Bias in population growth rate estimation: sparse data,partial life cycle analysis and Jensen's inequality

Demographic matrix models have become an integral part of population viability analysis for threatened and endangered species, but their use is often limited by data availability. A common solution to this problem is to assume constant annual rates within a multi‐year stage. Partial life cycle analysis (PLC), which incorporates only juvenile and adult stages, is a noteworthy example of this approach because it has been described in the literature as a reliable approximation of age‐structured populations. However, we predict from Jensen's Inequality that the required lumping of age classes leads to over‐ or underestimation of population fitness when survival rates are truly age‐dependent. We illuminate this problem by comparing fitness estimates from Leslie matrix and PLC models for theoretical populations having different levels of age‐dependence in their survival rates. We also propose a modification of the PLC approach to address this problem and demonstrate its applicability using data from a published long‐term study of red deer Cervus elephas.     

Plant Invasion Patches – Reconstructing Pattern and Process by Means of Herb-chronology

An understanding of the patterns of spread of invasive plant species requires analysis of the major dispersal mechanisms and of the patch structure of suitable habitats, both of which may be scale-dependent. On a larger scale, information from herbarium or literature records has proved useful for the reconstruction of past spread of invasive plants. The objective of this study is to investigate population development of invasive forbs at the scale of a site or stand (the population scale) by using herb-chronology. The feasibility of this approach has been largely disregarded until now because of the perceived difficulties in determining the age of perennial herbs. However, recent findings suggest that most of the dicotyledonous perennial herbs in the seasonal climates develop annual rings in the roots or subterraneous stems and thus demonstrate a high potential of the method in studies on plant invasions that went almost unnoticed. The spatial position and age (by means of analysis of annual rings) of individual plants were determined in invasion patches of five species of perennial forbs in Germany and in the USA. The data thus obtained revealed different spatio-temporal patterns of population development that are consistent with distinct models of (local) plant spread, including diffuse invasion and front-like invasion patterns, and thus suggest different processes at work in the course of invasion. The results suggest that analysis of spatial age structures is useful (i) to estimate rates of patch expansion, (ii) to distinguish between dispersal- and microsite-limited population development, (iii) to evaluate how different site conditions affect population development, and (iv) to help understand metapopulation dynamics.     

Growth and population dynamic model of the reef coral Fungia granulosa Klunzinger, 1879 at Eilat, northern Red Sea

The lack of population dynamic information for most species of stony corals is due in part to their complicated life histories that may include fission, fusion and partial mortality of colonies, leading to an uncoupling of coral age and size. However, some reef-building corals may produce compact upright or free-living individuals in which the above processes rarely occur, or are clearly detectable. In some of these corals, individual age may be determined from size, and standard growth and population dynamic models may be applied to gain an accurate picture of their life history. We measured long-term growth rates (up to 2.5 years) of individuals of the free-living mushroom coral Fungia granulosa Klunzinger, 1879 at Eilat, northern Red Sea, and determined the size structure of a population on the shallow reef slope. We then applied growth and population models to the data to obtain estimates of coral age, mortality rate, and life expectancy in members of this species. In the field, few F. granulosa polyps suffered partial mortality of >10% of their tissues. Thus, the majority of polyps grew isometrically and determinately, virtually ceasing growth by about 30-40 years of age. Coral ages as revealed by skeletal growth rings were similar to those estimated from a growth curve based on field data. The frequency of individuals in each age class on the reef slope decreased exponentially with coral age, indicating high mortality rates when corals were young. The maximum coral age observed in the field population (31 years) was similar to that estimated by application of a population dynamic model (30 years). Calculated rates of growth, mortality and life expectancy for F. granulosa were within the range of those known for other stony corals. Our results reveal a young, dynamic population of this species on Eilat reefs, with high turnover rates and short lifespans. Such information is important for understanding recovery of coral reefs from disturbances, and for application to the management of commercially exploited coral populations.     

Partial life cycle analysis: a model for pre-breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.     

Demography and population models of ticks of the genus Ixodes with long-term life cycles

The most important parameters necessary for creation of population models for three-host species with long-term life cycles are discussed by the example of the ticks Ixodes persulcatus and I. ricinus. In these species, specimens of the same biological age may belong to different age cohorts and their calendar age may differ by several months or even years. Accurate estimation of the calendar age of singular individuals is difficult; it is based on the extrapolation by its possible biological age and by belonging to the certain age cohort of a natural population. In order to possess a prognostic value, the population models must predict a single-moment population density of activated hungry specimens of all the three developmental stages and the probability of host-finding in hungry ticks during the questing period. The daily mortality of ticks of different developmental stages and phases of each stage (questing, feeding, preparation for molting, and diapause) should also be known. The population density of questing hungry ticks in the ecosystem is determined by the balance between the recruitment of new individuals into the population, their removal from the population by hosts, the dying of ticks from starvation, and the consumption of ticks by predators. At present, unfortunately, only some of these parameters are sufficiently known.     

Age and growth differences in two populations of the edible marine gastropod Buccinanops globulosus

Buccinanops globulosus is an edible marine gastropod that is being captured by artisanal fishermen without management regulations. As basic knowledge on population features is required in case a sustained commercial exploitation of this species is established, we estimated and compared the age and growth of B. globulosus in two populations separated by 16?km, inhabiting similar physical environments but different anthropic influence. Our results, based on stable oxygen isotope analysis and best fitted models by likelihood ratio tests, detected differences in age and growth between both samplings. Maximum shell marks suggest one year difference between populations. Richards was chosen as the best fitting model for both sampling sites and significant differences were observed between them. Human activities could be causing the age reduction among other irreversible effects on the population under anthropic influence. This should be considered when developing sustainable management regulations for this fishery resource, especially those based on shell/age size.     

The influence of patch demographics on metapopulations, with particular reference to successional landscapes

The classical view of metapopulations relates the regional abundance of a species to the balance between the extinction and colonization dynamics of identical local populations. Species in successional landscapes may represent the most appropriate examples of classical metapopulations. However, Levins‐type metapopulation models do not explicitly separate population loss due to successional habitat change from other causes of extinction. A further complication is that the chance of population loss due to successional habitat change may be related to the age of a patch. I developed simple patch occupancy models to include succession and included consideration of patch age structure to address two related questions: what are the implications of changes in patch demographic rates and when is a move to a structured patch occupancy model justified? Age‐related variation in patch demography could increase or decrease the equilibrium fraction of the available habitat occupied by a species when compared to the predictions of an unstructured model. Metapopulation persistence was enhanced when the age class of patches with the highest species occupancy suffered relatively low losses to habitat succession. Conversely, when the age class of patches with the highest species occupancy also had relatively high successional loss rates, extinction thresholds were higher that would be predicted by a simple unstructured model. Hence age‐related variation in patch successional rate introduces biases into the predictions of simple unstructured models. Such biases can be detected from field surveys of the fraction of occupied and unoccupied patches in each age class. Where a bias is demonstrated, unstructured models will not be adequate for making predictions about the effects of changing parameters on metapopulation size. Thinking in successional terms emphasizes how landscapes might be managed to enhance or reduce the patch occupancy by any particular metapopulation     

具有年龄结构的食蚜蝇-蚜虫捕食模型

现有的具有年龄结构的捕食-食饵模型总是假设只有成年捕食者捕食猎物,这与实际情况不符。建立了一个幼年捕食者捕食食饵的具有年龄结构的食蚜蝇-蚜虫模型,应用微分方程定性理论,讨论了系统平衡点及其稳定性:其中平衡点E1(0,0,0)为不稳定的;满足一定条件时,边界平衡点E2(K,0,0)及正平衡点E3(x*,y1*,y2*)为局部渐近稳定的;且应用一致持续生存理论得到了系统永久持续生存的条件,为有害生物综合治理提供了理论依据。     

Approximation of a physiologically structured population model with seasonal reproduction by a stage-structured biomass model

Seasonal reproduction causes, due to the periodic inflow of young small individuals in the population, seasonal fluctuations in population size distributions. Seasonal reproduction furthermore implies that the energetic body condition of reproducing individuals varies over time. Through these mechanisms, seasonal reproduction likely affects population and community dynamics. While seasonal reproduction is often incorporated in population models using discrete time equations, these are not suitable for size-structured populations in which individuals grow continuously between reproductive events. Size-structured population models that consider seasonal reproduction, an explicit growing season and individual-level energetic processes exist in the form of physiologically structured population models. However, modeling large species ensembles with these models is virtually impossible. In this study, we therefore develop a simpler model framework by approximating a cohort-based size-structured population model with seasonal reproduction to a stage-structured biomass model of four ODEs. The model translates individual-level assumptions about food ingestion, bioenergetics, growth, investment in reproduction, storage of reproductive energy, and seasonal reproduction in stage-based processes at the population level. Numerical analysis of the two models shows similar values for the average biomass of juveniles, adults, and resource unless large-amplitude cycles with a single cohort dominating the population occur. The model framework can be extended by adding species or multiple juvenile and/or adult stages. This opens up possibilities to investigate population dynamics of interacting species while incorporating ontogenetic development and complex life histories in combination with seasonal reproduction.     

Density-dependent age at first reproduction in the eastern kingbird

Theory predicts that maximal fitness is obtained by individuals who begin to breed immediately upon reaching sexual maturity. However, delayed breeding occurs regularly in some taxa, and in birds and mammals is most often associated with long lifespan and/or limited access to suitable habitats. Delayed breeding is not expected among relatively short-lived species such as migratory passerine birds, but this assumption remains untested in many species. Here we quantify age at first reproduction in an eastern kingbird Tyrannus tyrannus population breeding in an ecological island, and through both observational and experimental approaches, investigate the potential causes for the high frequency of delayed breeding that occurs in this population. Nearly half of the fledged nestlings that returned to the breeding grounds did not breed in their first potential breeding season. Some non-breeders occupied territories, for at least some period, but most remained as non-territorial 'floaters'. Parentage analysis failed to show any reproductive success for female floaters, and only limited success for male floaters, indicating that floating was not a successful reproductive tactic. On the other hand, a strong negative relationship existed between population size and the proportion of young birds that bred in their first year, and non-territorial birds of both sexes quickly filled territory vacancies created by experimental adult removals. Limited breeding habitat and territorial behavior of older birds thus appear to be the main causes of delayed breeding in kingbirds. The frequency of delayed breeding in most species is unknown but of potential significance because failure to incorporate accurate estimates of age at first reproduction in population models may lead to flawed population projections.     

Demography and population models of ticks of the genus Ixodes with long-term life cycles

The most important parameters necessary for the creation of population models for threehost species with long-term life cycles are discussed with an example of ticks Ixodes persulcatus and I. ricinus. In these species, specimens of the same biological age may belong to different age cohorts and their calendar age may differ by several months or even years. Accurate estimation of the calendar age of separate individuals is dofficult; it is based on the extrapolation by its possible biological age and by belonging to the certain age cohort of a natural population. Population models that can predict simultaneous abundance of activated hungry specimens of all the three developmental stages and probability of host-finding in hyngry ticks during questing period possess the prognostic value. Daily mortality of ticks of different developmental stages and phases of each stage (questing, feeding, preparation for molting, and diapause) must also be known. The abundance of questing hungry ticks in the ecosystem is determined by the balance between recruitment of the population with new individuals, their selection by hosts, dying of ticks from starvation, and consumption of ticks by predators. At present, unfortunately, only some of these parameters are known rather sufficiently.     

COMPARATIVE POPULATION DYNAMICS OF FUR SEALS

The population sizes, trends, exploitation, and life history parameters for the ten fur seal species and subspecies are summarized. The largest population is that of Arctocephalus pusillus pusillus with approximately two million seals, and the smallest is A. townsendi with approximately 7,000 individuals. Most populations are legally protected, although controlled harvesting may occur. None of the fur seal populations is currently known to be decreasing. Data are presented for parameters related to the survival of pups, juveniles, adults, and territorial males, and to reproduction, including the age of attainment of territorial status, aggregation sizes, age of first parturition, pregnancy rates, sex ratios of young animals, and information on the birth seasons of the different species. Since pinnipeds are often of concern in fisheries management, their daily consumption rates are of importance, and consequently data on body masses are summarized and the paucity of data on consumption rates as a function of body mass noted. A simplified age-structured model is developed, and the results of this model are compared with results from more detailed models based on two published life tables for Callorhinus ursinus. This comparison shows that the use of the simplified age-structured model is justified to explore changes in population growth rate. However, the simplified model does show exaggerated age structure effects compared to the more detailed models. This model is used to compare the population dynamics of those species for which sufficient data are available. Areas in which limited, or no, data are available for the different fur seal species are highlighted.     

A simple persistence condition for structured populations

The fundamental question in both basic and applied population biology of whether a species will increase in numbers is often investigated by finding the population growth rate as the largest eigenvalue of a deterministic matrix model. For a population classified only by age, and not stage or size, a simpler biologically interpretable condition can be used, namely whether R ₀, the mean number of offspring per newborn, is greater than one. However, for the many populations not easily described using only age classes, stage-structured models must be used for which there is currently no quantity like R ₀. We determine analogous quantities that must be greater than one for persistence of a general structured population model that have a similar useful biological interpretation. Our approach can be used immediately to determine the magnitude of changes and interactions that would either allow population persistence or would ensure control of an undesirable species.     

Demographic responses of Pinguicula ionantha to prescribed fire: a regression-design LTRE approach

This study describes the use of periodic matrix analysis and regression-design life table response experiments (LTRE) to investigate the effects of prescribed fire on demographic responses of Pinguicula ionantha, a federally listed plant endemic to the herb bog/savanna community in north Florida. Multi-state mark–recapture models with dead recoveries were used to estimate survival and transition probabilities for over 2,300 individuals in 12 populations of P. ionantha. These estimates were applied to parameterize matrix models used in further analyses. P. ionantha demographics were found to be strongly dependent on prescribed fire events. Periodic matrix models were used to evaluate season of burn (either growing or dormant season) for fire return intervals ranging from 1 to 20 years. Annual growing and biannual dormant season fires maximized population growth rates for this species. A regression design LTRE was used to evaluate the effect of number of days since last fire on population growth. Maximum population growth rates calculated using standard asymptotic analysis were realized shortly following a burn event (<2 years), and a regression design LTRE showed that short-term fire-mediated changes in vital rates translated into observed increases in population growth. The LTRE identified fecundity and individual growth as contributing most to increases in post-fire population growth. Our analyses found that the current four-year prescribed fire return intervals used at the study sites can be significantly shortened to increase the population growth rates of this rare species. Understanding the role of fire frequency and season in creating and maintaining appropriate habitat for this species may aid in the conservation of this and other rare herb bog/savanna inhabitants. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Variation in northern bobwhite demography along two temporal scales

Quantification and understanding of demographic variation across intra- and inter-annual temporal scales can benefit from the development of theoretical models of evolution and applied conservation of species. We used long-term survey data for northern bobwhites (Colinus virginianus) collected at the northern and southern extent of its geographic range to develop matrix population models which would allow investigation of intra- and inter-annual patterns in bobwhite population dynamics. We first evaluated intra-annual patterns in the importance of a seasonal demographic rate to asymptotic population growth rate with prospective perturbation analysis (elasticity analysis). We then conducted retrospective analysis (life table response experiments) of inter-annual patterns in the contribution of observed changes in demography to the observed change in population growth rate. Survival in the earliest age class during the nonbreeding season had the greatest potential influence in both the northern and southern populations. Examination of inter-annual variation in demography indicated that variation in nonbreeding season survival in the earliest age class contributed the most to observed changes in population growth rate in the northern population. In contrast, changes in fertility in the earliest age class in the southern population had the greatest influence on changes in population growth rate. Prospective elasticity analyses highlight the similarities in bobwhite demography throughout different parts of its geographic range, while retrospective life table response experiments revealed important patterns in the temporal differences of bobwhite life history at the northern and southern extent of its geographic range.     

Potential Applications of Population Viability Analysis to Risk Assessment for Invasive Species

Population viability analysis, the use of ecological models to assess a population's risk of extinction, plays an important role in contemporary conservation biology. The premise of this review is that models, concepts, and data analyses that yield results on extinction risk of threatened and endangered species can also tell us about establishment risks of potentially invasive species. I briefly review important results for simple unstructured models, demographic models, and spatial models, giving examples of the application of each type of model to invasive species, and general conclusions about the applicability of each type of model to risk analysis for invasive species. The examples illustrate a portion of the range of potential applications of such models to invasive species, and some of the types of predictions that they can provide. They also highlight some of the limitations of such models. Finally, I present several conjectures and open research questions concerning the application of population viability analyses to risk analysis and control of invasive species.     

Age estimation in a long‐lived seabird (Ardenna tenuirostris) using DNA methylation‐based biomarkers

Age structure is a fundamental aspect of animal population biology. Age is strongly related to individual physiological condition, reproductive potential and mortality rate. Currently, there are no robust molecular methods for age estimation in birds. Instead, individuals must be ringed as chicks to establish known‐age populations, which is a labour‐intensive and expensive process. The estimation of chronological age using DNA methylation (DNAm) is emerging as a robust approach in mammals including humans, mice and some non‐model species. Here, we quantified DNAm in whole blood samples from a total of 71 known‐age Short‐tailed shearwaters (Ardenna tenuirostris) using digital restriction enzyme analysis of methylation (DREAM). The DREAM method measures DNAm levels at thousands of CpG dinucleotides throughout the genome. We identified seven CpG sites with DNAm levels that correlated with age. A model based on these relationships estimated age with a mean difference of 2.8 years to known age, based on validation estimates from models created by repeated sampling of training and validation data subsets. Longitudinal observation of individuals re‐sampled over 1 or 2 years generally showed an increase in estimated age (6/7 cases). For the first time, we have shown that epigenetic changes with age can be detected in a wild bird. This approach should be of broad interest to researchers studying age biomarkers in non‐model species and will allow identification of markers that can be assessed using targeted techniques for accurate age estimation in large population studies.     

Life history of a wide‐ranging deepwater lantern shark in the north‐east Atlantic,Etmopterus spinax (Chondrichthyes: Etmopteridae), with implications for conservation

In this paper, the population biology of the velvet belly lanternshark Etmopterus spinax was studied and life‐history coefficients determined. Age was estimated from sections of the second dorsal spine and validated by marginal increment analysis. Males attained a maximum age of 8 years while 11 year‐old females were found. Several growth models were fitted and compared for both size‐at‐age and mass‐at‐age data, showing that even though this is a small‐sized species, it has a relatively slow growth rate. This species matures late, specifically at 49·6 and 42·5% of the maximum observed ages for males and females, respectively. It has a low fecundity, with a mean ovarian fecundity of 9·94 oocytes and a mean uterine fecundity of 7·59 embryos per reproductive cycle. This species seems to have a long reproductive cycle, and even though no conclusive data were obtained, a 2–3 year cycle is possible. The estimated coefficients indicate that this species has a vulnerable life cycle, typical of deepwater squalid sharks. Given the high fishing pressures that it is suffering in the north‐east Atlantic, this fish may already be facing severe declines or in risk of facing them in the near future.