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We explored the use of time series data to study the recent demographics of Bald Eagles in the northeast United States. We merged a stage-structured population matrix model (PMM) into a combinatorial optimization algorithm (COA) to estimate stage-wise abundances, survival rates, and other dynamics of the eagle population from 1990 to 2018. The PMM includes biannual periods (breeding and nonbreeding), density dependence, and stage-wise migration and dispersal. We demonstrated theoretically that the information contained in 17 yr of time series data of the reproductive stage would have contained all the information for the entire population. We then tested the performance of the COA in estimating transient and asymptotic population dynamics using simulated data. Finally, we assessed the COA under differing modeling conditions (without stage-wise dispersal and in the absence of biological assumptions). We found that time series data of the reproductive class, the skeleton of the PMM, and COA are sole ingredients to reliably estimate stage-wise abundances and survival, but additional biological assumptions are helpful in honing precision of the estimates. The algorithm is a tool to assess population dynamics when demographic data are limited to counts of reproductive adults. [ABSTRACT FROM AUTHOR]

Solar flare emission at X-ray and extreme ultraviolet (EUV) energies can cause substantial enhancements in the electron density in the Earth's lower ionosphere. It has now become clear that flares exhibit quasi-periodic pulsations with timescales of minutes at X-ray energies, but to date, it has not been known if the ionosphere is sensitive to this variability. Here using a combination of very low frequency (24 kHz) measurement together with space-based X-ray and EUV observations, we report pulsations of the ionospheric D region, which are synchronized with a set of pulsating flare loops. Modeling of the ionosphere show that the D region electron density varies by up to an order of magnitude over the timescale of the pulsations (∼ 20 min). Our results reveal that the Earth's ionosphere is more sensitive to small-scale changes in solar soft X-ray flux than previously thought and implies that planetary ionospheres are closely coupled to small-scale changes in solar/stellar activity. [ABSTRACT FROM AUTHOR]

The Gompertz state-space (GSS) model is a stochastic model for analyzing time-series observations of population abundances. The GSS model combines density dependence, environmental process noise, and observation error toward estimating quantities of interest in biological monitoring and population viability analysis. However, existing methods for estimating the model parameters apply only to population data with equal time intervals between observations. In the present paper, we extend the GSS model to data with unequal time intervals, by embedding it within a state-space version of the Ornstein- Uhlenbeck process, a continuous-time model of an equilibrating stochastic system. Maximum likelihood and restricted maximum likelihood calculations for the Ornstein-Uhlenbeck statespace model involve only numerical maximization of an explicit multivariate normal likelihood, and so the extension allows for easy bootstrapping, yielding confidence intervals for model parameters, statistical hypothesis testing of density dependence, and selection among sub-models using information criteria. Ecologists and managers previously drawn to models lacking density dependence or observation error because such models accommodated unequal time intervals (for example, due to missing data) now have an alternative analysis framework incorporating density dependence, process noise, and observation error. [ABSTRACT FROM AUTHOR]

Anthropogenic infrastructure is a mortality source for many vertebrate species. Mortality is often measured using periodic counts of carcasses or remains at infrastructure segments, and bias from carcass removal is estimated via field experiments with wildlife carcasses. We describe a model for combining removal experiment and carcass count data to estimate underlying process parameters using joint likelihood. In the model, the instantaneous number of carcasses present is a stochastic birth-death process with Poisson arrivals (carcass addition) and proportional deaths (removal of carcasses). The approach accommodates modeling heterogeneity in the addition and removal processes using generalized regression. Results of fitting the model to a Greater Sage-Grouse (Centrocercus urophasianus) fence collision data set show that order of magnitude differences in expected carcass counts can be a function of spatial differences in removal and suggest caution for interpretation of many published studies. While the model assumption of negligible detection error may be tenable for some systems, the modeling framework provides a starting point for future state-space versions incorporating detection error. [ABSTRACT FROM AUTHOR]

Twenty-first-century ecology requires statistical fluency. Ecological observations now include diverse types of data collected across spatial scales ranging from the microscopic to the global, and at temporal scales ranging from nanoseconds to millennia. Ecological experiments now use designs that account for multiple linear and non-linear interactions, and ecological theories incorporate both predictable and random processes. Ecological debates often revolve around issues of fundamental statistical theory and require the development of novel statistical methods. In short, if we want to test hypotheses, model data, and forecast future environmental conditions in the 21st century, we must move beyond basic statistical literacy and attain statistical fluency: the ability to apply statistical principles and adapt statistical methods to non-standard questions. Our prescription for attaining statistical fluency includes two semesters of standard calculus, calculus-based statistics courses, and, most importantly, a commitment to using calculus and post-calculus statistics in ecology and environmental science courses. [ABSTRACT FROM AUTHOR]

Observation or sampling error in population monitoring can cause serious degradation of the inferences, such as estimates of trend or risk, that ecologists and managers frequently seek to make with time-series observations of population abundances. We show that replicating the sampling process can considerably improve the information obtained from population monitoring. At each sampling time the sampling method would be repeated, either simultaneously or within a short time. In this study we examine the potential value of replicated sampling to population monitoring using a density-dependent population model. We modify an existing population time-series model, the Gompertz state-space model, to incorporate replicated sampling, and we develop maximum-likelihood and restricted maximum-likelihood estimates of model parameters. Depending on sampling protocols, replication may or may not entail substantial extra cost. Some sampling programs already have replicated samples, but the samples are aggregated or pooled into one estimate of population abundance; such practice of aggregating samples, according to our model, loses considerable information about model parameters. The gains from replicated sampling are realized in substantially improved statistical inferences about model parameters, especially inferences for sorting out the contributions of process noise and observation error to observed population variability. [ABSTRACT FROM AUTHOR]

Estimation of a population trend from a time series of abundance data is an important task in ecology, yet such estimation remains logistically and conceptually challenging in practice. First, the extent to which unequal intervals in the time series, due to missing observations or irregular sampling, compromise trend estimation is not well-known. Furthermore, the predominant trend estimation method (loglinear regression of abundance data against time) ignores the possibility of process noise, while an alternative method (the ‘diffusion approximation’) ignores observation error in the abundance data. State-space models that account for both process noise and observation error exist but have been little used. We study an adaptation of the exponential growth state-space (EGSS) model for use with missing data in the time series, and we compare its trend estimation to the status quo methods. The EGSS model provides superior estimates of trend across wide ranges of time series length and sources of variation. The performance of the EGSS model even with half of the counts in the time series missing implies that trend estimates may be improved by diverting effort away from annual monitoring and towards increasing time series length or improving precision of the abundance estimates for years that data are collected. [ABSTRACT FROM AUTHOR]

We conducted laboratory trials to determine the transit time and survival of New Zealand mudsnails Potamopyrgus antipodarum in the gastrointestinal tract of rainbow trout Oncorhynchus mykiss. To assess the rate of snail passage, we force-fed groups of fish a known quantity of snails and then held them in tanks. At selected intervals we removed individual fish from the test tanks and recorded the number of snails, their condition (live or dead), and their location in the gastrointestinal tract (stomach, anterior intesline, and posterior intestine). Feces were removed from tanks and examined for live snails. We repeated evaluations of passage rate and snail survival to determine the effects of varying the number of snails ingested, fish size, snail size, and feeding a commercial diet to fish after snail ingestion. We plotted and modeled gut evacuation using a stochastic model for ordinal data to consider each test variable. Snail passage rates were faster in fish that were fed smaller snails. Surprisingly, fish fed snails and then administered rations of commercial fish feed retained the snails longer in their stomach than did fish that were not administered fish feeds after being fed snails. Increased retention rime of snails in the stomach decreased the probability of snail survival when voided in fecal material. Snails that passed through the gastrointestinal tract within 12-24 h of ingestion were often recovered live in fecal samples. However, no live snails were recovered from the posterior intestine or fecal material collected 24 h after ingestion. Using our results we propose potential management options that could reduce the risks of introducing live snails into new locations when stocking fish from infested hatcheries. [ABSTRACT FROM AUTHOR]

Allee effects are an important dynamic phenomenon believed to be manifested in several population processes, notably extinction and invasion. Though widely cited in these contexts, the evidence for their strength and prevalence has not been critically evaluated. We review results from 91 studies on Allee effects in natural animal populations. We focus on empirical signatures that are used or might be used to detect Allee effects, the types of data in which Allee effects are evident, the empirical support for the occurrence of critical densities in natural populations, and differences among taxa both in the presence of Allee effects and primary causal mechanisms. We find that conclusive examples are known from Mollusca, Arthropoda, and Chordata, including three classes of vertebrates, and are most commonly documented to result from mate limitation in invertebrates and from predator–prey interactions in vertebrates. More than half of studies failed to distinguish component and demographic Allee effects in data, although the distinction is crucial to most of the population-level dynamic implications associated with Allee effects (e.g., the existence of an unstable critical density associated with strong Allee effects). Thus, although we find conclusive evidence for Allee effects due to a variety of mechanisms in natural populations of 59 animal species, we also find that existing data addressing the strength and commonness of Allee effects across species and populations is limited; evidence for a critical density for most populations is lacking. We suggest that current studies, mainly observational in nature, should be supplemented by population-scale experiments and approaches connecting component and demographic effects. [ABSTRACT FROM AUTHOR]

The article offers information on the relevance of integrating Bayesian methods and principles to various hierarchical approaches in gathering data for ecological research purposes. It elucidates the usefulness of these hierarchical models in collecting data for various scientific research, particularly in ecological studies. Meanwhile, several scientists and researchers have used Bayesian analysis and its approaches to solve and unlock the complexness of every scientific research they have conducted.

Hierarchical statistical models are increasingly being used to describe complex ecological processes. The data cloning (DC) method is a new general technique that uses Markov chain Monte Carlo (MCMC) algorithms to compute maximum likelihood (ML) estimates along with their asymptotic variance estimates for hierarchical models. Despite its generality, the method has two inferential limitations. First, it only provides Wald-type confidence intervals, known to be inaccurate in small samples. Second, it only yields ML parameter estimates, but not the maximized likelihood values used for profile likelihood intervals, likelihood ratio hypothesis tests, and information-theoretic model selection. Here we describe how to overcome these inferential limitations with a computationally efficient method for calculating likelihood ratios via data cloning. The ability to calculate likelihood ratios allows one to dó hypothesis tests, construct accurate confidence intervals and undertake information-based model selection with hierarchical models in a frequentist context. To demonstrate the use of these tools with complex ecological models, we reanalyze part of Gause's classic Paramecium data with state-space population models containing both environmental noise and sampling error. The analysis results include improved confidence intervals for parameters, a hypothesis test of laboratory replication, and a comparison of the Beverton-Holt and the Ricker growth forms based on a model selection index. [ABSTRACT FROM AUTHOR]

High-resolution data collected over the past 60 years by a single family of Siberian scientists on Lake Baikal reveal significant warming of surface waters and long-term changes in the basal food web of the world's largest, most ancient lake. Attaining depths over 1.6 km, Lake Baikal is the deepest and most voluminous of the world's great lakes. Increases in average water temperature (1.21 °C since 1946), chlorophyll a (300% since 1979), and an influential group of zooplankton grazers (335% increase in cladocerans since 1946) may have important implications for nutrient cycling and food web dynamics. Results from multivariate autoregressive (MAR) modeling suggest that cladocerans increased strongly in response to temperature but not to algal biomass, and cladocerans depressed some algal resources without observable fertilization effects. Changes in Lake Baikal are particularly significant as an integrated signal of long-term regional warming, because this lake is expected to be among those most resistant to climate change due to its tremendous volume. These findings highlight the importance of accessible, long-term monitoring data for understanding ecosystem response to large-scale stressors such as climate change. [ABSTRACT FROM AUTHOR]

We introduce a new statistical computing method, called data cloning, to calculate maximum likelihood estimates and their standard errors for complex ecological models. Although the method uses the Bayesian framework and exploits the computational simplicity of the Markov chain Monte Carlo (MCMC) algorithms, it provides valid frequentist inferences such as the maximum likelihood estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The data cloning method is easily implemented using standard MCMC software. Data cloning is particularly useful for analysing ecological situations in which hierarchical statistical models, such as state-space models and mixed effects models, are appropriate. We illustrate the method by fitting two nonlinear population dynamics models to data in the presence of process and observation noise. [ABSTRACT FROM AUTHOR]

We describe a discrete-time, stochastic population model with density dependence, environmental-type process noise, and lognormal observation or sampling error. The model, a stochastic version of the Gompertz model, can be transformed into a linear Gaussian state-space model (Kalman. filter) for convenient fitting to time series data. The model has a multivariate normal likelihood function and is simple enough for a variety of uses ranging from theoretical study of parameter estimation issues to routine data analyses in population monitoring. A special case of the model is the discrete-time, stochastic exponential growth model (density independence) with environmental-type process error and lognormal observation error. We describe two methods for estimating parameters in the Gompertz state-space model, and we compare the statistical qualities of the methods with computer simulations. The methods are maximum likelihood based on observations and restricted maximum likelihood based on first differences. Both offer adequate statistical properties. Because the likelihood function is identical to a repeated-measures analysis of variance model with a random time effect, parameter estimates can be calculated using PROC MIXED of SAS. We use the model to analyze a data set from the Breeding Bird Survey. The fitted model suggests that over 70% of the noise in the population's growth rate is due to observation error. The model describes the autocovariance properties of the data especially well. While observation error and process noise variance parameters can both be estimated from one time series, multimodal likelihood functions can and do occur. For data arising from the model, the statistically consistent parameter estimates do not necessarily correspond to the global maximum in the likelihood function. Maximization, simulation, and bootstrapping programs must accommodate the phenomenon of multimodal likelihood functions to produce statistically valid results. [ABSTRACT FROM AUTHOR]

A scaling rule of ecological theory, accepted but lacking experimental confirmation, is that the magnitude of fluctuations in population densities due to demographic stochasticity scales inversely with the square root of population numbers. This supposition is based on analyses of models exhibiting exponential growth or stable equilibria. Using two quantitative measures, we extend the scaling rule to situations in which population densities fluctuate due to nonlinear deterministic dynamics. These measures are applied to populations of the flour beetle Tribolium castaneum that display chaotic dynamics in both 20-g and 60-g habitats. Populations cultured in the larger habitat exhibit a clarification of the deterministic dynamics, which follows the inverse square root rule. Lattice effects, a deterministic phenomenon caused by the discrete nature of individuals, can cause deviations from the scaling rule when population numbers are small. The scaling rule is robust to the probability distribution used to model demographic variation among individuals. [ABSTRACT FROM AUTHOR]

Interest in facilitative predator-plant interactions has focused upon above-ground systems. Underground physical conditions are distinctive, however, and we provide evidence that bush lupine, Lupinus arboreus, facilitates the survival of the predatory nematode Heterorhabditis marelatus. Because H. mare/at us is prone to desiccation and lupines maintain a zone of moist soil around their taproots even during dry periods, we hypothesized that dry-season nematode survival under lupines might be higher than in the surrounding grasslands. We performed field surveys and measured nematode survival in lupine and grassland rhizospheres under wet- and dry-season conditions. Nematodes survived the crucial summer period better under lupines than in grasslands; however, this advantage disappeared in wet, winter soils. Modeling the probability of nematode population extinction showed that, while even large nematode cohorts were likely to go extinct in grasslands, even small cohorts in lupine rhizospheres were likely to survive until the arrival of the next prey generation. Because this nematode predator has a strong top-down effect on lupine survival via its effect on root-boring larvae of the ghost moth Hepialus californicus, this facilitative interaction may enable a belowground trophic cascade. Similar cases of predator facilitation in seasonally stressful environments are probably common in nature. [ABSTRACT FROM AUTHOR]

1. It was once assumed that commercially important fish make up significant portions of seal diets. As a consequence, prior to the 1970s many seal populations were dramatically reduced by rampant slaughtering. Today, seals and other marine mammals are valued components of marine ecosystems and their numbers are carefully managed. To facilitate management, government statutes mandate the systematic monitoring of seal populations. Population estimates are based on counts of hauled-out seals obtained by aerial survey and radio and satellite telemetry; hence, considerable effort has been devoted to finding optimal times for such counts. We have developed a predictive mathematical model of seal haul-out to assist resource managers in the selection of optimal census times.2. Haul-out depends on a number of environmental variables. Some of these variables, such as wind speed, can be obtained only as historical data or short-range predictions. Others, such as tide height, are deterministic and can be obtained as long-range predictions.3. We used deterministic environmental variables to develop mathematical models that describe haul-out dynamics of harbour sealsPhoca vitulinaduring the pupping season at a site in Washington, USA. A list of alternative hypotheses for environmental cues gave rise to a suite of competing models. We used information–theoretic model selection techniques to choose the best model. The selected model was a function of tide height and current direction, and explained 40% of the variability in hourly census data.4. An assumption that the system recovers rapidly after disturbance introduced two time scales. This allowed the differential equation model to be reduced to an algebraic equation.5. Synthesis and applications.This study demonstrates that resource managers can use a simple algebraic equation based on deterministic environmental variables to predict times at which to census maximal haul-out in harbour seals. At the Washington site, maximal daily haul-outs during pupping season are predicted to occur during receding tides, approximately midway between high and low tides. The largest maximal daily haul-outs during the pupping season are predicted to occur in the last week of July. The environmental factors correlated with haul-out are, however, site-specific; therefore the model developed for the Washington site will not necessarily hold for other haul-out areas. Managers should carry out the model selection procedure separately for each monitored haul-out site. The general methodology employed in this study can be used to make long-range predictions of diurnal movements for a variety of marine birds and mammals.Journal of Applied Ecology(2005)42, 108 –117doi: 10.1111/j.1365-2664.2005.00999.x [ABSTRACT FROM AUTHOR]

Time series of population abundance estimates often are the only data avail- able for evaluating the prospects for persistence of a species of concern. With such data, it is possible to perform a population viability analysis (PVA) with diffusion approximation methods using estimates of the mean population trend and the variance of the trend, the so-called process variation. Sampling error in the data, however, may bias estimators of process variation derived by simple methods. We develop a restricted maximum likelihood (REML)-based method for estimating trend, process variation, and sampling error from a single time series based on a discrete-time model of density-independent growth coupled with a model of the sampling process. Transformation of the data yields a conventional linear mixed model, in which the variance components are functions of the process variation and sampling error. Simulation results show essentially unbiased estimators of trend, process variation, and sampling error over a range of process variation/sampling error combinations. Example data analyses are provided for the Whooping Crane (Grus americana), grizzly bear (Ursus arctos horribilis), California Condor (Gymnogyps californianus), and Puerto Rican Parrot (Amazona vittata). This REML-based method is useful for PVA methods that depend on accurate estimation of process variation from time-series data. [ABSTRACT FROM AUTHOR]

Summary 1. In this journal 35 years ago, P. H. Leslie, T. Park and D. B. Mertz reported competitive exclusion data for two Tribolium species. It is less well-known that they also reported ‘difficult to interpret’ coexistence data. We suggest that the species exclusion and the species coexistence are consequences of a stable coexistence two-cycle in the presence of two stable competitive exclusion equilibria. 2. A stage-structured insect population model for two interacting species forecasts that as interspecific interaction is increased there occurs a sequence of dynamic changes (bifurcations) in which the classic Lotka–Volterra-type scenario with two stable competitive exclusion equilibria is altered abruptly to a novel scenario with three locally stable entities; namely, two competitive exclusion equilibria and a stable coexistence cycle. This scenario is novel in that it predicts the competitive coexistence of two nearly identical species on a single limiting resource and does so under circumstances of increased interspecific competition. This prediction is in contradiction to classical tenets of competition theory. [ABSTRACT FROM AUTHOR]

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that “chaos” should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets. [ABSTRACT FROM AUTHOR]

Presents information on a study which used a time series model of density dependent population growth as the basis for predicting mule deer population trends in south-central Oregon. Methods; Results; Discussion.

We use the concepts of composite maps, basins of attraction, basin switching, and saddle fly-by's to make the ecological hypothesis of the existence of multiple attractors more accessible to experimental scrutiny. Specifically, in a periodically forced insect population growth model we identify multiple attractors, namely, two locally stable 4-cycles. Using the model-predicted basins of attraction, we examine data time series from a Tribolium experiment for evidence of the multiple attractors. We conclude that the multiple attractor hypothesis together with demographic stochasticity accounts for the experimental observations. [ABSTRACT FROM AUTHOR]

Investigates the Allee effect or inverse density dependence at low population size on preservation and management of biological populations. Concept of the Allee effect; Ability of the Allee effect to alter population dynamics; Implications for empirically observed threshold in the number of organisms necessary to establish viable populations in species.

Tests a model-predicted sequence of transitions in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle Tribolium. Lyapunov exponents; Bootstrap parameter vectors; Responses of population to experimental demographic behavior.

We used small perturbations in adult numbers to control large fluctuations in the chaotic demographic dynamics of laboratory populations of the flour beetle Tribolium castaneum. A nonlinear mathematical model was used to identify a sensitive region of phase space where the addition of a few adult insects would result in a dampening of the life stage fluctuations. Three experimental treatments were applied: one in which perturbations were made whenever the populations were inside the sensitive region (“in-box treatment”), another where perturbations were made whenever the populations were outside the sensitive region (“out-box treatment”), and an unperturbed control. The in-box treatment caused a stabilization of insect densities at numbers well below the peak values exhibited by the out-box and control populations. This study demonstrates how small perturbations can be used to influence the chaotic dynamics of an ecological system. [ABSTRACT FROM AUTHOR]

Presents a study which analyzed the time-series abundance of San Joaquin kit fox estimated during 1983 to 1995 on the Naval Petroleum Reserves in California. Methodology; Results of the study; Discussion.

Presents a study on the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to understanding nonlinear ecological dynamics. Biologically based dynamical model; Mathematical and statistical analysis; Biological experiments.

Presents a statistical test for detecting density dependence in univariate time series observations of population abundances. Likelihood ratio test based on a discrete time stochastic logistic model; Stochastic exponential growth, stochastic exponential decline or random walk in the population.

1. We identify an unstable equilibrium with a two-dimensional stable manifold and a one-dimensional unstable manifold in a three-state variable (larva, pupa, adult) insect population growth model. 2. The saddle node forecasts that the time series of some initial numbers of larvae, pupae and adults are drawn closely to the unstable equilibrium before approaching the asymptotic stable attractor (a two-cycle), while the time series of other initial points are not. 3. Using two quantitative indices, we examine time series from a Tribolium experiment for evidence of the predicted saddle node. We conclude that a saddle node accounts for the transient dynamics in these data and for the differences between the transient behaviour of different replicates of the same experiment. [ABSTRACT FROM AUTHOR]

Focuses on the difficulties inherent in making convincing scientific arguments with Bayesian methods. Required changes under Bayesianism; Randomization; Sample surveys; Confidence intervals; Randomization tests; Purported advantages of Bayesianism; Bayesian statistics in ecology.

We used the symbolic solution for the roots of a cubic polynomial to derive expressions for the eigenvalues of a three‐stage population projection matrix. As well, we obtained expressions for eigenvectors, moduli, damping ratios, sensitivities, and elasticities. The equations reveal the existence of "superparameters;" natural groupings of vital rates that drive population dynamics. We show that growth rates can be calculated using (at most) three superparameters in place of as many as nine original vital rates, potentially simplifying data collection. Necessary and sufficient conditions for extinction can be summarized equivalently by three superparameter inequalities. For a common life history (Noon & Biles, 1990, J Wildl Manag, 54, 18–27) four vital rate parameters are reduced to two superparameters. The results are applicable to population viability and recovery analysis and harvest planning. Two recommendations for resource managers: Superparameters can be estimated to determine population growth rates.Superparameters can be used to conduct rapid assessments of extinction risk. [ABSTRACT FROM AUTHOR]

We present a nonstandard discretization method related to the methods of Mickens and Elaydi and Sacker for converting single species population models from continuous time to discrete time. The method falls in the category of the so‐called nonstandard discretization schemes, that is more advantageous than the classical discretization methods (such as adaptive step‐size) since it allows large step sizes. For instance, a large step size could better represent a generation time or a time interval between empirical measurements. Examples of single‐species models with and without negative density dependence, with an Allee effect, and with an alternative positive stable equilibrium (predator pit) are studied. Comparative analyses of bifurcations of ordinary differential equations and difference equations show how the new discretization proposed here preserves the dynamical properties of the continuous‐time models. Recommendations for Resource Managers: The discretization method we propose preserves the original dynamic properties of the continuous model, in the sense of equilibria, their stability, and bifurcation characteristics. Unlike the traditional numerical methods that are widely used in ecology, the dynamical consistency of our method does not depend on the size of the step size used.The discretization method we propose produces solution trajectories in a remarkable agreement with those of the corresponding continuous models irrespective of the size of the time interval used.Results presented here will be important to future ecological studies that seek to evaluate the pervasiveness and strength of negative density dependence as well as Allee effects, along with the prospects of alternative stable states, in natural populations. [ABSTRACT FROM AUTHOR]

A response by Brian Dennis and Aaron Ellison to a letter to the editor regarding their article on the course-based solutions for graduate students in ecological sciences.