Integral projection models for finite populations in a stochastic environment

Continuous types of population structure occur when continuous variables such as body size or habitat quality affect the vital parameters of individuals. These structures can give rise to complex population dynamics and interact with environmental conditions. Here we present a model for continuously structured populations with finite size, including both demographic and environmental stochasticity in the dynamics. Using recent methods developed for discrete age-structured models we derive the demographic and environmental variance of the population growth as functions of a continuous state variable. These two parameters, together with the expected population growth rate, are used to define a one-dimensional diffusion approximation of the population dynamics. Thus, a substantial reduction in complexity is achieved as the dynamics of the complex structured model can be described by only three population parameters. We provide methods for numerical calculation of the model parameters and demonstrate the accuracy of the diffusion approximation by computer simulation of specific examples. The general modeling framework makes it possible to analyze and predict future dynamics and extinction risk of populations with various types of structure, and to explore consequences of changes in demography caused by, e.g., climate change or different management decisions. Our results are especially relevant for small populations that are often of conservation concern.     

Estimation of population parameters from aerial counts of North American mallards: a cautionary tale.

The accuracy of population estimates strongly interferes with our ability to obtain unbiased estimates of population parameters based on analyses of time series of population fluctuations. Here we use long-term data on fluctuations in the size of Mallard populations collected as part of the May Breeding Waterfowl Survey covering a large section of North America. We assume a log-linear model of density dependence and use a hierarchical Bayesian state-space approach in which all parameters are assumed to be realizations from a common underlying distribution. Thus, parameters for different populations are not allowed to vary independently of each other. We then simulated independent time series of aerial counts, using the estimated parameters and adding various levels of observation error. These simulations showed that the estimates of stochastic population growth rate and strength of density dependence were biased even when moderate sampling errors were present. In contrast, the estimates of the environmental stochasticity and the carrying capacity were unbiased even for short time series and large observation error. Our results underline the importance of reducing the magnitude of sampling error in the design of large-scale monitoring programs of population fluctuations.     

A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove

The spread of invasive species is a long studied subject that garners much interest in the ecological research community. Historically the phenomenon has been approached using a purely deterministic mathematical framework (usually involving differential equations of some form). These methods, while scientifically meaningful, are generally highly simplified and fail to account for uncertainty in the data and process, of which our knowledge could not possibly exist without error. We propose a hierarchical Bayesian model for population spread that accommodates data sources with errors, dependence structures between population dynamics parameters, and takes into account prior scientific understanding via non-linear relationships between model parameters and space-time response variables. We model the process (i.e., the bird population in this case) as a Poisson response with spatially varying diffusion coefficients as well as a logistic population growth term using a common reaction-diffusion equation that realistically mimics the ecological process. We focus the application on the ongoing invasion of the Eurasian Collared-Dove.     

Hierarchical demographic approaches for assessing invasion dynamics of non-indigenous species: An example using northern snakehead (Channa argus)

Models of species’ demographic features are commonly used to understand population dynamics and inform management tactics. Hierarchical demographic models are ideal for the assessment of non-indigenous species because our knowledge of non-indigenous populations is usually limited, data on demographic traits often come from a species’ native range, these traits vary among populations, and traits are likely to vary considerably over time as species adapt to new environments. Hierarchical models readily incorporate this spatiotemporal variation in species’ demographic traits by representing demographic parameters as multi-level hierarchies. As is done for traditional non-hierarchical matrix models, sensitivity and elasticity analyses are used to evaluate the contributions of different life stages and parameters to estimates of population growth rate. We applied a hierarchical model to northern snakehead (Channa argus), a fish currently invading the eastern United States. We used a Monte Carlo approach to simulate uncertainties in the sensitivity and elasticity analyses and to project future population persistence under selected management tactics. We gathered key biological information on northern snakehead natural mortality, maturity and recruitment in its native Asian environment. We compared the model performance with and without hierarchy of parameters. Our results suggest that ignoring the hierarchy of parameters in demographic models may result in poor estimates of population size and growth and may lead to erroneous management advice. In our case, the hierarchy used multi-level distributions to simulate the heterogeneity of demographic parameters across different locations or situations. The probability that the northern snakehead population will increase and harm the native fauna is considerable. Our elasticity and prognostic analyses showed that intensive control efforts immediately prior to spawning and/or juvenile-dispersal periods would be more effective (and probably require less effort) than year-round control efforts. Our study demonstrates the importance of considering the hierarchy of parameters in estimating population growth rate and evaluating different management strategies for non-indigenous invasive species.     

A population viability analysis for the Island Fox on Santa Catalina Island,California

The island fox (Urocyon littoralis) on Santa Catalina Island is among the most imperiled species on the Channel Islands due to a recent outbreak of canine distemper virus (CDV). The western subpopulation, which was not exposed to CDV, is a crucial element in the recovery of foxes by providing a source of animals for translocation and captive breeding. Using the program VORTEX, we developed a population viability analysis for the Santa Catalina Island fox to (1) address the likelihood of population persistence, (2) estimate the current susceptibility of the population to catastrophic events, and (3) evaluate the efficacy of current restoration strategies of releasing captive bred foxes and transplanting wild animals. Overall, we found the population to be susceptible to catastrophic events; a 50% increase in mortality every 20 years was sufficient to elevate the extinction risk above 5%. Current management activities entail the transplanting of 12 juvenile foxes annually, which may reduce the viability of the western subpopulation. A minimum population size of at least 150 foxes should be maintained in each subpopulation to reduce the risk of extinction due to demographic stochasticity. Releases of translocated and captive bred animals affect the speed of recovery on the eastern half of Catalina Island, but not the probability of extinction, which is near zero under current conditions. We conducted a sensitivity analysis for demographic parameters by incrementally varying survival, fecundity and density-dependence parameters, while holding all other parameters constant. Sensitivity analyses identified mortality and mean litter size as the most sensitive parameters, while the implementation of density-dependence and environmental variation of model parameters did not seem to affect population performance. We conclude that the population of island foxes on Santa Catalina is currently at a critically low population level, but recovery of the species appears possible.     

A generalized estimating equations approach for capture–recapture closed population models

The estimation of population density animal population parameters, such as capture probability, population size, or population density, is an important issue in many ecological applications. Capture–recapture data may be considered as repeated observations that are often correlated over time. If these correlations are not taken into account then parameter estimates may be biased, possibly producing misleading results. We propose a generalized estimating equations (GEE) approach to account for correlation over time instead of assuming independence as in the traditional closed population capture–recapture studies. We also account for heterogeneity among observed individuals and over-dispersion, modelling capture probabilities as a function of covariates. The GEE versions of all closed population capture–recapture models and their corresponding estimating equations are proposed. We evaluate the effect of accounting for correlation structures on capture–recapture model selection based on the quasi-likelihood information criterion (QIC). An example is used for an illustrative application and for comparison to currently used methodology. A Horvitz–Thompson-like estimator is used to obtain estimates of population size based on conditional arguments. A simulation study is conducted to evaluate the performance of the GEE approach in capture-recapture studies. The GEE approach performs well for estimating population parameters, particularly when capture probabilities are high. The simulation results also reveal that estimated population size varies on the nature of the existing correlation among capture occasions.     

The Impact of Increased Environmental Stochasticity Due to Climate Change on the Dynamics of Asiatic Wild Ass

Abstract:  Theory proposes that increased environmental stochasticity negatively impacts population viability. Thus, in addition to the directional changes predicted for weather parameters under global climate change (GCC), the increase in variance of these parameters may also have a negative effect on biodiversity. As a case study, we assessed the impact of interannual variance in precipitation on the viability of an Asiatic wild ass ( Equus hemionus ) population reintroduced in Makhtesh Ramon Nature Reserve, Israel. We monitored the population from 1985 to 1999 to determine what environmental factors affect reproductive success. Annual precipitation during the year before conception, drought conditions during gestation, and population size determined reproductive success. We used the parameters derived from this model to assess population performance under various scenarios in a Leslie matrix type model with demographic and environmental stochasticity. Specifically, we used a change in the precipitation regime in our study area to formulate a GCC scenario and compared the simulated dynamics of the population with a no-change scenario. The coefficient of variation in population size under the global change scenario was 30% higher than under the no-change scenario. Minor die-offs (≥15%) following droughts increased extinction probability nearly 10-fold. Our results support the idea that an increase in environmental stochasticity due to GCC may, in itself, pose a significant threat to biodiversity.     

Uncertainty analysis of transient population dynamics

Two types of demographic analyses, perturbation analysis and uncertainty analysis, can be conducted to gain insights about matrix population models and guide population management. Perturbation analysis studies how the perturbation of demographic parameters (survival, growth, and reproduction parameters) may affect the population projection, while uncertainty analysis evaluates how much uncertainty there is in population dynamic predictions and where the uncertainty comes from. Previously, both perturbation analysis and uncertainty analysis were conducted on the long-term population growth rate. However, the population may not reach its equilibrium state, especially when there is management by harvesting or hunting. Recently, there has been an increased interest in short-term transient dynamics, which can differ from asymptotic long-term dynamics. There are currently techniques to conduct perturbation analyses of short-term transient dynamics, but no techniques have been proposed for uncertainty analysis of such dynamics. In this study, we introduced an uncertainty analysis technique, the general Fourier Amplitude Sensitivity Test (FAST), to study uncertainties in transient population dynamics. The general FAST is able to identify the amount of uncertainty in transient dynamics and contributions by different demographic parameters. We applied the general FAST to a mountain goat (Oreamnos americanus) matrix population model to give a clear illustration of how uncertainty analysis can be conducted for transient dynamics arising from matrix population models.     

On the use of growth rate parameters for projecting population sizes: Application to aphids

This paper extends the application of the cumulative size based mechanistic model, which has previously been shown to describe diverse aphid population size data well. The mechanistic model is reviewed with a focus on the explanatory role of the birth and death rate formulation. An analysis of two data sets, one on the mustard aphid and the other on the pecan aphid, indicates that multiple linear regression equations based on the estimated birth and death rate parameters alone account for nearly all (R² > 0.95) of the variability in two key population attributes, namely the peak count and the cumulative density. This indicates that population size variables may be projected directly from the growth rate parameters using linear equations. Such linear relationships based on the birth and death rate parameters are shown to hold also for certain generalized mechanistic models for which the analytical solution is not available. The birth and death rate coefficients, therefore, constitute a new succinct set of variables that could be included in the predictive modeling of aphid populations, as well as other insect and animal populations with local collapse which follow similar growth dynamics.     

Assessing plausible rates of population growth in humpback whales from life-history data

The rate of growth of any population is a quantity of interest in conservation and management and is constrained by biological factors. In this study, recent data on life-history parameters influencing rates of population growth in humpback whales, including survival, age at first parturition and calving rate are reviewed. Monte Carlo simulations are used to compute a distribution of rates of increase (ROIs) taking into account uncertainty in biological parameter estimates. Two approaches for computing juvenile survival are proposed, which taken into account along with other life-history data, resulted in the following estimates of the rate of population growth: Approach A: mean of 7.3%/year (95% CI = 3.5–10.5%/year) and Approach B: mean of 8.6%/year (95% CI = 5.0–11.4%/year). It is proposed that the upper 99% quantile of the resulting distribution of the ROI for Approach B (11.8%/year) be established as the maximum plausible ROI for humpback whales and be used in population assessment of the species. Possible sources of positive and negative biases in the present estimates are presented and include measurement error in estimation of life-history parameters, changes in the environment within the period these quantities are measured, density dependence or other natural factors. However, it is difficult to evaluate potential biases without additional data. The methods presented in this study can be applied to other species for which life-history parameters are available and are useful in assessing plausibility in the estimation of population growth rates from time series of abundance estimates.     

Incorporating Uncertainty of Management Costs in Sensitivity Analyses of Matrix Population Models

The importance of accounting for economic costs when making environmental‐management decisions subject to resource constraints has been increasingly recognized in recent years. In contrast, uncertainty associated with such costs has often been ignored. We developed a method, on the basis of economic theory, that accounts for the uncertainty in population‐management decisions. We considered the case where, rather than taking fixed values, model parameters are random variables that represent the situation when parameters are not precisely known. Hence, the outcome is not precisely known either. Instead of maximizing the expected outcome, we maximized the probability of obtaining an outcome above a threshold of acceptability. We derived explicit analytical expressions for the optimal allocation and its associated probability, as a function of the threshold of acceptability, where the model parameters were distributed according to normal and uniform distributions. To illustrate our approach we revisited a previous study that incorporated cost‐efficiency analyses in management decisions that were based on perturbation analyses of matrix population models. Incorporating derivations from this study into our framework, we extended the model to address potential uncertainties. We then applied these results to 2 case studies: management of a Koala (Phascolarctos cinereus) population and conservation of an olive ridley sea turtle (Lepidochelys olivacea) population. For low aspirations, that is, when the threshold of acceptability is relatively low, the optimal strategy was obtained by diversifying the allocation of funds. Conversely, for high aspirations, the budget was directed toward management actions with the highest potential effect on the population. The exact optimal allocation was sensitive to the choice of uncertainty model. Our results highlight the importance of accounting for uncertainty when making decisions and suggest that more effort should be placed on understanding the distributional characteristics of such uncertainty. Our approach provides a tool to improve decision making.     

Using Modeling to Improve Monitoring of Structured Populations: Are We Collecting the Right Data?

Abstract:  Population monitoring is central to most demographic studies and conservation efforts, but it may not always be directed at the most appropriate life stage. We used stochastic simulation modeling to evaluate the effectiveness of a monitoring program for a well-studied population of Eastern Imperial Eagles ( Aquila heliaca ) in Kazakhstan. Specifically, we asked whether the most appropriate data were being collected to understand system state and population dynamics. Our models were parameterized with data collected over the course of 25 years of study of this population. We used the models to conduct simulation experiments to evaluate relationships between monitored or potentially monitored parameters and the demographic variables of interest—population size ( N ) and population growth (λ). Static analyses showed that traditional territory-based monitoring was a poor indicator of eagle population size and growth and that monitoring survivorship would provide more information about these parameters. Nevertheless, these same traditionally monitored territory-based parameters had greater power to detect long-term changes in population size than did survivorship or population structure. Regardless of the taxa considered, threats can have immediate impacts on population size and growth or longer-term impacts on population dynamics. Prudently designed monitoring programs for any species will detect the demographic effects of both types of threats.     

Sensitivity analysis of North American bird population estimates

The Partners in Flight North American Landbird Conservation Plan provided estimates of population sizes for 448 landbird species using a multiplicative model. Input parameters in this calculation included the area of state × Bird Conservation Region polygons, area-specific mean Breeding Bird Survey counts circa 1995, and adjustment factors for the distance over which species may presumably be correctly counted, the assumed pairing of singing males with non-singing females, and variability in the propensity of birds to sing over the course of the survey day. I assessed the sensitivity of this population calculation to changes in the input parameters. I assessed both local and global sensitivity of the model to changes in the parameters with Monte Carlo one-at-a-time simulations and the Fourier amplitude sensitivity test (FAST). Monte Carlo simulations were an estimate of local model sensitivity whereas FAST estimated global model sensitivity, accommodating the potential shared variance between model parameters. Monte Carlo simulations suggested population estimates were 39% more sensitive to changes in the detection distance adjustment than to the other parameters; the other parameters were nearly equal in their contribution to model sensitivity. Conversely, FAST analysis determined that each of the input variables aside from the pair adjustment provided roughly equal contributions to variability in population estimates. The most efficient means for improving continental population estimates for birds surveyed by the Breeding Bird Survey will be through increased scrutiny of the species-specific distance detection and time-of-day adjustments and improved understanding in the spatial and temporal variability in the mean Breeding Bird Survey count.     

Exploring Population Dynamics Patterns in a Rare Fish, Zingel asper, through Capture-Mark-Recapture Methods

Abstract:  We performed a capture-mark-recapture study on one of the last populations of Zingel asper , an endemic percid species of the Rhône River basin in France. The distribution of Z. asper has decreased dramatically during the last century. We sampled three sites in suitable habitats in the Beaume River. No impact of individual tagging on survival was found. The demography of the population was analyzed using capture-recapture methods that allow the estimation of survival, recruitment, and demographic growth rates. Annual survival rates were low (0.35–0.50). The level of transience was high (5% to 25%), suggesting that a significant number of individuals were highly mobile or shifted to suboptimal habitats. Seniority rates suggested random highly variable recruitment between years. The three sites had similar variation patterns in all demographic parameters, indicating broad spatial covariation in population dynamics. We found some local differences in demographic parameters, which could be linked to local habitat quality. Individual tagging allowed for the estimation of demographic parameters that improved our understanding of Z. asper population dynamics and revealed mechanisms that may affect population persistence, such as stochastic recruitment, low survival, and frequent dispersal. The fragmentation of habitat through river damming inhibits dispersal and represents a threat to the persistence of Z. asper in the Rhône basin. Our results offer evidence of the importance of dispersal in nonmigratory fishes and confirm the usefulness of individual tagging methods in rare fish demography.     

Hidden process models for animal population dynamics.

Hidden process models are a conceptually useful and practical way to simultaneously account for process variation in animal population dynamics and measurement errors in observations and estimates made on the population. Process variation, which can be both demographic and environmental, is modeled by linking a series of stochastic and deterministic subprocesses that characterize processes such as birth, survival, maturation, and movement. Observations of the population can be modeled as functions of true abundance with realistic probability distributions to describe observation or estimation error. Computer-intensive procedures, such as sequential Monte Carlo methods or Markov chain Monte Carlo, condition on the observed data to yield estimates of both the underlying true population abundances and the unknown population dynamics parameters. Formulation and fitting of a hidden process model are demonstrated for Sacramento River winter-run chinook salmon (Oncorhynchus tshawytsha).     

马拉硫磷急性暴露对斑马鱼的行为毒性研究

以斑马鱼为受试生物,通过水质毒性生物监测仪记录行为轨迹,研究了亚致死浓度的马拉硫磷急性暴露下斑马鱼游动行为和群体分布等多项行为参数的变化。结果表明:斑马鱼对环境变化响应快速,游动速度短时间内急剧增大,之后下降再调整稳定至一定范围,变化趋势符合环境压力模型。游泳高度不断增大,暴露1 h后斑马鱼几乎全部集中到水箱上部,与暴露前水平差异显著。通讯行为参数平均距离和分散度在暴露后短时间内减小之后恢复到暴露前水平。通过解析斑马鱼的行为变化可以实现水体有机磷农药突发污染的早期预警。     

Sensitivity of population viability to spatial and nonspatial parameters using GRIP.

Metapopulation dynamics are influenced by spatial parameters including the amount and arrangement of suitable habitat, yet these parameters may be uncertain when deciding how to manage species or their habitats. Sensitivity analyses of population viability analysis (PVA) models can help measure relative parameter influences on predictions, identify research priorities for reducing uncertainty, and evaluate management strategies. Few spatial PVAs, however, include sensitivity analyses of both spatial and nonspatial parameters, perhaps because computationally efficient tools for such analyses are lacking or inaccessible. We developed GRIP, a program to facilitate sensitivity analysis of spatial and nonspatial input parameters for PVAs created in RAMAS Metapop, a widely applied software program. GRIP creates random sets of input files by varying parameters specified in the PVA model including vital rates and their correlations among populations, the number and configuration of populations, dispersal rates, dispersal survival, initial population abundances, carrying capacities, and the probability, intensity, and spatial extent of catastrophes, while drawing on specified parameter distributions. We evaluated GRIP's performance as a tool for sensitivity analysis of spatial PVAs and explored the consequences of varying spatial input parameters for predictions of a published PVA model of the sand lizard (Lacerta agilis). We used GRIP output to generate standardized regression coefficients (SRCs) and nonparametric correlation coefficients as indices of the relative sensitivity of predicted conservation status to input parameters. GRIP performed well; with a single analysis we were able to rank the relative influence of input parameters identified as influential by the PVA's original author, S. A. Berglind, who used three separate forms of sensitivity analysis. Our analysis, however, also underscored the value of exploring the relative influence of spatial parameters on PVA predictions; both SRCs and correlation coefficients indicated that the most influential parameters in the sand lizard model were spatial in nature. We provide annotated code so that GRIP may be modified to reflect particular species biology, customized for more complex spatial PVA models, upgraded to incorporate features added in newer versions of RAMAS Metapop, used as a template to develop similar programs, or used as it is for computationally efficient sensitivity analyses in support of conservation planning.     

Predicting local population distributions around a central shelter based on a predation risk-growth trade-off

Animals face trade-offs between predation risk and foraging success depending on their location in the landscape; for example, individuals that remain near a common shelter may be safe from predation but incur stronger competition for resources. Despite a long tradition of theoretical exploration of the relationships among foraging success, conspecific competition, predation risk, and population distribution in a heterogeneous environment, the scenario we describe here has not been explored theoretically. We construct a model of habitat use rules to predict the distribution of a local population (prey sharing a common shelter and foraging across surrounding habitats). Our model describes realized habitat quality as a ratio of density- and location-dependent mortality to density-dependent growth. We explore how the prey distribution around a shelter is expected to change as the parameters governing the strength of density dependence, landscape characteristics, and local abundance vary. Within the range of parameters where prey spend some time away from shelter but remain site-attached, the prey density decreases away from shelter. As the distance at which prey react to predators increases, the population range generally increases. At intermediate reaction distances, however, increases in the reaction distance lead to decreases in the maximum foraging distance because of increased evenness in the population distribution. As total abundance increases, the population range increases, average population density increases, and realized quality decreases. The magnitude of these changes differs in, for example, ‘high-’ and ‘low-visibility’ landscapes where prey can detect predators at different distances.     

Sampling variability and estimates of density dependence: a composite-likelihood approach

It is well known that sampling variability, if not properly taken into account, affects various ecologically important analyses. Statistical inference for stochastic population dynamics models is difficult when, in addition to the process error, there is also sampling error. The standard maximum-likelihood approach suffers from large computational burden. In this paper, I discuss an application of the composite-likelihood method for estimation of the parameters of the Gompertz model in the presence of sampling variability. The main advantage of the method of composite likelihood is that it reduces the computational burden substantially with little loss of statistical efficiency. Missing observations are a common problem with many ecological time series. The method of composite likelihood can accommodate missing observations in a straightforward fashion. Environmental conditions also affect the parameters of stochastic population dynamics models. This method is shown to handle such nonstationary population dynamics processes as well. Many ecological time series are short, and statistical inferences based on such short time series tend to be less precise. However, spatial replications of short time series provide an opportunity to increase the effective sample size. Application of likelihood-based methods for spatial time-series data for population dynamics models is computationally prohibitive. The method of composite likelihood is shown to have significantly less computational burden, making it possible to analyze large spatial time-series data. After discussing the methodology in general terms, I illustrate its use by analyzing a time series of counts of American Redstart (Setophaga ruticilla) from the Breeding Bird Survey data, San Joaquin kit fox (Vulpes macrotis mutica) population abundance data, and spatial time series of Bull trout (Salvelinus confluentus) redds count data.     

The Reliability of Using Population Viability Analysis for Risk Classification of Species

I examine whether or not it is appropriate to use extinction probabilities generated by population viability analyses, based on best estimates for model parameters, as criteria for listing species in Red Data Book categories as recently proposed by the World Conservation Union. Such extinction probabilities are influenced by how accurately model parameters are estimated and by how accurately the models depict actual population dynamics. I evaluate the effect of uncertainty in parameter estimation through simulations. Simulations based on Steller sea lions were used to evaluate bias and precision in estimates of probability of extinction and to consider the performance of two proposed classification schemes. Extinction time estimates were biased (because of violation of the assumption of stable age distribution) and underestimated the variability of probability of extinction for a given time (primarily because of uncertainty in parameter estimation). Bias and precision in extinction probabilities are important when these probabilities are used to compare the risk of extinction between species. Suggestions are given for population viability analysis techniques that incorporate parameter uncertainty. I conclude that testing classification schemes with simulations using quantitative performance objectives should precede adoption of quantitative listing criteria.