Predicting [corrected] extinction risk in spite of predator-prey oscillations.

Most population viability analyses (PVA) assume that the effects of species interactions are subsumed by population-level parameters. We examine how robust five commonly used PVA models are to violations of this assumption. We develop a stochastic, stage-structured predator-prey model and simulate prey population vital rates and abundance. We then use simulated data to parameterize and estimate risk for three demographic models (static projection matrix, stochastic projection matrix, stochastic vital rate matrix) and two time series models (diffusion approximation [DA], corrupted diffusion approximation [CDA]). Model bias is measured as the absolute deviation between estimated and observed quasi-extinction risk. Our results highlight three generalities about the application of single-species models to multi-species conservation problems. First, our collective model results suggest that most single-species PVA models overestimate extinction risk when species interactions cause periodic variation in abundance. Second, the DA model produces the most (conservatively) biased risk forecasts. Finally, the CDA model is the most robust PVA to population cycles caused by species interactions. CDA models produce virtually unbiased and relatively precise risk estimates even when populations cycle strongly. High performance of simple time series models like the CDA owes to their ability to effectively partition stochastic and deterministic sources of variation in population abundance.     

Improved estimates of certainty in stable-isotope-based methods for tracking migratory animals.

The use of stable-hydrogen isotopes (deltaD) has become a common tool for estimating geographic patterns of movement in migratory animals. This method relies on broad and relatively predictable geographic patterning in deltaD values of precipitation, but these patterns are not estimated without error. In addition, deltaD measurements are relatively imprecise, particularly for organic tissue. Most models for estimating geographic locations have ignored these sources of error. Common modeling approaches include regression, range-matching, and likelihood-based assignment tests (including discriminant analysis). Here, we show the benefits of a simple stochastic extension to likelihood-based assignment tests that incorporates two estimable sources of error and describe the resulting influence on the certainty of assigning breeding origins for wintering American Redstarts (Setophaga ruticilla), a small Nearctic-Neotropical migratory bird. Through simulation, we incorporated both spatial interpolation error associated with models of deltaD in precipitation and analytical error associated with the measurement of deltaD in tissue samples. In general, assignments that did not include these sources of error fell within the ranges of the stochastic results, but the difference in proportion of birds assigned to any one breeding region varied by as much as 54%. To explore how the distribution of assignments generated from error models influenced the application of these results, we developed a simple model of winter habitat loss. We removed the proportion of Redstarts wintering at a particular site from the global population and then used the isotope-based assignments to predict the resulting population declines for each breeding region. This gave distributions of change in population sizes, some of which included no change or even a population increase. The sources of error we modeled may challenge the degree of certainty in the use of stable-isotope-based data on connectivity to predict population dynamics of migratory animals. We suggest that stronger inference will result from incorporating these sources of error into future studies that use deltaD or other stable isotopes to infer the geographic origin of individuals.     

A comparison of models for predicting population persistence

We consider a range of models that may be used to predict the future persistence of populations, particularly those based on discrete-state Markov processes. While the mathematical theory of such processes is very well-developed, they may be difficult to work with when attempting to estimate parameters or expected times to extinction. Hence, we focus on diffusion and other approximations to these models, presenting new and recent developments in parameter estimation for density dependent processes, and the calculation of extinction times for processes subject to catastrophes. We illustrate these and other methods using data from simulated and real time series. We give particular attention to a procedure, due to Ross et al. [Ross, J.V., Taimre, T., Pollett, P.K. On parameter estimation in population models, Theor. Popul. Biol., in press], for estimating the parameters of the stochastic SIS logistic model, and demonstrate ways in which these parameters may be used to estimate expected extinction times. Although the stochastic SIS logistic model is strictly density dependent and allows only for birth and death events, it nonetheless may be used to predict extinction times with some accuracy even for populations that are only weakly density dependent, or that are subject to catastrophes.     

Invasion with stage-structured coupled map lattices: Application to the spread of scentless chamomile

Two fundamental aspects of invasion dynamics are population growth and population spread. These quantities have been subject of study in biological invasions and can be used to study management and control of organisms. In this paper we derive formulae to calculate wave speed and rates of spread for coupled map lattices. Coupled map lattice models are dynamical models where space and time are discrete. We also show how wave speed and rate of spread can be calculated for structured population coupled map lattices in deterministic, stochastic environments and heterogeneous landscapes. Coupled map lattices are simple mathematical models that can be easily linked to landscape data to study invading organisms control strategies.     

Environment-specific elasticity and sensitivity analysis of the stochastic growth rate

For matrix population models, analyses of how sensitive the population growth rate is to changes in vital rates (i.e. perturbations) are important for studies of life history evolution as well as for management and conservation of threatened species. There are two types of sensitivity analyses corresponding to absolute (sensitivity) or relative (elasticity) changes in the vital rates and both types can be applied to both deterministic and stochastic matrix population models. To date, most empirical studies of elasticity and sensitivity of the stochastic growth rate have examined the response to perturbations in the vital rates in a complete set of possible environments. However, it is often of interest to examine the response to perturbations occurring in only a subset of the possible environments. This has been done for periodic time-varying models elsewhere, but here we describe a recently published method for calculating the environment-specific sensitivity and elasticity of the stochastic growth rate and apply this method to data. These environment-specific perturbation analyses provide a logical way of dividing the sensitivity and elasticity among the environments. They give important insight into the selection regime in different environments and can provide valuable information for making management decisions and management evaluations in stochastic environments.     

Climate change threatens polar bear populations: a stochastic demographic analysis

The polar bear (Ursus maritimus) depends on sea ice for feeding, breeding, and movement. Significant reductions in Arctic sea ice are forecast to continue because of climate warming. We evaluated the impacts of climate change on polar bears in the southern Beaufort Sea by means of a demographic analysis, combining deterministic, stochastic, environment-dependent matrix population models with forecasts of future sea ice conditions from IPCC general circulation models (GCMs). The matrix population models classified individuals by age and breeding status; mothers and dependent cubs were treated as units. Parameter estimates were obtained from a capture-recapture study conducted from 2001 to 2006. Candidate statistical models allowed vital rates to vary with time and as functions of a sea ice covariate. Model averaging was used to produce the vital rate estimates, and a parametric bootstrap procedure was used to quantify model selection and parameter estimation uncertainty. Deterministic models projected population growth in years with more extensive ice coverage (2001-2003) and population decline in years with less ice coverage (2004-2005). LTRE (life table response experiment) analysis showed that the reduction in lambda in years with low sea ice was due primarily to reduced adult female survival, and secondarily to reduced breeding. A stochastic model with two environmental states, good and poor sea ice conditions, projected a declining stochastic growth rate, log lambdas, as the frequency of poor ice years increased. The observed frequency of poor ice years since 1979 would imply log lambdas approximately - 0.01, which agrees with available (albeit crude) observations of population size. The stochastic model was linked to a set of 10 GCMs compiled by the IPCC; the models were chosen for their ability to reproduce historical observations of sea ice and were forced with "business as usual" (A1B) greenhouse gas emissions. The resulting stochastic population projections showed drastic declines in the polar bear population by the end of the 21st century. These projections were instrumental in the decision to list the polar bear as a threatened species under the U.S. Endangered Species Act.     

Testing a stochastic version of the Beddington–DeAngelis functional response in foraging shore crabs

Current behaviour-based interference models assume that the predator population is infinitely large and that interference is weak. While the realism of the first assumption is questionable, the second assumption conflicts with the purpose of interference models. Here, we tested a recently developed stochastic version of the Beddington–DeAngelis functional response—which applies to a finite predator population without assuming weak interference—against experimental data of shore crabs (Carcinus maenas) foraging on mussels (Mytilus edulis). We present an approximate maximum likelihood procedure for parameter estimation when only one focal individual is observed, and introduce ‘correction factors’ that capture the average behaviour of the competing but unobserved individuals. We used the method to estimate shore crab handling time, interaction time, and searching rates for prey and competitor. Especially the searching rates were sensitive to variation in prey and competitor density. Incorporating constant parameter values in the model and comparing observed and predicted feeding rates revealed that the predictive power of the model is high. Our stochastic version of the Beddington–DeAngelis model better reflects reality than current interference models and is also amenable for modelling effects of interference on predator distributions.     

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.     

Optimal annual routines: new tools for conservation biology?

Many applied problems in ecology and conservation require prediction, and population models are important tools for that purpose. Formerly, the majority of predictive population models were based on matrix models. As the limitations of classical matrix models have become clearer, the use of individual-based models has increased. These models use behavioral rules imposed at the level of the individual to establish the emergent consequences of those rules at the population level. Individual behaviors in such models use an array of different rule types, from empirically derived probabilities to long-term fitness considerations. There has been surprisingly little discussion of the strengths and weaknesses of these different rule types. Here, we consider different strategies for modeling individual behaviors, together with some problems associated with individual-based models. We propose a novel approach based on modeling optimal annual routines. Annual routines allow individual behaviors to be predicted over a whole annual cycle within the context of long-term fitness considerations. Temporal trade-offs between different behaviors are automatically included in annual routine models, overcoming some of the primary limitations of other individual-based models. Furthermore, as well as population predictions, individual behaviors and indices of condition are emergent features of annual routine models. We show that these can be more sensitive to environmental change than population size, offering alternative, repeatable metrics for monitoring population status. Annual routine models provide no panacea for the problems of data limitations in predictive population modeling. However, as a result of their ability to deal with life-history trade-offs, as well as their potential for relatively rapid and accurate validation and parameterization, we suggest that annual routine models have strong potential for predictive population modeling in applied conservation settings.     

Identifying key species in ecosystems with stochastic sensitivity analysis

The development of approaches to estimate the vulnerability of biological communities and ecosystems to extirpations and reductions of species is a central challenge of conservation biology. One key aim of this challenge is to develop quantitative approaches to estimate and rank interaction strengths and keystoneness of species and functional groups, i.e. to quantify the relative importance of species. Network analysis can be a powerful tool for this because certain structural aspects of ecological networks are good indicators of the mechanisms that maintain co-evolved, biotic interactions. A static view of ecological networks would lead us to focus research on highly-central species in food webs (topological key players in ecosystems). There are a variety of centrality indices, developed for several types of ecological networks (e.g. for weighted and un-weighted webs). However, truly understanding extinction and its community-wide effects requires the use of dynamic models. Deterministic dynamic models are feasible when population sizes are sufficiently large to minimize noise in the overall system. In models with small population sizes, stochasticity can be modelled explicitly. We present a stochastic simulation-based ecosystem model for identification of “dynamic key species” in situations where stochastic models are appropriate. To demonstrate this approach, we simulated ecosystem dynamics and performed sensitivity analysis using data from the Prince William Sound, Alaska ecosystem model. We then compare these results to those of purely topological analyses and deterministic dynamic (Ecosim) studies. We present the relationships between various topological and dynamic indices and discuss their biological relevance. The trophic group with the largest effect on others is nearshore demersals, the species mostly sensitive to others is halibut, and the group of both considerable effect on and sensitivity to others is juvenile herring. The most important trophic groups in our dynamical simulations appear to have intermediate trophic levels.     

Unveiling human-assisted dispersal mechanisms in invasive alien insects: Integration of spatial stochastic simulation and phenology models

Capturing the spread of biological invasions in heterogeneous landscapes is a complex modelling task where information on both dispersal and population dynamics needs to be integrated. Spatial stochastic simulation and phenology models have rarely been combined to assist in the study of human-assisted long-distance dispersal events.Here we develop a process-based spatially explicit landscape-extent simulation model that considers the spread and detection of invasive insects. Natural and human-assisted dispersal mechanisms are modelled with an individual-based approach using negative exponential and negative power law dispersal kernels and gravity models. The model incorporates a phenology sub-model that uses daily temperature grids for the prediction and timing of the population dynamics in each habitat patch. The model was applied to the study of the invasion by the important maize pest western corn rootworm (WCR) Diabrotica virgifera ssp. virgifera in Europe. We parameterized and validated the model using maximum likelihood and simulation methods from the historical invasion of WCR in Austria.WCR was found to follow stratified dispersal where international transport networks in the Danube basin played a key role in the occurrence of long-distance dispersal events. Detection measures were found to be effective and altitude had a significant effect on limiting the spread of WCR. Spatial stochastic simulation combined with phenology models, maximum likelihood methods and predicted versus observed regression showed a high degree of flexibility that captured the salient features of WCR spread in Austria. This modelling approach is useful because it allows to fully exploit and the often limited and heterogeneous information available regarding the population dynamics and dispersal of alien invasive insects.     

A Comparison of Linear Demographic Models and Fraction of Lifetime Egg Production for Assessing Sustainability in Sharks

Conventional methods for management of data‐rich fisheries maintain sustainable populations by assuring that lifetime reproduction is adequate for individuals to replace themselves and accounting for density‐dependent recruitment. Fishing is not allowed to reduce relative lifetime reproduction, the fraction of current egg production relative to unfished egg production (FLEP), below a sustainable level. Because most shark fisheries are data poor, other representations of persistence status have been used, including linear demographic models, which incorporate life‐history characteristics in age‐structured models with no density dependence. We tested how well measures of sustainability from 3 linear demographic methods (rebound potential, stochastic growth rate, and potential population increase) reflect actual population persistence by comparing values of these measures with FLEP for 26 shark species. We also calculated the value of fishing mortality (F) that would allow all 26 species to maintain an accepted precautionary threshold for sharks of FLEP = 60%, expressing F as a fraction of natural mortality (M). Values of stochastic growth rate and potential population growth did not covary in rank order with FLEP (p = 0.057 and p = 0.077, respectively) and neither was significantly correlated with FLEP. Ordinal ranking of rebound potential positively covaried with FLEP (p = 0.00013), but the relative rankings of some species were substantially out of order. Adopting a sustainable limit of F = 0.16M would maintain all 26 species above the precautionary minimum value of FLEP (60%). We concluded that shark‐fishery and conservation policies should rely on calculation of replacement (i.e., FLEP), and that sharks should be fished at a precautionary level that would protect all stocks (i.e., F< 0.16M). Comparación entre Modelos Demográficos Lineales y la Fracción de Producción de Huevos a lo Largo de la Vida para Estudiar la Sustentabilidad en Tiburones Resumen     

Estimating the functional form for the density dependence from life history data

Two contrasting approaches to the analysis of population dynamics are currently popular: demographic approaches where the associations between demographic rates and statistics summarizing the population dynamics are identified; and time series approaches where the associations between population dynamics, population density, and environmental covariates are investigated. In this paper, we develop an approach to combine these methods and apply it to detailed data from Soay sheep (Ovis aries). We examine how density dependence and climate contribute to fluctuations in population size via age- and sex-specific demographic rates, and how fluctuations in demographic structure influence population dynamics. Density dependence contributes most, followed by climatic variation, age structure fluctuations and interactions between density and climate. We then simplify the density-dependent, stochastic, age-structured demographic model and derive a new phenomenological time series which captures the dynamics better than previously selected functions. The simple method we develop has potential to provide substantial insight into the relative contributions of population and individual-level processes to the dynamics of populations in stochastic environments.     

Fitting complex population models by combining particle filters with Markov chain Monte Carlo

We show how a recent framework combining Markov chain Monte Carlo (MCMC) with particle filters (PFMCMC) may be used to estimate population state-space models. With the purpose of utilizing the strengths of each method, PFMCMC explores hidden states by particle filters, while process and observation parameters are estimated using an MCMC algorithm. PFMCMC is exemplified by analyzing time series data on a red kangaroo (Macropus rufus) population in New South Wales, Australia, using MCMC over model parameters based on an adaptive Metropolis-Hastings algorithm. We fit three population models to these data; a density-dependent logistic diffusion model with environmental variance, an unregulated stochastic exponential growth model, and a random-walk model. Bayes factors and posterior model probabilities show that there is little support for density dependence and that the random-walk model is the most parsimonious model. The particle filter Metropolis-Hastings algorithm is a brute-force method that may be used to fit a range of complex population models. Implementation is straightforward and less involved than standard MCMC for many models, and marginal densities for model selection can be obtained with little additional effort. The cost is mainly computational, resulting in long running times that may be improved by parallelizing the algorithm.     

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.     

How sensitive are elasticities of long-run stochastic growth to how environmental variability is modelled?

A variable environment leaves a signature in a population's dynamics. Deriving statistical and mathematical models of how environmental variability affects population projections has - in the wake of reports of substantial climatic fluctuations - received much recent attention. If the model changes, then so too does the population projection. This is because a different model of environmental variability changes estimates of long-run stochastic growth, which is a function of demographic rates and their temporal sequence. Decomposing elasticities of long-run stochastic growth into constituent parts can assess the relative influence of different components. Here, we investigate the consequences of changing the environmental state definition, and therefore altering the shape of demographic rate distributions and their temporal sequence, by using age-structured matrix models to project vertebrate populations into the future under a range of environmental scenarios. The identity of the most influential demographic rate was consistent among all approaches that perturbed only the mean, but was not when only the variance was perturbed. Furthermore, the influence of each demographic rate fluctuated among projections by up to factors of six and two for changes to the variance and mean, respectively. These changes in influence depend in part upon how environmental variability - in particular, the color of environmental noise - is incorporated. In the light of predictions of increasing climatic variability in the future, these results suggest caution when drawing quantitative conclusions from stochastic population projections.     

Stochastic population dynamics in populations of western terrestrial garter snakes with divergent life histories

Comparative evaluations of population dynamics in species with temporal and spatial variation in life-history traits are rare because they require long-term demographic time series from multiple populations. We present such an analysis using demographic data collected during the interval 1978-1996 for six populations of western terrestrial garter snakes (Thamnophis elegans) from two evolutionarily divergent ecotypes. Three replicate populations from a slow-living ecotype, found in mountain meadows of northeastern California, were characterized by individuals that develop slowly, mature late, reproduce infrequently with small reproductive effort, and live longer than individuals of three populations of a fast-living ecotype found at lakeshore locales. We constructed matrix population models for each of the populations based on 8-13 years of data per population and analyzed both deterministic dynamics based on mean annual vital rates and stochastic dynamics incorporating annual variation in vital rates. (1) Contributions of highly variable vital rates to fitness (lambda(s)) were buffered against the negative effects of stochastic variation, and this relationship was consistent with differences between the meadow (M-slow) and lakeshore (L-fast) ecotypes. (2) Annual variation in the proportion of gravid females had the greatest negative effect among all vital rates on lambda(s). The magnitude of variation in the proportion of gravid females and its effect on lambda(s) was greater in M-slow than L-fast populations. (3) Variation in the proportion of gravid females, in turn, depended on annual variation in prey availability, and its effect on lambda(s) was 4 23 times greater in M-slow than L-fast populations. In addition to differences in stochastic dynamics between ecotypes, we also found higher mean mortality rates across all age classes in the L-fast populations. Our results suggest that both deterministic and stochastic selective forces have affected the evolution of divergent life-history traits in the two ecotypes, which, in turn, affect population dynamics. M-slow populations have evolved life-history traits that buffer fitness against direct effects of variation in reproduction and that spread lifetime reproduction across a greater number of reproductive bouts. These results highlight the importance of long-term demographic and environmental monitoring and of incorporating temporal dynamics into empirical studies of life-history evolution.     

Minimise long-term loss or maximise short-term gain?: Optimal translocation strategies for threatened species

Translocation is a useful management option for conservation of threatened animal species. It can be used to increase the range of a species, augment the numbers in a critical population, or establish new populations and hence spread the risk of extinction through local catastrophes. As it is an important and expensive conservation tool, translocation management decisions must be carefully considered, with the objective of the translocation project in mind. By analysing the translocation problem within a decision-theory framework, we find optimal management decisions that are rational and transparent. We illustrate our approach using a case study of the bridled nailtail wallaby (Onychogalea fraenata). Our particular translocation question is: if we have a set number of wallabies to translocate in each time period and two translocation sites, how many wallabies should we put at each site given the state of each population to maximise the benefit to the species? We model the translocated populations with first-order Markov chain stochastic population models, and use stochastic dynamic programming to determine the optimal management decisions. We look at two sites with different growth rates – one increasing and one decreasing – and compare the optimal strategies for two different objective functions. The first is a long-term persistence objective function, which maximises the persistence of translocated populations a large number of time steps after the end of the translocation program. The second maximises total population size at the end of the translocation program. Although these objective functions are similar, they generate surprisingly different optimal translocation strategies. When maximising the long-term persistence of the translocated populations, translocation decisions are not important as long as an increasing population is established. This indicates that site quality – rather than the number and timing of translocations – primarily determines the long-term persistence of populations. When maximising total population size, the optimal strategy is to add to the increasing population unless it is above a size where it is likely to reach its carrying capacity over the planning timeframe. As translocation decisions are important in fulfilling the objective, this objective function is more useful in creating practical advice for translocation managers. The discrepancy between the optimal strategies given by the two objectives demonstrates the importance of careful consideration when specifying the goals of a project. This observation applies not only to translocation programs, but any project where clear decision-making is needed.     

Population Viability Estimates: Theory and Practice for a Wild Gorilla Population

The logic of demographic modeling, the apparent simplicity of its quantifiably substantiated answers, and the ready availability of software correlate with increasing use of demographic modeling as the means of applying biology to the conservation of potentially endangered populations. I investigated that use by considering a small population (about 300 individuals) of a large, forest-dwelling mammal of the tropics, the Virunga gorilla ( Gorilla gorilla ) of Zaire, Uganda, and Rwanda. Because censuses of forest populations are so inaccurate and data on variance of some parameters takes so long to collect, models might not be broadly applicable. Therefore, simple demographic indices of potential extinction should replace sophisticated models. The current best index could be problematic, however, because it is based on detecting adult mortality, perhaps the most difficult demographic parameter to measure. Models of the Virunga gorilla population that incorporate aspects of demographic heterogeneity valuably indicate genetic and demographic persistence for several hundred years. Deterministic change in habitat is a greater threat than stochastic demographic variation, and yet our ecological ignorance is such that we could not begin to model the consequences of removal of even the main food plant. We must add to our ability to model outcomes of demographic perturbation a far greater understanding of the processes by which the perturbations occur. Demography allows us to model demographic response to demographic change, but we usually need ecology to tell us how the threat produced the demographic change in the first place. In a time of change, accurate prediction requires ecological understanding of process as well as demographic understanding of outcome.