JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Ecological Society of America is collaborating with JSTOR to digitize, preserve and extend access to Ecology. Abstract.Most of the fundamental elements of ecology, ranging from individual behavior to species abundance, diversity, and population dynamics, exhibit spatial variation. Partial differential equation models provide a means of melding organism movement with population processes and have been used extensively to elucidate the effects of spatial variation on populations. While there has been an explosion of theoretical advances in partial differential equation models in the past two decades, this work has been generally neglected in mathematical ecology textbooks. Our goal in this paper is to make this literature accessible to experimental ecologists.Partial differential equations are used to model a variety of ecological phenomena; here we discuss dispersal, ecological invasions, critical patch size, dispersal-mediated coexistence, and diffusion-driven spatial patterning. These models emphasize that simple organism movement can produce striking large-scale patterns in homogeneous environments, and that in heterogeneous environments, movement of multiple species can change the outcome of competition or predation. In the classical applications of PDE models to population ecology, organisms are assumed to have Brownian random motion, the rate of which is invariant in time and space. This assumption leads to the diffusion model (Okubo 1980, Edelstein-Keshet 1986, Murray 1989): au (x, y, t) (02U a2U _ _ _ _= LITERATURE CITED Allee, W. C. 1931. Animal aggregations, a study on general sociology. University of Chicago Press, Chicago, Illinois, USA. Alt, W. 1985. Models for mutual attraction and aggregation of motile individuals. Pages 33-38 in V. Capasso, E. Grosso, and S. L. Paveri-Fontana, editors. Lecture Notes in Biomathematics Number 57. Ammerman, A. J., and L. L. Cavalli-Sforza. 1984. The Neolithic transition and the genetics of populations in Europe. . 1988. Analyzing field studies of insect dispersal using two-dimensional transport equations. Environmental Entomology 17:815-820. Bertsch, M., M. E. Gurtin, D. Hilhorst, and L. A. Peletier. 1984. On interacting populations that disperse to avoid crowding: the effect of a sedentary colony. Journal of Mathematical Biology 19:1-12. Bertsch, M., M. E. Gurtin, D. Hilhorst, and L. A. Peletier. 1985. On interacting populations that disperse to avoid crowding: preservation of segregation. Journal of Mathematical Biology 23:1-13. Bradford, E., and J. P. Philip. 1970a. Stability of steady distributions of asocial populations in one dimension. Journal of Theoretical Biology 29:13-26.

MARSS is a package for fitting multivariate autoregressive state-space models to time-series data. The MARSS package implements state-space models in a maximum likelihood framework. The core functionality of MARSS is based on likelihood maximization using the Kalman filter/smoother, combined with an EM algorithm. To make comparisons with other packages available, parameter estimation is also permitted via direct search routines available in 'optim'. The MARSS package allows data to contain missing values and allows a wide variety of model structures and constraints to be specified (such as fixed or shared parameters). In addition to model-fitting, the package provides bootstrap routines for simulating data and generating confidence intervals, and multiple options for calculating model selection criteria (such as AIC).The MARSS package (Holmes et al., 2012) is an R package for fitting linear multivariate autoregressive state-space (MARSS) models with Gaussian errors to time-series data. This class of model is extremely important in the study of linear stochastic dynamical systems, and these models are used in many different fields, including economics, engineering, genetics, physics and ecology. The model class has different names in different fields; some common names are dynamic linear models (DLMs) and vector autoregressive (VAR) state-space models. There are a number of existing R packages for fitting this class of models, including sspir (Dethlefsen et al., 2009) for univariate data and dlm (Petris, 2010), dse (Gilbert, 2009), KFAS (Helske, 2011) and FKF (Luethi et al., 2012) for multivariate data. Additional packages are available on other platforms, such as SsfPack (Durbin and Koopman, 2001), EViews (www.eviews.com) and Brodgar (www.brodgar.com). Except for Brodgar and sspir, these packages provide maximization of the likelihood surface (for maximum-likelihood parameter estimation) via quasi-Newton or Nelder-Mead type algorithms. The MARSS package was developed to provide an alternative maximization algorithm, based instead on an Expectation-Maximization (EM) algorithm and to provide a standardized modelspecification framework for fitting different model structures.The MARSS package was originally developed for researchers analyzing data in the natural and environmental sciences, because many of the problems often encountered in these fields are not commonly encountered in disciplines like engineering or finance. Two typical problems are high fractions of irregularly spaced missing observations and observation error variance that cannot be estimated or known a priori (Schnute, 1994). Packages developed for other fields did not always allow estimation of the parameters of interest to ecologists because these parameters are always fixed in the package authors' field or application. The MARSS package was developed to address these issues and its three main differences are summarized as follows.First, maximum-likelihood optimization in most packages for fitting state-space models relies on quasi-Newton or Nelde...

We developed a database of 10 wild vertebrate populations whose declines to extinction were monitored over at least 12 years. We quantitatively characterized the final declines of these well-monitored populations and tested key theoretical predictions about the process of extinction, obtaining two primary results. First, we found evidence of logarithmic scaling of time-to-extinction as a function of population size for each of the 10 populations. Second, two lines of evidence suggested that these extinction-bound populations collectively exhibited dynamics akin to those theoretically proposed to occur in extinction vortices. Specifically, retrospective analyses suggested that a population size of n individuals within a decade of extinction was somehow less valuable to persistence than the same population size was earlier. Likewise, both year-to-year rates of decline and year-to-year variability increased as the time-to-extinction decreased. Together, these results provide key empirical insights into extinction dynamics, an important topic that has received extensive theoretical attention.

Census data on endangered species are often sparse, error-ridden, and confined to only a segment of the population. Estimating trends and extinction risks using this type of data presents numerous difficulties. In particular, the estimate of the variation in year-to-year transitions in population size (the ''process error'' caused by stochasticity in survivorship and fecundities) is confounded by the addition of high sampling error variation. In addition, the year-to-year variability in the segment of the population that is sampled may be quite different from the population variability that one is trying to estimate. The combined effect of severe sampling error and age-or stage-specific counts leads to severe biases in estimates of population-level parameters. I present an estimation method that circumvents the problem of age-or stage-specific counts and is markedly robust to severe sampling error. This method allows the estimation of environmental variation and population trends for extinction-risk analyses using corrupted census counts-a common type of data for endangered species that has hitherto been relatively unusable for these analyses.

Diffusion models of animal movement are often criticized because they assume animals have infinite velocity and completely random motion. To investigate the impact of these assumptions, I compared a diffusion model with a telegraph model of dispersal The telegraph model assumes organisms have finite velocity and tend to maintain their direction. I compared the models in two settings: (i) as models for dispersal of nonreproducing organisms and (ii) as models for range expansion of organisms that simultaneously disperse and reproduce (so-called reaction-diffusion or reaction-telegraph models). Both models predict very similar dispersal patterns for nonreproducing organisms. In the case of reproducing organisms, however, they predict grossly different rates of range expansion for all but a small range of parameter values. The disparity is greatest for organisms with high population growth and low movement rates. To assess the magnitude of disparity for different organisms, I used published data to estimate model parameters for the cabbage butterfly (Pieris rapae), gypsy moth (Lymantria dispar), European starling (Sturnus vulgaris), collared turtledove (Streptopelia decaocto), Black Death (Yersinia pestis), and for rabies. All six cases fell within the narrow parameter range where the diffusion and telegraph models yield indistinguishable predictions regarding the rate of range expansion.

Twelve salmonid evolutionarily significant units (ESUs) throughout the Columbia River Basin are currently listed as threatened or endangered under the Endangered Species Act; these ESUs are affected differentially by a variety of human activities. We present a standardized quantitative status and risk assessment for 152 listed salmonid stocks in these ESUs and 24 nonlisted stocks. Using data from 1980-2000, which represents a time of stable conditions in the Columbia River hydropower system and a period of ocean conditions generally regarded as poor for Columbia Basin salmonids, we estimated the status of these stocks under two different assumptions: that hatchery-reared spawners were not reproducing during the period of the censuses, or that hatchery-reared spawners were reproducing and thus that reproduction from hatchery inputs was masking population trends. We repeated the analyses using a longer time period containing both ''good' ' and ''bad'' ocean conditions (1965-2000) as a first step toward determining whether recent apparent declines are a result of sampling a period of poor ocean conditions.All the listed ESUs except Columbia River chum showed declining trends with estimated long-term population growth rates ('s) ranging from 0.85 to 1.0, under the assumption that hatchery fish were not reproducing and not masking the true . If hatchery fish were reproducing, the estimated 's ranged from 0.62 to 0.89, indicating extremely low natural reproduction and survival. For most ESUs, there was no significant decline in population growth rates calculated for the 1980-2000 vs. 1965-2000 time periods, suggesting that the current population status for most ESUs is not solely a result of changes in ocean conditions, and that without other changes, risks will persist even during upturns in ocean conditions. However, estimated population growth rates for the Snake River spring-summer chinook salmon and steelhead ESUs were significantly lower during the longer time period. This difference may be due to a period of dam building on the Snake River during the 1960s and 1970s. For 33 stocks and seven ESUs, the probability of extinction could be estimated. The estimates were generally low for all ESUs with the exception of Upper Columbia River spring chinook and Upper Willamette River steelhead. The probability of 90% decline could be estimated for all stocks. The mean probability of 90% decline in 50 years was highest for Upper Columbia River spring chinook (95% mean probability across all stocks within the ESU) and Lower Columbia River steelhead (80% mean probability).We estimated the effects of two different management actions on long-term growth rates for the ESUs. Harvest reductions offer a means to mitigate risks for ESUs that bear substantial harvest pressure, but they are unlikely to increase population growth rates enough to produce stable or increasing trends for all ESUs. Similarly, anticipated improvements to passage survival through the Snake and mainstem Columbia hydropower systems may be important, but add...

Summary 1.Management decisions for threatened and endangered species require risks to be identified and prioritized, based on the degree to which they influence population dynamics. The potential for recovery of small populations at risk may be determined by multiple factors, including intrinsic population characteristics (inbreeding, sex ratios) and extrinsic variables (prey availability, disease, human disturbance). Using Bayesian statistical methods, the impact of each of these risk factors on demographic rates can be quantified and assigned probabilities to express uncertainty. 2.We assessed the impact of a wide range of factors on the fecundity of two threatened populations of killer whales Orcinus orca , specifically whether killer whale production is limited by availability of Chinook salmon Oncorhynchus tshawytscha . Additional variables included anthropogenic factors, climate variables, temporal effects, and population variables (population size, number of males, female age). 3. Our results indicate that killer whale fecundity is highly correlated with the abundance of Chinook salmon. For example, the probability of a female calving differed by 50% between years of low salmon abundance and high salmon abundance. Weak evidence exists for linking fecundity to other variables, such as sea surface temperature. 4. There was strong data support for reproductive senescence in female killer whales. This pattern of rapid maturity and gradual decline of fecundity with age commonly seen in terrestrial mammals has been documented in few marine mammal species. Maximum production for this species occurs between ages 20-22, and reproductive performance declines gradually to menopause over a period of 25 years. 5. Synthesis and applications . Our results provide strong evidence for reproductive senescence in killer whales, and more importantly, that killer whale fecundity is strongly tied to the abundance of Chinook salmon, a species that is susceptible to environmental variation and has high commercial value to fisheries. This strong predator-prey relationship highlights the importance of understanding which salmon populations overlap with killer whales seasonally and spatially, so that those salmon populations important as prey for killer whales can be identified and targeted for conservation efforts.

Short‐term forecasts based on time series of counts or survey data are widely used in population biology to provide advice concerning the management, harvest and conservation of natural populations. A common approach to produce these forecasts uses time‐series models, of different types, fit to time series of counts. Similar time‐series models are used in many other disciplines, however relative to the data available in these other disciplines, population data are often unusually short and noisy and models that perform well for data from other disciplines may not be appropriate for population data. In order to study the performance of time‐series forecasting models for natural animal population data, we assembled 2379 time series of vertebrate population indices from actual surveys. Our data were comprised of three vastly different types: highly variable (marine fish productivity), strongly cyclic (adult salmon counts), and small variance but long‐memory (bird and mammal counts). We tested the predictive performance of 49 different forecasting models grouped into three broad classes: autoregressive time‐series models, non‐linear regression‐type models and non‐parametric time‐series models. Low‐dimensional parametric autoregressive models gave the most accurate forecasts across a wide range of taxa; the most accurate model was one that simply treated the most recent observation as the forecast. More complex parametric and non‐parametric models performed worse, except when applied to highly cyclic species. Across taxa, certain life history characteristics were correlated with lower forecast error; specifically, we found that better forecasts were correlated with attributes of slow growing species: large maximum age and size for fishes and high trophic level for birds. Synthesis Evaluating the data support for multiple plausible models has been an integral focus of many ecological analyses. However, the most commonly used tools to quantify support have weighted models’ hindcasting and forecasting abilities. For many applications, predicting the past may be of little interest. Concentrating only on the future predictive performance of time series models, we performed a forecasting competition among many different kinds of statistical models, applying each to many different kinds of vertebrate time series of population abundance. Low‐dimensional (simple) models performed well overall, but more complex models did slightly better when applied to time series of cyclic species (e.g. salmon).