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

Interconnection between density-regulation and stability in competitive ecological network

In natural ecosystems, species can be characterized by the nonlinear density-dependent self-regulation of their growth profile. Species of many taxa show a substantial density-dependent reduction for low population size. Nevertheless, many show the opposite trend; density regulation is minimal for small populations and increases significantly when the population size is near the carrying capacity. The theta-logistic growth equation can portray the intraspecific density regulation in the growth profile, theta being the density regulation parameter. In this study, we examine the role of these different growth profiles on the stability of a competitive ecological community with the help of a mathematical model of competitive species interactions. This manuscript deals with the random matrix theory to understand the stability of the classical theta-logistic models of competitive interactions. Our results suggest that having more species with strong density dependence, which self-regulate at low densities, leads to more stable communities. With this, stability also depends on the complexity of the ecological network. Species network connectance (link density) shows a consistent trend of increasing stability, whereas community size (species richness) shows a context-dependent effect. We also interpret our results from the aspect of two different life history strategies: r and K-selection. Our results show that the stability of a competitive network increases with the fraction of r-selected species in the community. Our result is robust, irrespective of different network architectures.

Distilling Professional Opinion to Gauge Vulnerability of Guam Avifauna to Brown Treesnake Predation

The avifauna of Guam was devastated by the introduction of the Brown Treesnake, and the restoration of native birds would need to address the problem with eradication or suppression of BTS. With eradication of the snake unlikely in the near term, and suppression capabilities limited to specific finite areas, key information for reintroductions is how low BTS abundance will likely need to be for each bird species to be re-established based on their vulnerability to BTS predation. Here, we estimate vulnerability, which can no longer be measured directly, so biologists who are familiar with one or more of seven Guam birds were surveyed to obtain their knowledge and produce quantitative vulnerability estimates. As is typical of birds adapted to islands devoid of predators, respondents judged that our focal species exhibit few predator avoidance and tolerance traits, leaving body size as the prime determinant of vulnerability. Respondent opinion also holds that any behavior that reduces the likelihood of an encounter by BTS, e.g., roosting/nesting in palm crowns, cavity nesting, and in particular urban dwelling, substantially reduces vulnerability. Our results can help inform species-specific decisions about when it may be safe to consider the release of birds on Guam depending on the relative vulnerability of each species to predation by BTS.

Evidence for r- and K-selection in a wild bird population: a reciprocal link between ecology and evolution

Understanding the variation in selection pressure on key life-history traits is crucial in our rapidly changing world. Density is rarely considered as a selective agent. To study its importance, we partition phenotypic selection in fluctuating environments into components representing the population growth rate at low densities and the strength of density dependence, using a new stochastic modelling framework. We analysed the number of eggs laid per season in a small song-bird, the great tit, and found balancing selection favouring large clutch sizes at small population densities and smaller clutches in years with large populations. A significant interaction between clutch size and population size in the regression for the Malthusian fitness reveals that those females producing large clutch sizes at small population sizes also are those that show the strongest reduction in fitness when population size is increased. This provides empirical support for ongoing r- and K-selection in this population, favouring phenotypes with large growth rates r at small population sizes and phenotypes with high competitive skills when populations are close to the carrying capacity K This selection causes long-term fluctuations around a stable mean clutch size caused by variation in population size, implying that r- and K-selection is an important mechanism influencing phenotypic evolution in fluctuating environments. This provides a general link between ecological dynamics and evolutionary processes, operating through a joint influence of density dependence and environmental stochasticity on fluctuations in population size.

A case study of bats and white-nose syndrome demonstrating how to model population viability with evolutionary effects

Ecological factors generally affect population viability on rapid time scales. Traditional population viability analyses (PVA) therefore focus on alleviating ecological pressures, discounting potential evolutionary impacts on individual phenotypes. Recent studies of evolutionary rescue (ER) focus on cases in which severe, environmentally induced population bottlenecks trigger a rapid evolutionary response that can potentially reverse demographic threats. ER models have focused on shifting genetics and resulting population recovery, but no one has explored how to incorporate those findings into PVA. We integrated ER into PVA to identify the critical decision interval for evolutionary rescue (DIER) under which targeted conservation action should be applied to buffer populations undergoing ER against extinction from stochastic events and to determine the most appropriate vital rate to target to promote population recovery. We applied this model to little brown bats (Myotis lucifugus) affected by white-nose syndrome (WNS), a fungal disease causing massive declines in several North American bat populations. Under the ER scenario, the model predicted that the DIER period for little brown bats was within 11 years of initial WNS emergence, after which they stabilized at a positive growth rate (λ = 1.05). By comparing our model results with population trajectories of multiple infected hibernacula across the WNS range, we concluded that ER is a potential explanation of observed little brown bat population trajectories across multiple hibernacula within the affected range. Our approach provides a tool that can be used by all managers to provide testable hypotheses regarding the occurrence of ER in declining populations, suggest empirical studies to better parameterize the population genetics and conservation-relevant vital rates, and identify the DIER period during which management strategies will be most effective for species conservation.

Effects of density and fire on the vital rates and population growth of a perennial goldenaster

In a novel analysis, a regression-design life-table response experiment was used to determine how the interaction of fire and density affected vital rates of the perennial composite Pityopsis aspera, and ultimately how these changes in vital rates contributed to differences in estimated population growth rates.

Intraspecific density effects are generally associated with other factors, like disturbance. Therefore, the ways in which density effects might interact with disturbance to modify the relationships between vital rates and population growth must be understood. I quantified the effects of density on the life-history stages of the perennial composite Pityopsis aspera over 3 years, the span of which included years in which fire did and did not occur. In an experimental study, I estimated the survival, growth and reproduction for shoots in plots established across a natural range of densities in Florida, USA. In a novel analysis, a regression-design life-table response experiment was used to determine which transitions were associated with density, how they contributed to differences in estimated population growth rates and how this relationship differed as a result of fire. The shape of the relationship between population growth rate (λ) and density was modified by fire, primarily as a result of contributions from adult flowering stasis and survival, and first-year survival probabilities. Fire modified and even reversed the effect of extreme densities on adult flowering stasis and survival and of first-year survival, resulting in more positive contributions from these transitions to λ at the lowest and highest density values. These results demonstrate the first application of a regression-design life-table response experiment to elucidating the interactive effects of density and fire. They highlight the utility of this approach for both capturing the complex dynamics of populations and establishing a means of determining how vital rates might contribute to differences in demography across densities.

Allowable levels of take for the trade in Nearctic songbirds

The take of Nearctic songbirds for the caged-bird trade is an important cultural and economic activity in Mexico, but its sustainability has been questioned. We relied on the theta-logistic population model to explore options for setting allowable levels of take for 11 species of passerines that were subject to legal take in Mexico in 2010. Because estimates of population size necessary for making-periodic adjustments to levels of take are not routinely available, we examined the conditions under which a constant level of take might contribute to population depletion (i.e., a population below its level of maximum net productivity). The chance of depleting a population is highest when levels of take are based on population sizes that happen to be much lower or higher than the level of maximum net productivity, when environmental variation is relatively high and serially correlated, and when the interval between estimation of population size is relatively long (> or = 5 years). To estimate demographic rates of songbirds involved in the Mexican trade we relied on published information and allometric relationships to develop probability distributions for key rates, and then sampled from those distributions to characterize the uncertainty in potential levels of take. Estimates of the intrinsic rate of growth (r) were highly variable, but median estimates were consistent with those expected for relatively short-lived, highly fecund species. Allowing for the possibility of nonlinear density dependence generally resulted in allowable levels of take that were lower than would have been the case under an assumption of linearity. Levels of take authorized by the Mexican government in 2010 for the 11 species we examined were small in comparison to relatively conservative allowable levels of take (i.e., those intended to achieve 50% of maximum sustainable yield). However, the actual levels of take in Mexico are unknown and almost certainly exceed the authorized take. Also, the take of Nearctic songbirds in other Latin American and Caribbean countries ultimately must be considered in assessing population-level impacts.

Minimum viable populations: is there a 'magic number' for conservation practitioners?

Establishing species conservation priorities and recovery goals is often enhanced by extinction risk estimates. The need to set goals, even in data-deficient situations, has prompted researchers to ask whether general guidelines could replace individual estimates of extinction risk. To inform conservation policy, recent studies have revived the concept of the minimum viable population (MVP), the population size required to provide some specified probability of persistence for a given period of time. These studies conclude that long-term persistence requires ≥5000 adult individuals, an MVP threshold that is unaffected by taxonomy, life history or environmental conditions. Here, we re-evaluate this suggestion. We find that neither data nor theory supports its general applicability, raising questions about the utility of MVPs for conservation planning.

Richards growth model and viability indicators for populations subject to interventions

In this work we study the problem of modeling identification of a population employing a discrete dynamic model based on the Richards growth model. The population is subjected to interventions due to consumption, such as hunting or farming animals. The model identification allows us to estimate the probability or the average time for a population number to reach a certain level. The parameter inference for these models are obtained with the use of the likelihood profile technique as developed in this paper. The identification method here developed can be applied to evaluate the productivity of animal husbandry or to evaluate the risk of extinction of autochthon populations. It is applied to data of the Brazilian beef cattle herd population, and the the population number to reach a certain goal level is investigated.

Likelihood ridges and multimodality in population growth rate models

A central problem in population ecology is to use time series data to estimate the form of density dependence in the per capita growth rate (pgr). This is often accomplished with phenomenological models such as the theta-Ricker or generalized Beverton-Holt. Using the theta-Ricker model as a simple but flexible description of density dependence, we apply theory and simulations to show how multimodality and ridges in the likelihood surface can emerge even in the absence of model misspecification or observation error. The message for model fitting of real data is to consider the likelihood surface in detail, check whether the best-fit model is located on a likelihood ridge and, if so, evaluate predictive differences of biologically plausible models along the ridge. We present a detailed analysis of a focal data set showing how multimodality and ridges emerge in practice for fits of several parametric models, including a state-space model with explicit accommodation of observation error. Best-fit models for these data are biologically dubious beyond the range of the data, and likelihood ratio confidence regions include a wide range of more biologically plausible models. We demonstrate the broad relevance of these findings by presenting analyses of 25 additional data sets spanning a wide range of taxa. The results here are relevant to information-theoretic and Bayesian methods, which also rely on likelihoods. Beyond presentation of best-fit models and confidence regions around individual parameters, effort toward understanding features of the likelihood surface will help ensure the most robust translation from statistical analysis to biological interpretation.

Climate and spatio-temporal variation in the population dynamics of a long distance migrant, the white stork

1. A central question in ecology is to separate the relative contribution of density dependence and stochastic influences to annual fluctuations in population size. Here we estimate the deterministic and stochastic components of the dynamics of different European populations of white stork Ciconia ciconia. We then examined whether annual changes in population size was related to the climate during the breeding period (the 'tap hypothesis' sensu Saether, Sutherland & Engen (2004, Advances in Ecological Research, 35, 185 209) or during the nonbreeding period, especially in the winter areas in Africa (the 'tube hypothesis'). 2. A general characteristic of the population dynamics of this long-distance migrant is small environmental stochasticity and strong density regulation around the carrying capacity with short return times to equilibrium. 3. Annual changes in the size of the eastern European populations were correlated by rainfall in the wintering areas in Africa as well as local weather in the breeding areas just before arrival and in the later part of the breeding season and regional climate variation (North Atlantic Oscillation). This indicates that weather influences the population fluctuations of white storks through losses of sexually mature individuals as well as through an effect on the number of individuals that manages to establish themselves in the breeding population. Thus, both the tap and tube hypothesis explains climate influences on white stork population dynamics. 4. The spatial scale of environmental noise after accounting for the local dynamics was 67 km, suggesting that the strong density dependence reduces the synchronizing effects of climate variation on the population dynamics of white stork. 5. Several climate variables reduced the synchrony of the residual variation in population size after accounting for density dependence and demographic stochasticity, indicating that these climate variables had a synchronizing effect on the population fluctuations. In contrast, other climatic variables acted as desynchronizing agents. 6. Our results illustrate that evaluating the effects of common environmental variables on the spatio-temporal variation in population dynamics require estimates and modelling of their influence on the local dynamics.

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.

The extended Moran effect and large-scale synchronous fluctuations in the size of great tit and blue tit populations

1. Synchronous fluctuations of geographically separated populations are in general explained by the Moran effect, i.e. a common influence on the local population dynamics of environmental variables that are correlated in space. Empirical support for such a Moran effect has been difficult to provide, mainly due to problems separating out effects of local population dynamics, demographic stochasticity and dispersal that also influence the spatial scaling of population processes. Here we generalize the Moran effect by decomposing the spatial autocorrelation function for fluctuations in the size of great tit Parus major and blue tit Cyanistes caeruleus populations into components due to spatial correlations in the environmental noise, local differences in the strength of density regulation and the effects of demographic stochasticity. 2. Differences between localities in the strength of density dependence and nonlinearity in the density regulation had a small effect on population synchrony, whereas demographic stochasticity reduced the effects of the spatial correlation in environmental noise on the spatial correlations in population size by 21.7% and 23.3% in the great tit and blue tit, respectively. 3. Different environmental variables, such as beech mast and climate, induce a common environmental forcing on the dynamics of central European great and blue tit populations. This generates synchronous fluctuations in the size of populations located several hundred kilometres apart. 4. Although these environmental variables were autocorrelated over large areas, their contribution to the spatial synchrony in the population fluctuations differed, dependent on the spatial scaling of their effects on the local population dynamics. We also demonstrate that this effect can lead to the paradoxical result that a common environmental variable can induce spatial desynchronization of the population fluctuations. 5. This demonstrates that a proper understanding of the ecological consequences of environmental changes, especially those that occur simultaneously over large areas, will require information about the spatial scaling of their effects on local population dynamics.

Density-dependence as a size-independent regulatory mechanism

The growth function of populations is central in biomathematics. The main dogma is the existence of density-dependence mechanisms, which can be modelled with distinct functional forms that depend on the size of the population. One important class of regulatory functions is the theta-logistic, which generalizes the logistic equation. Using this model as a motivation, this paper introduces a simple dynamical reformulation that generalizes many growth functions. The reformulation consists of two equations, one for population size, and one for the growth rate. Furthermore, the model shows that although population is density-dependent, the dynamics of the growth rate does not depend either on population size, nor on the carrying capacity. Actually, the growth equation is uncoupled from the population size equation, and the model has only two parameters, a Malthusian parameter rho and a competition coefficient theta. Distinct sign combinations of these parameters reproduce not only the family of theta-logistics, but also the van Bertalanffy, Gompertz and Potential Growth equations, among other possibilities. It is also shown that, except for two critical points, there is a general size-scaling relation that includes those appearing in the most important allometric theories, including the recently proposed Metabolic Theory of Ecology. With this model, several issues of general interest are discussed such as the growth of animal population, extinctions, cell growth and allometry, and the effect of environment over a population.

Extinction in relation to demographic and environmental stochasticity in age-structured models

The demographic variance of an age-structured population is defined. This parameter is further split into components generated by demographic stochasticity in each vital rate. The applicability of these parameters are investigated by checking how an age-structured population process can be approximated by a diffusion with only three parameters. These are the deterministic growth rate computed from the expected projection matrix and the environmental and demographic variances. We also consider age-structured populations where the fecundity at any stage is either zero or one, and there is neither environmental stochasticity nor dependence between individual fecundity and survival. In this case the demographic variance is uniquely determined by the vital rates defining the projection matrix. The demographic variance for a long-lived bird species, the wandering albatross in the southwestern part of the Indian Ocean, is estimated. We also compute estimates of the age-specific contributions to the total demographic variance from survival, fecundity and the covariance between survival and fecundity.

Environmental stochasticity in dispersal areas can explain the "mysterious" disappearance of breeding populations

We present the results of an individual-based simulation model, showing that increasing the mortality of non-breeding dispersers within settlement areas can lead to the extinction of species and (meta)populations in a subtle way. This is because the areas where dispersers settle are generally unknown or difficult to detect. Consequently, fewer efforts are devoted to the conservation of these sites than to the conservation of breeding territories. Additionally, high mortality rates affecting the floater sector of a population become evident in the breeding sector only after several of years, when it is too difficult or too late to halt the decline. As a result, because most conservation projects on endangered species and populations mainly focus on breeding areas, many current efforts may be wasted in locations other than those in which conservation would be really necessary and effective.

Life-History Variation Predicts the Effects of Demographic Stochasticity on Avian Population Dynamics

Comparative analyses of avian population fluctuations have shown large interspecific differences in population variability that have been difficult to relate to variation in general ecological characteristics. Here we show that interspecific variation in demographic stochasticity, caused by random variation among individuals in their fitness contributions, can be predicted from a knowledge of the species' position along a "slow-fast" gradient of life-history variation, ranging from high reproductive species with short life expectancy at one end to species that often produce a single offspring but survive well at the other end of the continuum. The demographic stochasticity decreased with adult survival rate, age at maturity, and generation time or the position of the species toward the slow end of the slow-fast life-history gradient. This relationship between life-history characteristics and demographic stochasticity was related to interspecific differences in the variation among females in recruitment as well as to differences in the individual variation in survival. Because reproductive decisions in birds are often subject to strong natural selection, our results provide strong evidence for adaptive modifications of reproductive investment through life-history evolution of the influence of stochastic variation on avian population dynamics.

Two complementary paradigms for analysing population dynamics

To understand why population growth rate is sometimes positive and sometimes negative, ecologists have adopted two main approaches. The most common approach is through the density paradigm by plotting population growth rate against population density. The second approach is through the mechanistic paradigm by plotting population growth rate against the relevant ecological processes affecting the population. The density paradigm is applied a posteriori, works sometimes but not always and is remarkably useless in solving management problems or in providing an understanding of why populations change in size. The mechanistic paradigm investigates the factors that supposedly drive density changes and is identical to Caughley's declining population paradigm of conservation biology. The assumption that we can uncover invariant relationships between population growth rate and some other variables is an article of faith. Numerous commercial fishery applications have failed to find the invariant relationships between stock and recruitment that are predicted by the density paradigm. Environmental variation is the rule, and non-equilibrial dynamics should force us to look for the mechanisms of population change. If multiple factors determine changes in population density, there can be no predictability in either of these paradigms and we will become environmental historians rather than scientists with useful generalizations for the population problems of this century. Defining our questions clearly and adopting an experimental approach with crisp alternative hypotheses and adequate controls will be essential to building useful generalizations for solving the practical problems of population management in fisheries, wildlife and conservation.

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.

Demographic characteristics and population dynamical patterns of solitary birds

In birds and many other animals, there are large interspecific differences in the magnitude of annual variation in population size. Using time-series data on populations of solitary bird species, we found that fluctuations in population size of solitary birds were affected by the deterministic characteristics of the population dynamics as well as the stochastic factors. In species with highly variable populations, annual variation in recruitment was positively related to the return rate of adults between successive breeding seasons. In stable populations, more recruits were found in years with low return rates of breeding adults. This identifies a gradient, associated with the position of the species along a "slow-fast" continuum of life history variation, from highly variable populations with a recruitment-driven demography to stable, strongly density-regulated populations with a survival-restricted demography. These results suggest that patterns in avian population fluctuations can be predicted from a knowledge of life-history characteristics and/or temporal variation in certain demographic traits.

Understanding the persistence of measles: reconciling theory, simulation and observation

Ever since the pattern of localized extinction associated with measles was discovered by Bartlett in 1957, many models have been developed in an attempt to reproduce this phenomenon. Recently, the use of constant infectious and incubation periods, rather than the more convenient exponential forms, has been presented as a simple means of obtaining realistic persistence levels. However, this result appears at odds with rigorous mathematical theory; here we reconcile these differences. Using a deterministic approach, we parameterize a variety of models to fit the observed biennial attractor, thus determining the level of seasonality by the choice of model. We can then compare fairly the persistence of the stochastic versions of these models, using the 'best-fit' parameters. Finally, we consider the differences between the observed fade-out pattern and the more theoretically appealing 'first passage time'.

Variation among Individuals and Reduced Demographic Stochasticity

Population viability analysis ( PVA) is a technique that employs stochastic demographic models to predict extinction risk. All else being equal, higher variance in a demographic rate leads to a greater extinction risk. Demographic stochasticity represents variance due to differences among individuals. Current implementations of PVAs, however, assume that the expected fates of all individuals are identical. For example, demographic stochasticity in survival is modeled as a random draw from a binomial distribution. We developed a simple conceptual model showing that if there is variation among individuals in expected survival, then existing PVA models overestimate the variance due to demographic stochasticity in survival. This is a consequence of Jensen's inequality and the fact that the binomial demographic variance is a concave function of mean survival. The effect of variation among individuals on demographic stochasticity in fecundity depends on the mean-variance relationship for individual reproductive success, which is not presently known. If fecundity patterns mirror those of survival, then variation among individuals will reduce the extinction risk of small populations.