the COMADRE Animal Matrix Database. The first two errors were found in 34% and 62%, respectively, of the published studies; nearly all were in models that used a “postbreeding census” representation of the life cycle (in which newborns—eggs, neonates, fledglings, etc.—are explicitly included). Of the studies where stages may last longer than one time step, 53% constructed the growth rate using inappropriate formulas for estimating the asymptotic population growth rate or its sensitivity to demographic parameters. These results suggest that further efforts may be required to educate biologists on the construction of MPMs, perhaps in concert with the development of new software tools. Furthermore, the conclusions of many studies that are based on MPMs may need to be re-examined, and synthetic studies using the COMADRE Database need to be accompanied by careful examination of the underlying studies.

Let {Fn } n ≧ 0 be a sequence of c.d.f. and let {Rn } n ≧ 1 be the sequence of record values in a non-stationary record model where after the (n − 1)th record the population is distributed according to Fn. Then the equidistribution of the nth population and the record increment Rn – Rn – 1 (i.e. Rn – Rn – 1 ~ Fn ) characterizes Fn to have an exponentially decreasing hazard function. To be more precise Fn is the exponential distribution if the support of Rn – 1 generates a dense subgroup in and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chosen arbitrarily. Setting λ = – log(1 – F(1)), Fn (x) = 1 – F(x – [x])exp(–λ [x]) is an admissible solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that Fn is either exponential or geometric are given. Similar results for exponential or geometric tail distributions based on the independence of Rn – 1 and Rn – Rn – 1 are proved.

Let {Fn}n ≧ 0 be a sequence of c.d.f. and let {Rn}n ≧ 1 be the sequence of record values in a non-stationary record model where after the (n − 1)th record the population is distributed according to Fn. Then the equidistribution of the nth population and the record increment Rn – Rn– 1 (i.e. Rn – Rn– 1~ Fn) characterizes Fn to have an exponentially decreasing hazard function. To be more precise Fn is the exponential distribution if the support of Rn– 1 generates a dense subgroup in and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chosen arbitrarily. Setting λ = – log(1 – F(1)), Fn(x) = 1 – F(x – [x])exp(–λ [x]) is an admissible solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that Fn is either exponential or geometric are given. Similar results for exponential or geometric tail distributions based on the independence of Rn– 1 and Rn – Rn– 1 are proved.

Matrix population models are widely used to describe the growth of stage-structured populations. The variability in stage duration among individuals is one of the important parameters affecting population growth. Despite that importance, studies frequently focus only on the average duration and (perhaps unknowingly) make specific assumptions regarding the variation in stage duration that likely contradicts with the data. Furthermore, although there are methods for modeling variable stage duration, they are not sufficiently flexible to encompass the diversity of potential distributions. This study discusses problems associated with the existing methods and describes approaches to deal with the problems.

Cross-temporal data on suicide for the period 1962 to 1981 from the National Center for Health Statistics were analyzed. These data were used to estimate period and cohort suicide rates for the four middle-aged and elderly groups (ages 45 to 64, 65 to 74, 75 to 84, and 85 and older) by sex and race. Statistical procedures included multiple decrement life table analyses and cause elimination life table analyses for each year 1968 to 1981. Results from an age, period, cohort analysis of cohort trends 1962 to 1981 also were presented. The analysis showed that suicide continues to be a serious problem in later life especially among the "oldest-old" (those aged 85 and over) and among nonwhite males. It also showed important differences in cohort risks that may strongly affect future suicide risks among elderly adults.

Matrix population models are a standard tool for studying stage‐structured populations, but they are not flexible in describing stage duration distributions. This study describes a method for modeling various such distributions in matrix models. The method uses a mixture of two negative binomial distributions (parametrized using a maximum likelihood method) to approximate a target (true) distribution. To examine the performance of the method, populations consisting of two life stages (juvenile and adult) were considered. The juvenile duration distribution followed a gamma distribution, lognormal distribution, or zero‐truncated (over‐dispersed) Poisson distribution, each of which represents a target distribution to be approximated by a mixture distribution. The true population growth rate based on a target distribution was obtained using an individual‐based model, and the extent to which matrix models can approximate the target dynamics was examined. The results show that the method generally works well for the examined target distributions, but is prone to biased predictions under some conditions. In addition, the method works uniformly better than an existing method whose performance was also examined for comparison. Other details regarding parameter estimation and model development are also discussed.

The successful development of phenology models from field studies depends on many factors, some of which are entirely under the control of pest managers. For example, one such factor is the choice of method for calculating thermal units. In this study, we have demonstrated that four methods for calculating thermal units provided for acceptable predictions of one phenological event of one insect species, while another method for calculating thermal units did not. The measure of central tendency (mean or median) that is used to estimate lower developmental temperatures and required thermal summations is another factor that pest managers can control when developing phenology models from field studies. Here, we show that predictions that were made when using phenology models based on median lower developmental temperatures and median required thermal summations were superior to predictions that were made when using phenology models based on mean lower developmental temperatures and mean required thermal summations. The use of bootstrap vs. non-bootstrap estimates of lower developmental temperatures and required thermal summations is yet another factor that pest managers can control when developing phenology models from field studies. In this study, we found that calculating and using bootstrap estimates of lower developmental temperatures and required thermal summations in phenology models did not improve the predictions of one phenological event for one insect species. The implications of these and other findings are discussed.