Population Prediction Intervals Research Articles

Critical parameters for predicting population fluctuations of some British passerines

1. One of the greatest challenges in ecology is to develop tools that can give reliable projections of future population fluctuations as well as to quantity uncertainties. The population prediction interval (PPI), i.e. the stochastic interval that includes a given population size with a certain probability, is affected by changes in expected population size, e.g. due to density regulation, fluctuations in population size because of demographic and environmental stochasticity, uncertainties and biases in parameter estimates, and observation error in estimates of population size. 2. The aim of this study was to examine how PPI can be used to obtain reliable projections of future population fluctuations. Our approach is to split long time series into two parts: the first part is used for parameter estimation and the second part is used for comparing population predictions with actual population sizes after a certain period of time. 3. Here we use the Common Birds Census - data from the UK for several species of passerines. Unbiased predictions will give a uniform distribution of the recorded population sizes across the PPI when transformed to a scale defined by the quantiles of the PPI. However, deviations from a uniform distribution reveal biases in the predictions. For instance, if there is a predominance of recorded population sizes in the upper quantiles of the PPI, this shows that our predictions underestimate future population sizes. 4. Unbiased predictions required models that included both partitioning of stochastic influences into demographic and environmental stochasticity as well as observation error. 5. Precision in the population predictions was improved when including observation error as well as density dependence. 6. We recommend that predictions of future fluctuations of small passerine populations are based on models that include density dependence as well as observation error and estimates of the demographic variance that is obtained from individual-based demographic data or based on species-specific life-history characteristics. 7. These results show that constructing PPI by stochastic simulations may be a useful tool for obtaining reliable population projections.

OP16.08: Accuracy of sonographic prediction of fetal weight discordance in twin pregnancies

twin pregnancies were analyzed using different modelling strategies including linear and non-linear two-level mixed effects models throughout the 12–38 weeks range. 90% population prediction intervals as well as individual fits were obtained through MonteCarlo simulations. Results: 409 twin pregnancies were included (329 dichorionic, 80 monochorionic) with a median number of measurements of 5 (IQR = 3–6) and 7 (IQR = 4–11) in dichorionics and monochorionics respectively. Among the different models, a 4parameter non-linear logistic function was found as the best fit. As an external validation of the model, the 90% prediction intervals for the whole population derived from mixed models closely matched the 5–95p interval derived from published cross-sectional studies. The modelling strategy allows for a parsimonious estimation of individual growth trajectories and may easily be used for conditional growth assessment. Conclusions: In twins, because a close biometric follow-up is needed, longitudinal modelling may be an important tool for growth monitoring, though widely used cross-sectional ranges have provided a good reference for serial measurements.

OC128: Evaluation of ventricular function in the recipient in twin to twin transfusion syndrome

second one was used for model validation and was a cross-sectional registry of twin birth weight in the same area and during the same period. A two-level mixed model was developed on the first database. 90% population prediction intervals were then computed by Monte-Carlo simulation and for gestational ages 28, 30, 32, 34, 36 weeks’ respectively. The upper and lower bounds of these prediction intervals were compared to the 5th and 95th percentiles obtained for the same gestational ages in the second set using a semi-parametric quantile smoother (LMS). Results: 434 consecutive twin pregnancies with a median of 4 measurements (IQR = 2–5) were included in the first set and 929 in the second. A cubic polynomial model was found as best fit. For each of the 5 selected gestational ages, the relative difference between the 90% population prediction bounds for EFW derived from longitudinal measurements and the corresponding 5th and 95th p. in the cross-sectional dataset was < 10%. Conclusions: Mixed effects models allow for parsimonious modelling of both population and individual growth paths in twin pregnancies as well as conditional norms. The precision of the presented model for prenatal EFW in twins was validated on an external cross-sectional birth-weight set.

OP16.07: Ultrasound estimation of fetal weight in twin pregnancies. How accurate are we?

twin pregnancies were analyzed using different modelling strategies including linear and non-linear two-level mixed effects models throughout the 12–38 weeks range. 90% population prediction intervals as well as individual fits were obtained through MonteCarlo simulations. Results: 409 twin pregnancies were included (329 dichorionic, 80 monochorionic) with a median number of measurements of 5 (IQR = 3–6) and 7 (IQR = 4–11) in dichorionics and monochorionics respectively. Among the different models, a 4parameter non-linear logistic function was found as the best fit. As an external validation of the model, the 90% prediction intervals for the whole population derived from mixed models closely matched the 5–95p interval derived from published cross-sectional studies. The modelling strategy allows for a parsimonious estimation of individual growth trajectories and may easily be used for conditional growth assessment. Conclusions: In twins, because a close biometric follow-up is needed, longitudinal modelling may be an important tool for growth monitoring, though widely used cross-sectional ranges have provided a good reference for serial measurements.

OP16.09: Fetal weight discordance in twin pregnancies: Lessons from a longitudinal study

twin pregnancies were analyzed using different modelling strategies including linear and non-linear two-level mixed effects models throughout the 12–38 weeks range. 90% population prediction intervals as well as individual fits were obtained through MonteCarlo simulations. Results: 409 twin pregnancies were included (329 dichorionic, 80 monochorionic) with a median number of measurements of 5 (IQR = 3–6) and 7 (IQR = 4–11) in dichorionics and monochorionics respectively. Among the different models, a 4parameter non-linear logistic function was found as the best fit. As an external validation of the model, the 90% prediction intervals for the whole population derived from mixed models closely matched the 5–95p interval derived from published cross-sectional studies. The modelling strategy allows for a parsimonious estimation of individual growth trajectories and may easily be used for conditional growth assessment. Conclusions: In twins, because a close biometric follow-up is needed, longitudinal modelling may be an important tool for growth monitoring, though widely used cross-sectional ranges have provided a good reference for serial measurements.

OC126: Management and neonatal outcome in monochorionic twin pregnancies presenting with marked estimated weight discordance at midtrimester

second one was used for model validation and was a cross-sectional registry of twin birth weight in the same area and during the same period. A two-level mixed model was developed on the first database. 90% population prediction intervals were then computed by Monte-Carlo simulation and for gestational ages 28, 30, 32, 34, 36 weeks’ respectively. The upper and lower bounds of these prediction intervals were compared to the 5th and 95th percentiles obtained for the same gestational ages in the second set using a semi-parametric quantile smoother (LMS). Results: 434 consecutive twin pregnancies with a median of 4 measurements (IQR = 2–5) were included in the first set and 929 in the second. A cubic polynomial model was found as best fit. For each of the 5 selected gestational ages, the relative difference between the 90% population prediction bounds for EFW derived from longitudinal measurements and the corresponding 5th and 95th p. in the cross-sectional dataset was < 10%. Conclusions: Mixed effects models allow for parsimonious modelling of both population and individual growth paths in twin pregnancies as well as conditional norms. The precision of the presented model for prenatal EFW in twins was validated on an external cross-sectional birth-weight set.

ANATOMY OF A BOTTLENECK: DIAGNOSING FACTORS LIMITING POPULATION GROWTH IN THE PUERTO RICAN PARROT

The relative importance of genetic, demographic, environmental, and catastrophic processes that maintain population bottlenecks has received little consideration. We evaluate the role of these factors in maintaining the Puerto Rican Parrot (Amazona vittata) in a prolonged bottleneck from 1973 through 2000 despite intensive conservation efforts. We first conduct a risk analysis, then examine evidence for the importance of specific processes maintaining the bottleneck using the multiple competing hypotheses approach, and finally integrate these results through a sensitivity analysis of a demographic model using life‐stage simulation analysis (LSA) to determine the relative importance of genetic, demographic, environmental, and catastrophic processes on population growth. Annual population growth has been slow and variable (1.0 ± 5.2 parrots per year, or an average λ = 1.05 ± 0.19) from 16 parrots (1973) to a high of 40–42 birds (1997–1998). A risk analysis based on population prediction intervals (PPI) indicates great risk and large uncertainty, with a range of 22–83 birds in the 90% PPI only five years into the future. Four primary factors (reduced hatching success due to inbreeding, failure of adults to nest, nest failure due to nongenetic causes, and reduced survival of adults and juveniles) were responsible for maintaining the bottleneck. Egg‐hatchability rates were low (70.6% per egg and 76.8% per pair), and hatchability increased after mate changes, suggesting inbreeding effects. Only an average of 34% of the population nested annually, which was well below the percentage of adults that should have reached an age of first breeding (41–56%). This chronic failure to nest appears to have been caused primarily by environmental and/or behavioral factors, and not by nest‐site scarcity or a skewed sex ratio. Nest failure rates from nongenetic causes (i.e., predation, parasitism, and wet cavities) were low (29%) due to active management (protecting nests and fostering captive young into wild nests), diminishing the importance of nest failure as a limiting factor. Annual survival has been periodically reduced by catastrophes (hurricanes), which have greatly constrained population growth, but survival rates were high under non‐catastrophic conditions. Although the importance of factors maintaining the Puerto Rican Parrot bottleneck varied throughout the 30‐year period of study, we determined their long‐term influence using LSA simulations to correlate variation in demographic rates with variation in population growth (λ). The bottleneck appears to have been maintained primarily by periodic catastrophes (hurricanes) that reduced adult survival, and secondarily by environmental and/or behavioral factors that resulted in a failure of many adults to nest. The influence of inbreeding through reduced hatching success played a much less significant role, even when additional effects of inbreeding on the production and mortality of young were incorporated into the LSA. Management actions needed to speed recovery include (1) continued nest guarding to minimize the effects of nest failure due to nongenetic causes; (2) creating a second population at another location on the island—a process that was recently initiated—to reduce the chance that hurricane strikes will cause extinction; and (3) determining the causes of the low percentage of breeders in the population and ameliorating them, which would have a large impact on population growth.

Predicting fluctuations of reintroduced ibex populations: the importance of density dependence, environmental stochasticity and uncertain population estimates

1. Development of population projections requires estimates of observation error, parameters characterizing expected dynamics such as the specific population growth rate and the form of density regulation, the influence of stochastic factors on population dynamics, and quantification of the uncertainty in the parameter estimates. 2. Here we construct a Population Prediction Interval (PPI) based on Bayesian state space modelling of future population growth of 28 reintroduced ibex populations in Switzerland that have been censused for up to 68 years. Our aim is to examine whether the interpopulation variation in the precision of the population projections is related to differences in the parameters characterizing the expected dynamics, in the effects of environmental stochasticity, in the magnitude of uncertainty in the population parameters, or in the observation error. 3. The error in the population censuses was small. The median coefficient of variation in the estimates across populations was 5.1%. 4. Significant density regulation was present in 53.6% of the populations, but was in general weak. 5. The width of the PPI calculated for a period of 5 years showed large variation among populations, and was explained by differences in the impact of environmental stochasticity on population dynamics. 6. In spite of the high accuracy in population estimates, the uncertainty in the parameter estimates was still large. This uncertainty affected the precision in the population predictions, but it decreased with increasing length of study period, mainly due to higher precision in the estimates of the environmental variance in the longer time-series. 7. These analyses reveal that predictions of future population fluctuations of weakly density-regulated populations such as the ibex often become uncertain. Credible population predictions require that this uncertainty is properly quantified.

Demographic Characteristics of Extinction in a Small, Insular Population of House Sparrows in Northern Norway

In conservation ecology there is an urgent need for indicators that can be used to predict the risk of extinction of populations. Identifying extinction-prone populations has been difficult because few data sets on the demographic characteristics of the final stage to extinction are available and because of problems in separating out stochastic effects from changes in the expected dynamics. We documented the demographic changes that occurred during the period prior to extinction of a small island population of House Sparrows (Passer domesticus) after the end of permanent human settlement. A mark-recapture analysis revealed that this decline to extinction was mainly due to increased mortality after closure of the last farm that resulted in a negative long-term-specific growth rate. No change occurred in either the structural composition (breeding sex ratio and age distribution) of the population or in female recruitment. No male, however, recruits were produced on the island after the farm closure. Based on a simple, stochastic, density-dependent model we constructed a population prediction interval (PPI) to estimate the time to extinction. The 95% PPI slightly overestimated the time to extinction with large uncertainty in predictions, especially due to the influence of demographic stochasticity and parameter drift. Our results strongly emphasize the importance of access to data on temporal variation that can be used to parameterize simple population models that allow estimation of critical parameters for credible prediction of time to extinction.

Predicting the growth of a small introduced muskox population using population prediction intervals

Summary A key issue in ecology is the prediction of future population fluctuations. Such population predictions are fundamental for population‐viability analysis and are essential for assessing the implications of various management actions. Development of reliable population predictions is however, difficult because it requires estimation and modelling of the separate effects of the deterministic components of the population dynamics as well as the stochastic influences on the population fluctuations. Here we model the stochastic dynamics of an introduced population of muskox Ovibos moschatus in the Dovrefjell mountains of central Norway, using a simple model without density regulation. Our aim is to examine quantitatively factors affecting the accuracy of the population projections by applying the concept of Population Prediction Interval (PPI). The long‐term growth rate was % = 0·0511, assuming no density dependence. The environmental variance was relatively large (= 0·0159). This gives a deterministic growth rate of r = 0·0591. However, accounting for losses due to various kinds of human activities resulted in a nearly doubling of s (% = 0·0980). Autumn temperature and late winter snow depth were each able to explain a significant proportion of the annual variation in population growth rates. The impact of environmental stochasticity made the PPI wide after only a few years. Uncertainties in the estimates of the population parameters were quite small and had a minor impact on the PPI. A sensitivity analysis showed that ignoring demographic stochasticity led to an overestimate of the environmental variance , but that the impact on the width of the PPI was small. This study shows that reliable projections of future population growth, even based on simple population models without density regulation, are dependent on assessment of the accuracy in the population predictions that must be based on estimating and modelling the stochastic influences on the population dynamics.

Density dependence and stochastic variation in a newly established population of a small songbird

Models describing fluctuations in population size should include both density dependence and stochastic effects. We examine the relative contribution of variation in parameters of the expected dynamics as well as demographic and environmental stochasticity to fluctuations in a population of a small passerine bird, the pied flycatcher, that was newly established in a Dutch study area. Using the theta‐logistic model of density regulation, we demonstrate that the estimated quasi‐stationary distribution including demographic stochasticity is close to the stationary distribution ignoring demographic stochasticity, indicating a long expected time to extinction. We also show that the variance in the estimated quasi‐stationary distribution is especially sensitive to variation in the density regulation function. Reliable population projections must therefore account for uncertainties in parameter estimates which we do by using the population prediction interval (PPI). After 2 years the width of the 90% PPI was already larger than the corresponding estimated range of variation in the quasi‐stationary distribution. More precise prediction of future population size than can be derived from the quasi‐stationary distribution could only be made for a time span less than about five years.

Pattern of variation in avian population growth rates.

A central question in population ecology is to understand why population growth rates differ over time. Here, we describe how the long-term growth of populations is not only influenced by parameters affecting the expected dynamics, for example form of density dependence and specific population growth rate, but is also affected by environmental and demographic stochasticity. Using long-term studies of fluctuations of bird populations, we show an interaction between the stochastic and the deterministic components of the population dynamics: high specific growth rates at small densities r(1) are typically positively correlated with the environmental variance sigma(e)(2). Furthermore, theta, a single parameter describing the form of the density regulation in the theta-logistic density-regulation model, is negatively correlated with r(1). These patterns are in turn correlated with interspecific differences in life-history characteristics. Higher specific growth rates, larger stochastic effects on the population dynamics and stronger density regulation at small densities are found in species with large clutch sizes or high adult mortality rates than in long-lived species. Unfortunately, large uncertainties in parameter estimates, as well as strong stochastic effects on the population dynamics, will often make even short-term population projections unreliable. We illustrate that the concept of population prediction interval can be useful in evaluating the consequences of these uncertainties in the population projections for the choice of management actions.

Stochastic population dynamics and time to extinction of a declining population of barn swallows

Summary Time to extinction was predicted for a declining population of barn swallow Hirundo rustica in Denmark, using a model that includes demographic as well as environmental stochasticity and that takes the uncertainties in the parameter estimates into account. We apply the concept of population prediction interval (PPI), which is a stochastic interval that includes the unknown variable that may be the extinction time or the population size at some future point of time, with a given probability (1 − α). The lower bound of the upper one‐sided prediction interval for the extinction time for α = 0·10 was 22 years. Ignoring uncertainties in the parameter estimates led to a 41% increase in this statistic. Although the estimate of the demographic variance was small compared to other passerines (σd2 = 0·180), a sensitivity analysis showed that it strongly influenced the predicted time to extinction compared to the model ignoring demographic stochasticity. A similar effect on the prediction of the time to extinction was found for the environmental variance σe2. In addition, choosing σe2 = 0 strongly reduced the width of the prediction interval. This demonstrates that reliable population projections require modelling of the environmental as well as the demographic stochasticity, and that the uncertainty in the estimates of the model parameters must be taken into account.

Estimating the time to extinction in an island population of song sparrows.

We estimated and modelled how uncertainties in stochastic population dynamics and biases in parameter estimates affect the accuracy of the projections of a small island population of song sparrows which was enumerated every spring for 24 years. The estimate of the density regulation in a theta-logistic model (theta = 1.09 suggests that the dynamics are nearly logistic, with specific growth rate r1 = 0.99 and carrying capacity K = 41.54. The song sparrow population was strongly influenced by demographic (ŝigma2(d) = 0.66) and environmental (ŝigma2(d) = 0.41) stochasticity. Bootstrap replicates of the different parameters revealed that the uncertainties in the estimates of the specific growth rate r1 and the density regulation theta were larger than the uncertainties in the environmental variance sigma2(e) and the carrying capacity K. We introduce the concept of the population prediction interval (PPI), which is a stochastic interval which includes the unknown population size with probability (1 - alpha). The width of the PPI increased rapidly with time because of uncertainties in the estimates of density regulation as well as demographic and environmental variance in the stochastic population dynamics. Accepting a 10% probability of extinction within 100 years, neglecting uncertainties in the parameters will lead to a 33% overestimation of the time it takes for the extinction barrier (population size X = 1) to be included into the PPI. This study shows that ignoring uncertainties in population dynamics produces a substantial underestimation of the extinction risk.