Combining multistate capture-recapture data with tag recoveries to estimate demographic parameters

Matrix population models that allow an animal to occupy more than one state over time are important tools for population and evolutionary ecologists. Definition of state can vary, including location for metapopulation models and breeding state for life history models. For populations whose members can be marked and subsequently reencountered, multistate mark-recapture models are available to estimate the survival and transition probabilities needed to construct population models. Multistate models have proved extremely useful in this context, but they often require a substantial amount of data and restrict estimation of transition probabilities to those areas or states subjected to formal sampling effort. At the same time, for many species, there are considerable tag recovery data provided by the public that could be modeled in order to increase precision and to extend inference to a greater number of areas or states. Here we present a statistical model for combining multistate capture-recapture data (e.g., from a breeding ground study) with multistate tag recovery data (e.g., from wintering grounds). We use this method to analyze data from a study of Canada Geese (Branta canadensis) in the Atlantic Flyway of North America. Our analysis produced marginal improvement in precision, due to relatively few recoveries, but we demonstrate how precision could be further improved with increases in the probability that a retrieved tag is reported.     

Using Scalar Models for Precautionary Assessments of Threatened Species

Abstract:  Scalar population models, commonly referred to as count-based models, are based on time-series data of population sizes and may be useful for screening-level ecological risk assessments when data for more complex models are not available. Appropriate use of such models for management purposes, however, requires understanding inherent biases that may exist in these models. Through a series of simulations, which compared predictions of risk of decline of scalar and matrix-based models, we examined whether discrepancies may arise from different dynamics displayed due to age structure and generation time. We also examined scalar and matrix-based population models of 18 real populations for potential patterns of bias in population viability estimates. In the simulation study, precautionary bias (i.e., overestimating risks of decline) of scalar models increased as a function of generation time. Models of real populations showed poor fit between scalar and matrix-based models, with scalar models predicting significantly higher risks of decline on average. The strength of this bias was not correlated with generation time, suggesting that additional sources of bias may be masking this relationship. Scalar models can be useful for screening-level assessments, which should in general be precautionary, but the potential shortfalls of these models should be considered before using them as a basis for management decisions.     

Demographic heterogeneity, cohort selection, and population growth

Demographic heterogeneity--variation among individuals in survival and reproduction--is ubiquitous in natural populations. Structured population models address heterogeneity due to age, size, or major developmental stages. However, other important sources of demographic heterogeneity, such as genetic variation, spatial heterogeneity in the environment, maternal effects, and differential exposure to stressors, are often not easily measured and hence are modeled as stochasticity. Recent research has elucidated the role of demographic heterogeneity in changing the magnitude of demographic stochasticity in small populations. Here we demonstrate a previously unrecognized effect: heterogeneous survival in long-lived species can increase the long-term growth rate in populations of any size. We illustrate this result using simple models in which each individual's annual survival rate is independent of age but survival may differ among individuals within a cohort. Similar models, but with nonoverlapping generations, have been extensively studied by demographers, who showed that, because the more "frail" individuals are more likely to die at a young age, the average survival rate of the cohort increases with age. Within ecology and evolution, this phenomenon of "cohort selection" is increasingly appreciated as a confounding factor in studies of senescence. We show that, when placed in a population model with overlapping generations, this heterogeneity also causes the asymptotic population growth rate lambda to increase, relative to a homogeneous population with the same mean survival rate at birth. The increase occurs because, even integrating over all the cohorts in the population, the population becomes increasingly dominated by the more robust individuals. The growth rate increases monotonically with the variance in survival rates, and the effect can be substantial, easily doubling the growth rate of slow-growing populations. Correlations between parent and offspring phenotype change the magnitude of the increase in lambda, but the increase occurs even for negative parent-offspring correlations. The effect of heterogeneity in reproductive rate on lambda is quite different: growth rate increases with reproductive heterogeneity for positive parent-offspring correlation but decreases for negative parent-offspring correlation. These effects of demographic heterogeneity on lambda have important implications for population dynamics, population viability analysis, and evolution.     

Impact and Dynamics of Disease in Species Threatened by the Amphibian Chytrid Fungus, Batrachochytrium dendrobatidis

Abstract:  Estimating disease-associated mortality and transmission processes is difficult in free-ranging wildlife but important for understanding disease impacts and dynamics and for informing management decisions. In a capture–mark–recapture study, we used a PCR-based diagnostic test in combination with multistate models to provide the first estimates of disease-associated mortality and detection, infection, and recovery rates for frogs endemically infected with the chytrid fungus Batrachochytrium dendrobatidis (Bd), which causes the pandemic amphibian disease chytridiomycosis. We found that endemic chytridiomycosis was associated with a substantial reduction (approximately 38%) in apparent monthly survival of the threatened rainforest treefrog Litoria pearsoniana despite a long period of coexistence (approximately 30 years); detection rate was not influenced by disease status; improved recovery and reduced infection rates correlated with decreased prevalence, which occurred when temperatures increased; and incorporating changes in individuals' infection status through time with multistate models increased effect size and support (98.6% vs. 71% of total support) for the presence of disease-associated mortality when compared with a Cormack–Jolly–Seber model in which infection status was restricted to the time of first capture. Our results indicate that amphibian populations can face significant ongoing pressure from chytridiomycosis long after epidemics associated with initial Bd invasions subside, an important consideration for the long-term conservation of many amphibian species worldwide. Our findings also improve confidence in estimates of disease prevalence in wild amphibians and provide a general framework for estimating parameters in epidemiological models for chytridiomycosis, an important step toward better understanding and management of this disease.     

Demographic Processes Underlying Population Growth and Decline in Salamandra salamandra

Abstract:  Human activity commonly has negative impacts on wildlife. Often, however, only a single element of the life cycle is affected, and it is unclear whether such effects translate into effects on population growth. This is particularly true for research into the causes of global amphibian declines, where experimental research focuses primarily on the aquatic larval stages but theory suggests these stages have only minor importance for population growth. We used data from long-term mark-recapture studies of two natural populations of the salamander Salamandra salamandra to confirm the predictions of population models. One population remained stable (i.e., stationary) throughout the 20 years of the study whereas the other declined to local extinction. We used mark-recapture models to break down population growth rate into its two main components, recruitment and adult survival. Survival of postmetamorphic salamanders was constant over time in the stable population, whereas the declining population was characterized by a decrease in survival and constant recruitment. Population growth was most sensitive to variation in adult survival. Current amphibian research focuses on preadult stages, and researchers assume recruitment is the most important determinant of population growth. This may not be the case. A better understanding of amphibian population dynamics is possible only through the integration of experiments, theory, and data from natural populations. Our results also suggest that amphibian conservation efforts should focus on all stages of the life cycle and their associated habitats.     

A simulation model of population dynamics of the codling moth, Cydia pomonella

A simulation model has been developed that predicts numbers and phenology of a population of codling moth, Cydia pomonella (L.), in an apple orchard. The model is a general insect population model based on the interative-cohort technique. It operates at two time scales: a fine time scale (1 h) for temperature-dependent physiological processes, and a coarse time scale (1 day) for population processes. The population is divided into a specifiable number of stages, and each stage is described by four process functions, which may be of any convenient mathematical form, and may differ among stages. Each stage is divided into cohorts of individuals born or emerged on the same day, and individuals within a cohort are considered probabilistically identical. The model simulates the processes of development, transition among stages, and mortality by using probability distributions representing these processes, and incorporates the effects of pesticides on mortality of the insect. Model output was evaluated by comparison with records of pheromone trap catches of codling moths in commercial apple orchards in North Carolina. The model predicts timing of the first spring flight well, depending on the initial age distribution used. Time between peaks of numbers of adults in the model is about 15 days longer than the observed period between flight peaks in orchards. Sensitivity analysis indicates that this discrepancy may be related to differences between measured ambient temperature and tree canopy temperature. The sensitivities of numbers of insects produced by the model, and timing of peaks in numbers present were determined for each of the parameters in the model. The parameters with greatest effect on the model output were those which control the locations of developmental rate functions and survival functions on the temperature scale. In the model, pesticides had a much larger effect on numbers of adults present than records of moths caught in pheromone traps indicate actually occurred, suggesting that moths caught in traps in commercial orchards where effective pesticides are applied may be largely immigrants.     

Effects of surrounding land use and water depth on seagrass dynamics relative to a catastrophic algal bloom

Seagrasses are the foundation of many coastal ecosystems and are in global decline because of anthropogenic impacts. For the Indian River Lagoon (Florida, U.S.A.), we developed competing multistate statistical models to quantify how environmental factors (surrounding land use, water depth, and time [year]) influenced the variability of seagrass state dynamics from 2003 to 2014 while accounting for time‐specific detection probabilities that quantified our ability to determine seagrass state at particular locations and times. We classified seagrass states (presence or absence) at 764 points with geographic information system maps for years when seagrass maps were available and with aerial photographs when seagrass maps were not available. We used 4 categories (all conservation, mostly conservation, mostly urban, urban) to describe surrounding land use within sections of lagoonal waters, usually demarcated by land features that constricted these waters. The best models predicted that surrounding land use, depth, and year would affect transition and detection probabilities. Sections of the lagoon bordered by urban areas had the least stable seagrass beds and lowest detection probabilities, especially after a catastrophic seagrass die‐off linked to an algal bloom. Sections of the lagoon bordered by conservation lands had the most stable seagrass beds, which supports watershed conservation efforts. Our results show that a multistate approach can empirically estimate state‐transition probabilities as functions of environmental factors while accounting for state‐dependent differences in seagrass detection probabilities as part of the overall statistical inference procedure.     

Generation time and the maximum growth rate for populations with age-specific fecundities and unknown juvenile survival

In age-classified population models where all parameters are known, the generation time and growth rate are calculated in a straightforward manner. For many populations, some parameters, such as juvenile survival, are difficult to estimate accurately. In a simplified population model where fecundity and survival are constant from the onset of breeding, it is known that generation time may be calculated given only adult survival, age at first reproduction, and the population growth rate. However, the assumption of constant fecundity from the onset of breeding does not hold for many populations. An extended population model allows calculation of generation time with the additional knowledge of the ratio of age-specific fecundities compared to a maximum fecundity rate. When these relative fecundities are unknown, an ad hoc adjustment to the simplified model performs well.When the study population is in an ideal environment, the optimal generation time and maximum growth rate are linked, and both may be approximated knowing only adult survival, age at first reproduction, and the relative fecundities. The maximum growth rate has important conservation implications, and calculating it correctly is therefore important. Improper use of the simplified population model to calculate the maximum growth rate, combined with a simple decision rule, leads to an average overharvest of 36%, and >60% for three of six bird species studied, compared to the full population model. By comparison, using the approximation from the extended or adjusted models results in average overharvests of only 8% (extended model) and 5% (adjusted model), and <50% for all six species (either model).     

Modeling survival and mark loss in molting animals: recapture, dead recoveries, and exuvia recoveries

Capture-mark-recapture (CMR) analyses aim primarily at estimating relevant life history parameters, despite the fact that some individuals are not always recaptured, even if alive on the study site. Applying such approaches to species with a complex life cycle, such as insects, remains challenging because each change of stage tends to cause mark loss through molting. We developed a multistate model based on three exclusive events ("dead", "surviving and molting", and "surviving and staying in the same larval stage") to estimate probabilities of survival and mark loss. Estimates of biologically relevant parameters were derived from those of the probabilities of transition between these states. The model was applied to data from radio-tracking diodes glued on grasshoppers. The estimates of recapture probabilities decreased throughout the season for animals remaining alive, while the detection of dead animals and lost diodes was exhaustive. The survival probability was higher for larvae than for adults (0.98 vs. 0.96), and mark loss was stronger in larvae than in adults (0.09 vs. 0.06). We show that the survival rate of a species with a high rate of mark loss can be estimated using multistate models, provided that marks can be recovered after being lost. These models are flexible enough to test for several effects that potentially affect survival and mark loss probabilities.     

Estimating survival rates with time series of standing age-structure data

It has long been recognized that age-structure data contain useful information for assessing the status and dynamics of wildlife populations. For example, age-specific survival rates can be estimated with just a single sample from the age distribution of a stable, stationary population. For a population that is not stable, age-specific survival rates can be estimated using techniques such as inverse methods that combine time series of age-structure data with other demographic data. However, estimation of survival rates using these methods typically requires numerical optimization, a relatively long time series of data, and smoothing or other constraints to provide useful estimates. We developed general models for possibly unstable populations that combine time series of age-structure data with other demographic data to provide explicit maximum likelihood estimators of age-specific survival rates with as few as two years of data. As an example, we applied these methods to estimate survival rates for female bison (Bison bison) in Yellowstone National Park, USA. This approach provides a simple tool for monitoring survival rates based on age-structure data.     

Evolutionarily stable strategies of age-dependent sexual advertisement

In various models of sexual selection mediated by the viability indicator (“good genes”) mechanism, a sexually selected trait will truly reflect male quality if its expression is costly for the male. However, in long-lived species, the expression of a trait often increases with age while the genotype of the male remains unchanged. This fact may obscure the indicator mechanism. Hitherto, game theory models of honesty in sexual advertisement have not taken life-history effects into account, whereas life-history models of reproductive effort have only seldom considered the dependence of mating success on the actions of other individuals. Here, the two approaches are combined, and I examine whether honesty is maintained if males can divide their advertisement effort over their lifetime. The model shows that an increase in the expression of the sexually selected trait over several years is an evolutionarily stable strategy (ESS) under a wide range of situations, so that a correlated preference for old age can emerge through a viability indicator mechanism. Honesty in the strict sense is not preserved: an optimally behaving low-quality male will in some cases advertise more than a high-quality male of equal age, to the extent that the strongest advertisement found in the population can be associated with a low-quality male. Due to life-history trade-offs, however, honesty in an average sense holds true over the lifetime of individuals: “cheater” age classes will remain small enough, that a female will obtain a higher expected mate quality if she trusts in the trait as an indicator of viability. Received: 25 June 1996 / Accepted after revision: 15 April 1997     

Estimating age from recapture data: integrating incremental growth measures with ancillary data to infer age-at-length

Estimating the age of individuals in wild populations can be of fundamental importance for answering ecological questions, modeling population demographics, and managing exploited or threatened species. Significant effort has been devoted to determining age through the use of growth annuli, secondary physical characteristics related to age, and growth models. Many species, however, either do not exhibit physical characteristics useful for independent age validation or are too rare to justify sacrificing a large number of individuals to establish the relationship between size and age. Length-at-age models are well represented in the fisheries and other wildlife management literature. Many of these models overlook variation in growth rates of individuals and consider growth parameters as population parameters. More recent models have taken advantage of hierarchical structuring of parameters and Bayesian inference methods to allow for variation among individuals as functions of environmental covariates or individual-specific random effects. Here, we describe hierarchical models in which growth curves vary as individual-specific stochastic processes, and we show how these models can be fit using capture-recapture data for animals of unknown age along with data for animals of known age. We combine these independent data sources in a Bayesian analysis, distinguishing natural variation (among and within individuals) from measurement error. We illustrate using data for African dwarf crocodiles, comparing von Bertalanffy and logistic growth models. The analysis provides the means of predicting crocodile age, given a single measurement of head length. The von Bertalanffy was much better supported than the logistic growth model and predicted that dwarf crocodiles grow from 19.4 cm total length at birth to 32.9 cm in the first year and 45.3 cm by the end of their second year. Based on the minimum size of females observed with hatchlings, reproductive maturity was estimated to be at nine years. These size benchmarks are believed to represent thresholds for important demographic parameters; improved estimates of age, therefore, will increase the precision of population projection models. The modeling approach that we present can be applied to other species and offers significant advantages when multiple sources of data are available and traditional aging techniques are not practical.     

Estimating parameters of hidden Markov models based on marked individuals: use of robust design data

Development and use of multistate mark-recapture models, which provide estimates of parameters of Markov processes in the face of imperfect detection, have become common over the last 20 years. Recently, estimating parameters of hidden Markov models, where the state of an individual can be uncertain even when it is detected, has received attention. Previous work has shown that ignoring state uncertainty biases estimates of survival and state transition probabilities, thereby reducing the power to detect effects. Efforts to adjust for state uncertainty have included special cases and a general framework for a single sample per period of interest. We provide a flexible framework for adjusting for state uncertainty in multistate models, while utilizing multiple sampling occasions per period of interest to increase precision and remove parameter redundancy. These models also produce direct estimates of state structure for each primary period, even for the case where there is just one sampling occasion. We apply our model to expected-value data, and to data from a study of Florida manatees, to provide examples of the improvement in precision due to secondary capture occasions. We have also implemented these models in program MARK. This general framework could also be used by practitioners to consider constrained models of particular interest, or to model the relationship between within-primary-period parameters (e.g., state structure) and between-primary-period parameters (e.g., state transition probabilities).     

Adjusting age and stage distributions for misclassification errors

Ecologists often use samples from the age or stage structure of a population to make inferences about population-level processes and to parameterize matrix models. Typically, researchers make a simplifying assumption that age and stage classes are determined without error, when in fact some level of misclassification often can be expected. If unaccounted for, misclassification will lead to overly optimistic levels of precision and can cause biased estimates of age or stage structure. Although several studies have used information from known-age individuals to quantify errors in age or stage distribution, the problem of estimating the age or stage structure in face of such errors has received comparably little attention. In this paper, we describe a general statistical framework for estimating the true stage distribution of a sample when misclassification rates can be estimated. The estimation process requires auxiliary information on misclassification rates, such as data from individuals of known age. We analyze age-structured harvest records from black bears in Pennsylvania to illustrate how incorporating misclassification errors leads to changes in point estimates and provides a measure of precision.     

Effects of Social Structure and Prey Dynamics on Extinction Risk in Gray Wolves

Extinction models based on diffusion theory generally fail to incorporate two important aspects of population biology—social structure and prey dynamics. We include these aspects in an individual-based extinction model for small, isolated populations of the gray wolf (Canis lupus). Our model predicts mean times to extinction significantly longer than those predicted by more general (diffusion) models. According to our model, an isolated population of 50 wolves has a 95% chance of surviving just 9 years and only a 30% chance of surviving beyond 100 years. Reflecting the influence of social structure, a wolf population initially comprising 50 individuals is expected to persist only a few years longer, on average (71 years), than is a population initially comprising just a single reproductive pair (62 years). In contrast, substantially greater average prey abundance leads to dramatically longer expected persistence times. Autocorrelated prey dynamics result in a more complex distribution of extinction times than predicted by many extinction models. We contend that demographic stochasticity may pose the greatest threat to small, isolated wolf populations, although environmental stochasticity and genetic effects may compound this threat. Our work highlights the importance of considering social structure and resource dynamics in the development of population viability analyses.     

The influence of initial age structure on predator–prey interaction

Initial age structure influences the growth of a prey population and the outcome of the predator–prey interaction. In order to quantify that influence, we employed a simple numerical model using experimental data from the system Tetranychus urticae–Phytoseiulus persimilis. Four major points were drawn from the results: (1) A population created by young females grows much faster than a population created by the same number of females but distributed among the stable age structure. Final number of individuals after a few generations is then much higher than what a plant could support. Consequently, a stable age structure is probably never achieved under these conditions; (2) In the presence of a predator, such a population can persist for a sufficiently long time to overexploit its host plant and to produce enough individuals to allow dispersal; (3) The impact of the predator on the prey population is drastically different according to its own age structure at the beginning of the interaction; and (4) Predators disturb the prey age structure during the course of interactions and thus maintain the prey growth potential at a high level. These points constitute an important adaptation that determine the persistence of the prey and the predator at a metapopulation level. They bring a new insight on the adaptive characters of young female dispersal.     

Multimethod,multistate Bayesian hierarchical modeling approach for use in regional monitoring of wolves

In many cases, the first step in large‐carnivore management is to obtain objective, reliable, and cost‐effective estimates of population parameters through procedures that are reproducible over time. However, monitoring predators over large areas is difficult, and the data have a high level of uncertainty. We devised a practical multimethod and multistate modeling approach based on Bayesian hierarchical‐site‐occupancy models that combined multiple survey methods to estimate different population states for use in monitoring large predators at a regional scale. We used wolves (Canis lupus) as our model species and generated reliable estimates of the number of sites with wolf reproduction (presence of pups). We used 2 wolf data sets from Spain (Western Galicia in 2013 and Asturias in 2004) to test the approach. Based on howling surveys, the naïve estimation (i.e., estimate based only on observations) of the number of sites with reproduction was 9 and 25 sites in Western Galicia and Asturias, respectively. Our model showed 33.4 (SD 9.6) and 34.4 (3.9) sites with wolf reproduction, respectively. The number of occupied sites with wolf reproduction was 0.67 (SD 0.19) and 0.76 (0.11), respectively. This approach can be used to design more cost‐effective monitoring programs (i.e., to define the sampling effort needed per site). Our approach should inspire well‐coordinated surveys across multiple administrative borders and populations and lead to improved decision making for management of large carnivores on a landscape level. The use of this Bayesian framework provides a simple way to visualize the degree of uncertainty around population‐parameter estimates and thus provides managers and stakeholders an intuitive approach to interpreting monitoring results. Our approach can be widely applied to large spatial scales in wildlife monitoring where detection probabilities differ between population states and where several methods are being used to estimate different population parameters.     

Modeling the Persistence of Small Populations of Strongly Interdependent Species: Figs and Fig Wasps

Conservation problems are usually studied at the population or ecosystem levels. Formulating predictive theory for the domain in between has been difficult. Fig trees and their pollinating wasps, principally tropical groups of organisms, form pairs of obligate mutualists that provide unique opportunities for studying the influence of species interactions on the survival of small populations. Survival of each partner depends on that of the associated species. The pollinator population can be maintained only if figs are produced year-round. Because fig trees flower synchronously at the individual level, wasps have to locate a new individual host tree at each generation. We describe results of simulation models estimating the minimum number of trees required to maintain a wasp population using two levels of the criteria: (1) different probability of survival (50% and 99%) and (2) different time of survival (5 or 1000 years). We also examined how these different estimates are sensitive to differences in the seasonality of flowering period and in the length of the period of female receptivity in figs. Such estimates can be used to understand the potential effects of the reduction of fig population size via fragmentation. Unlike most studies on the effect of low population size on population viability, our paper focuses on maintenance of a biotic interaction, rather than on single-species dynamics. The biotic interaction on which we focus is important because figs in many tropical ecosystems may be keystone resources for frugivores that are in turn essential seed dispersal agents for other plants.     

General methods for sensitivity analysis of equilibrium dynamics in patch occupancy models

Sensitivity analysis is a useful tool for the study of ecological models that has many potential applications for patch occupancy modeling. Drawing from the rich foundation of existing methods for Markov chain models, I demonstrate new methods for sensitivity analysis of the equilibrium state dynamics of occupancy models. Estimates from three previous studies are used to illustrate the utility of the sensitivity calculations: a joint occupancy model for a prey species, its predators, and habitat used by both; occurrence dynamics from a well-known metapopulation study of three butterfly species; and Golden Eagle occupancy and reproductive dynamics. I show how to deal efficiently with multistate models and how to calculate sensitivities involving derived state variables and lower-level parameters. In addition, I extend methods to incorporate environmental variation by allowing for spatial and temporal variability in transition probabilities. The approach used here is concise and general and can fully account for environmental variability in transition parameters. The methods can be used to improve inferences in occupancy studies by quantifying the effects of underlying parameters, aiding prediction of future system states, and identifying priorities for sampling effort.     

Multistate modeling of habitat dynamics: factors affecting Florida scrub transition probabilities

Many ecosystems are influenced by disturbances that create specific successional states and habitat structures that species need to persist. Estimating transition probabilities between habitat states and modeling the factors that influence such transitions have many applications for investigating and managing disturbance-prone ecosystems. We identify the correspondence between multistate capture-recapture models and Markov models of habitat dynamics. We exploit this correspondence by fitting and comparing competing models of different ecological covariates affecting habitat transition probabilities in Florida scrub and flatwoods, a habitat important to many unique plants and animals. We subdivided a large scrub and flatwoods ecosystem along central Florida's Atlantic coast into 10-ha grid cells, which approximated average territory size of the threatened Florida Scrub-Jay (Aphelocoma coerulescens), a management indicator species. We used 1.0-m resolution aerial imagery for 1994, 1999, and 2004 to classify grid cells into four habitat quality states that were directly related to Florida Scrub-Jay source-sink dynamics and management decision making. Results showed that static site features related to fire propagation (vegetation type, edges) and temporally varying disturbances (fires, mechanical cutting) best explained transition probabilities. Results indicated that much of the scrub and flatwoods ecosystem was resistant to moving from a degraded state to a desired state without mechanical cutting, an expensive restoration tool. We used habitat models parameterized with the estimated transition probabilities to investigate the consequences of alternative management scenarios on future habitat dynamics. We recommend this multistate modeling approach as being broadly applicable for studying ecosystem, land cover, or habitat dynamics. The approach provides maximum-likelihood estimates of transition parameters, including precision measures, and can be used to assess evidence among competing ecological models that describe system dynamics.