On the latent state estimation of nonlinear population dynamics using Bayesian and non-Bayesian state-space models

Nonlinear state-space models have been increasingly applied to study population dynamics and data assimilation in environmental sciences. State-space models can account for process error and measurement error simultaneously to correct for the bias in the estimates of system state and model parameters. However, few studies have compared the performance of different nonlinear state-space models for reconstructing the state of population dynamics from noisy time series. This study compared the performance of the extended Kalman filter (EKF), unscented Kalman filter (UKF) and Bayesian nonlinear state-space models (BNSSM) through simulations. Synthetic population time series were generated using the theta logistic model with known parameters, and normally distributed process and measurement errors were introduced using the Monte Carlo simulations. At higher levels of nonlinearity, the UKF and BNSSM had lower root mean square error (RMSE) than the EKF. The BNSSM performed reliably across all levels of nonlinearity, whereas increased levels of nonlinearity resulted in higher RMSE of the EKF. The Metropolis–Hastings algorithm within the Gibbs algorithm was used to fit the theta logistic model to synthetic time series to estimate model parameters. The estimated posterior distribution of the parameter θ indicated that the 95% credible intervals included the true values of θ (=0.5 and 1.5), but did not include 1.0 and 0.0. Future studies need to incorporate the adaptive Metropolis algorithm to estimate unknown model parameters for broad applications of Bayesian nonlinear state-space models in ecological studies.     

Estimation methods for nonlinear state-space models in ecology

The use of nonlinear state-space models for analyzing ecological systems is increasing. A wide range of estimation methods for such models are available to ecologists, however it is not always clear, which is the appropriate method to choose. To this end, three approaches to estimation in the theta logistic model for population dynamics were benchmarked by Wang (2007). Similarly, we examine and compare the estimation performance of three alternative methods using simulated data. The first approach is to partition the state-space into a finite number of states and formulate the problem as a hidden Markov model (HMM). The second method uses the mixed effects modeling and fast numerical integration framework of the AD Model Builder (ADMB) open-source software. The third alternative is to use the popular Bayesian framework of BUGS. The study showed that state and parameter estimation performance for all three methods was largely identical, however with BUGS providing overall wider credible intervals for parameters than HMM and ADMB confidence intervals.     

Insect herbivores, density dependence, and the performance of the perennial herb Solanum carolinense

How insect herbivores affect plant performance is of central importance to basic and applied ecology. A full understanding of herbivore effects on plant performance requires understanding interactions (if any) of herbivore effects with plant density and size because these interactions will be critical for determining how herbivores influence plant population size. However, few studies have considered these interactions, particularly over a wide enough range of densities to detect nonlinear effects. Here we ask whether plant density and herbivores influence plant performance linearly or nonlinearly, how plant density affects herbivore damage, and how herbivores alter density dependence in transitions between plant size classes. In a large field experiment, we manipulated the density of the herbaceous perennial plant Solanum carolinense and herbivore presence in a fully crossed design. We measured plant size, sexual reproduction, and damage to plants in two consecutive years, and asexual reproduction of new stems in the second year, allowing us to characterize both plant performance and rates of transition between plant size classes across years. We found nonlinear effects of plant density on damage. Damage by herbivores and plant density both influenced sexual and asexual reproduction of S. carolinense; these effects were mostly mediated via effects on plant size. Importantly, we found that herbivores altered the pattern of linear density dependence in some transition rates (including survival and asexual reproduction) between plant size classes. These results suggest that understanding the ecological or evolutionary effects of herbivores on plant populations requires consideration of plant density and plant size, because feedbacks between density, herbivores, and plant size may complicate longer-term dynamics.     

Nutrient subsidies to belowground microbes impact aboveground food web interactions

Historically, terrestrial food web theory has been compartmentalized into interactions among aboveground or belowground communities. In this study we took a more synthetic approach to understanding food web interactions by simultaneously examining four trophic levels and investigating how nutrient (nitrogen and carbon) and detrital subsidies impact the ability of the belowground microbial community to alter the abundance of aboveground arthropods (herbivores and predators) associated with the intertidal cord grass Spartina alterniflora. We manipulated carbon, nitrogen, and detrital resources in a field experiment and measured decomposition rate, soil nitrogen pools, plant biomass and quality, herbivore density, and arthropod predator abundance. Because carbon subsidies impact plant growth only indirectly (microbial pathways), whereas nitrogen additions both directly (plant uptake) and indirectly (microbial pathways) impact plant primary productivity, we were able to assess the effect of both belowground soil microbes and nutrient availability on aboveground herbivores and their predators. Herbivore density in the field was suppressed by carbon supplements. Carbon addition altered soil microbial dynamics (net potential ammonification, litter decomposition rate, DON [dissolved organic N] concentration), which limited inorganic soil nitrogen availability and reduced plant size as well as predator abundance. Nitrogen addition enhanced herbivore density by increasing plant size and quality directly by increasing inorganic soil nitrogen pools, and indirectly by enhancing microbial nitrification. Detritus adversely affected aboveground herbivores mainly by promoting predator aggregation. To date, the effects of carbon and nitrogen subsidies on salt marshes have been examined as isolated effects on either the aboveground or the belowground community. Our results emphasize the importance of directly addressing the soil microbial community as a factor that influences aboveground food web structure by affecting plant size and aboveground plant nitrogen.     

Consequences of stage-structured predators: cannibalism, behavioral effects, and trophic cascades

Cannibalistic and asymmetrical behavioral interactions between stages are common within stage-structured predator populations. Such direct interactions between predator stages can result in density- and trait-mediated indirect interactions between a predator and its prey. A set of structured predator-prey models is used to explore how such indirect interactions affect the dynamics and structure of communities. Analyses of the separate and combined effects of stage-structured cannibalism and behavior-mediated avoidance of cannibals under different ecological scenarios show that both cannibalism and behavioral avoidance of cannibalism can result in short- and long-term positive indirect connections between predator stages and the prey, including "apparent mutualism." These positive interactions alter the strength of trophic cascades such that the system's dynamics are determined by the interaction between bottom-up and top-down effects. Contrary to the expectation of simpler models, enrichment increases both predator and prey abundance in systems with cannibalism or behavioral avoidance of cannibalism. The effect of behavioral avoidance of cannibalism, however, depends on how strongly it affects the maturation rate of the predator. Behavioral interactions between predator stages reduce the short-term positive effect of cannibalism on the prey density, but can enhance its positive long-term effects. Both interaction types reduce the destabilizing effect of enrichment. These results suggest that inconsistencies between data and simple models can be resolved by accounting for stage-structured interactions within and among species.     

Revisiting the classics: considering nonconsumptive effects in textbook examples of predator-prey interactions

Predator effects on prey dynamics are conventionally studied by measuring changes in prey abundance attributed to consumption by predators. We revisit four classic examples of predator-prey systems often cited in textbooks and incorporate subsequent studies of nonconsumptive effects of predators (NCE), defined as changes in prey traits (e.g., behavior, growth, development) measured on an ecological time scale. Our review revealed that NCE were integral to explaining lynx-hare population dynamics in boreal forests, cascading effects of top predators in Wisconsin lakes, and cascading effects of killer whales and sea otters on kelp forests in nearshore marine habitats. The relative roles of consumption and NCE of wolves on moose and consequent indirect effects on plant communities of Isle Royale depended on climate oscillations. Nonconsumptive effects have not been explicitly tested to explain the link between planktonic alewives and the size structure of the zooplankton, nor have they been invoked to attribute keystone predator status in intertidal communities or elsewhere. We argue that both consumption and intimidation contribute to the total effects of keystone predators, and that characteristics of keystone consumers may differ from those of predators having predominantly NCE. Nonconsumptive effects are often considered as an afterthought to explain observations inconsistent with consumption-based theory. Consequently, NCE with the same sign as consumptive effects may be overlooked, even though they can affect the magnitude, rate, or scale of a prey response to predation and can have important management or conservation implications. Nonconsumptive effects may underlie other classic paradigms in ecology, such as delayed density dependence and predator-mediated prey coexistence. Revisiting classic studies enriches our understanding of predator-prey dynamics and provides compelling rationale for ramping up efforts to consider how NCE affect traditional predator-prey models based on consumption, and to compare the relative magnitude of consumptive and NCE of predators.     

Predation, habitat complexity, and variation in density-dependent mortality of temperate reef fishes

Density dependence in demographic rates can strongly affect the dynamics of populations. However, the mechanisms generating density dependence (e.g., predation) are also dynamic processes and may be influenced by local conditions. Understanding the manner in which local habitat features affect the occurrence and/or strength of density dependence will increase our understanding of population dynamics in heterogeneous environments. In this study I conducted two separate field experiments to investigate how local predator density and habitat complexity affect the occurrence and form of density-dependent mortality of juvenile rockfishes (Sebastes spp.). I also used yearly censuses of rockfish populations on nearshore reefs throughout central California to evaluate mortality of juvenile rockfish at large spatial scales. Manipulations of predators (juvenile bocaccio, S. paucispinus) and prey (kelp, gopher, and black-and-yellow [KGB] rockfish, Sebastes spp.) demonstrated that increasing the density of predators altered their functional response and thus altered patterns of density dependence in mortality of their prey. At low densities of predators, the number of prey consumed per predator was a decelerating function, and mortality of prey was inversely density dependent. However, at high densities of predators, the number of prey killed per predator became an accelerating response, and prey mortality was directly density dependent. Results of field experiments and large-scale surveys both indicated that the strength of density-dependent mortality may also be affected by the structural complexity of the habitat. In small-scale field experiments, increased habitat complexity increased the strength of density-dependent mortality. However, at large scales, increasing complexity resulted in a decrease in the strength of density dependence. I suggest that these differences resulted from scale-dependent changes in the predatory response that generated mortality. Whether increased habitat complexity leads to an increase or a decrease in the strength of density-dependent mortality may depend on how specific predatory responses (e.g., functional or aggregative) are altered by habitat complexity. Overall, the findings of this study suggest that rates of demographic density dependence and the resulting dynamics of local populations may largely depend upon attributes of the local habitat.     

Population dynamics and control of invasive Spartina alterniflora: inference and forecasting under uncertainty

Managing invaded ecosystems entails making decisions about control strategies in the face of scientific uncertainty and ecological stochasticity. Statistical tools such as model selection and Bayesian decision analysis can guide decision-making by estimating probabilities of outcomes under alternative management scenarios, but these tools have seldom been applied in invasion ecology. We illustrate the use of model selection and Bayesian methods in a case study of smooth cordgrass (Spartina alterniflora) invading Willapa Bay, Washington. To address uncertainty in model structure, we quantified the weight of evidence for two previously proposed hypotheses, that S. alterniflora recruitment varies with climatic conditions (represented by sea surface temperature) and that recruitment is subject to an Allee effect due to pollen limitation. By fitting models to time series data, we found strong support for climate effects, with higher per capita seedling production in warmer years, but no evidence for an Allee effect based on either the total area invaded or the mean distance between neighboring clones. We used the best-supported model to compare alternative control strategies, incorporating uncertainty in parameter estimates and population dynamics. For a fixed annual removal effort, the probability of eradication in 10 years was highest, and final invaded area lowest, if removals targeted the smallest clones rather than the largest or randomly selected clones. The relationship between removal effort and probability of eradication was highly nonlinear, with a sharp threshold separating -0% and -100% probability of success, and this threshold was 95% lower in simulations beginning early rather than late in the invasion. This advantage of a rapid response strategy is due to density-dependent population growth, which produces alternative stable equilibria depending on the initial invasion size when control begins. Our approach could be applied to a wide range of invasive species management problems where appropriate data are available.     

Bottom-up effects of plant genotype on aphids, ants, and predators

Theory predicts that bottom-up ecological forces can affect community dynamics, but whether this extends to the effects of heritable plant variation on tritrophic communities is poorly understood. In a field experiment, I contrasted the effects of plant genotype (28 genotypes; 1064 plants), aphid density, and the presence/absence of mutualistic ants in affecting the per capita population growth of a specialist aphid herbivore, as well as the effects of plant genotype on the third trophic level. Plant genotype strongly affected aphid population growth rate, explaining 29% of the total variation in growth rate, whereas aphid density and ant-aphid interactions explained substantially less variation (< 2%) in aphid population growth rate. Plant genotype also had direct and indirect effects on the third trophic level, affecting the abundance of aphid-tending ants and the richness of predators. Multiple regression identified several heritable plant traits that explained 49% of the variation in aphid growth rate and 30% of the variation in ant abundance among plant genotypes. These bottom-up effects of plant genotype on tritrophic interactions were independent of the effects of either initial aphid density or the presence/absence of mutualistic ants. This study shows that plant genotype can be one of the most important ecological factors shaping tritrophic communities.     

Testing a new model of aphid abundance with sedentary and non-sedentary predators

Aphid population dynamics has been thoroughly investigated, especially in tree-dwelling aphids. Among the controls of the aphid rate of increase are the negative effects of antagonists, the positive effects of mutualists, the density-dependence of the aphid dynamics, and the non-stationary quality of plant tissues. Here we present a mechanistic model of aphid growth that considers most of these governing factors using a simple formulation. What is new in this model is that it considers two kinds of antagonists. The first kind is a guild of aphid predator specialists that includes ladybirds (Coleoptera: Coccinellidae), but also species of some families of Hemiptera, Diptera, and Neuroptera. The second kind of antagonists consists of omnivores or generalist predators and in this particular setting is exemplified by the European earwig Forficula auricularia (Dermaptera: Forficulidae). The model developed here compared the effects of these two different kinds of aphid predators, the second one always at the site (sedentary predators) and the first one that arrives in important numbers only once the aphid population has already developed to some degree (non-sedentary predators). Multiple model parameter sets, representing different hypotheses about controls on aphid populations, were evaluated within the Generalised Likelihood Uncertainty Estimation (GLUE) methodology. The model correctly reproduced the experimental data obtained in an organic citrus grove showing the important effect that sedentary predators as earwigs can have on the aphid populations. Low densities of sedentary predators or even low predation rates can have a disproportionate effect on the final aphid density, as they prey on small populations, when the per capita effect on the aphid population is higher. During the main spring peak of aphids the role of non-sedentary predators is secondary, as they track the aphid density rather than control it. However, these non-sedentary predators are important within the proposed model to keep the second autumn peak of aphids at low values.     

Density dependence in marine fish populations revealed at small and large spatial scales

Experimental manipulation of population density has frequently been used to demonstrate demographic density dependence. However, such studies are usually small scale and typically provide evidence of spatial (within-generation) density dependence. It is often unclear whether small-scale, experimental tests of spatial density dependence will accurately describe temporal (between-generation) density dependence required for population regulation. Understanding the mechanisms generating density dependence may provide a link between spatial experiments and temporal regulation of populations. In this study, I manipulated the density of recently settled kelp rockfish (Sebastes atrovirens) in both the presence and absence of predators to test for density-dependent mortality and whether predation was the mechanism responsible. I also examined mortality of rockfish cohorts within kelp beds throughout central California to evaluate temporal (between-generation) density dependence in mortality. Experiments suggested that short-term behavioral responses of predators and/or a shortage of prey refuges caused spatial density dependence. Temporal density dependence in mortality was also detected at larger spatial scales for several species of rockfish. It is likely that short-term responses of predators generated both spatial and temporal density dependence in mortality. Spatial experiments that describe the causal mechanisms generating density dependence may therefore be valuable in describing temporal density dependence and population regulation.     

Habitat saturation drives thresholds in stream subsidies

Understanding how abundance regulates the effects of organisms on their ecosystems remains a critical goal of ecology, especially for understanding inter-ecosystem transfers of energy and nutrients. Here we examined how territoriality and nest-digging by anadromous salmon mediate trophic subsidies to stream fishes. Salmon eggs become available for consumption primarily by the digging of salmon that superimpose their nests on previous nests. An individual-based model of spawning salmon predicted that territoriality and habitat saturation produce a nonlinear effect of salmon density on numbers of available eggs to resident predators. Field studies in Alaskan streams found that higher densities of salmon produce disproportionately more eggs in stream drift and in diets of resident fishes (Arctic grayling and rainbow trout). Bioenergetics model simulations indicated that these subsidies produce substantially enhanced growth rates of trout. These results demonstrate that small changes in salmon abundance can drive large changes in subsidies to stream food webs. Thus, the ecological consequences of population declines of keystone species, such as salmon, will be exacerbated when behavior generates nonlinear impacts.     

A stage-structured, Aedes albopictus population model

Aedes albopictus has been the fastest spreading invasive animal species in the world from the mid-1980s until the mid-2000s. In areas it infests, it disrupts native mosquito ecology and can potentially vector up to 21 viruses. To better understand the population dynamics of this species, we created a temperature dependent population model. A stage-structured model was chosen to allow each life-stage to have different temperature dependent mortality and development rates, and each stage was modeled with an ordinary differential equation. Model parameters and distributions were based upon literature values. Initially, a basic model was constructed. This model then had parameters that were forced based upon daily average temperatures. Several criteria were used to evaluate the model, including a comparison to field data from Lubbock, TX. In a stochastic version of the model, a 95% confidence limit contained 70.7% of the field data points. Based upon these results, we feel reasonably confident that we have captured the role of temperature in driving the population dynamics of Ae. albopictus.     

Bayesian modelling of extreme wind speed at Cape Town,South Africa

In the framework of generalized extreme value (GEV) distribution, the frequentist and Bayesian methods have been used to analyse the extremes of annual maxima wind speed recorded by automatic weather stations in Cape Town, Western Cape, South Africa. In the frequentist approach, the GEV distribution parameters were estimated using maximum likelihood, whereas in the Bayesian method the Markov Chain Monte Carlo technique with the Metropolis–Hastings algorithm was used. The results show that the GEV model with trend in the location parameter appears to be a better model for annual maxima data. The paper also discusses a method to construct informative priors empirically using historical data of the underlying process from other weather stations. The results from the Bayesian analysis show that posterior inference might be affected by the choice of priors and hence by the distance between a weather station used to formulate the priors and the point of interest.     

Seed dispersal by pulp consumers, not "legitimate" seed dispersers, increases Guettarda viburnoides population growth

We examined the effect of seed dispersal by Purplish Jays (Cyanocorax cyanomelas; pulp consumers) and the Chestnut-eared Ara?ari (Pteroglossus castanotis; "legitimate" seed dispersers) on population growth of the small tree Guettarda viburnoides (Rubiaceae) in northeastern Bolivian savannas. Because each bird species differs with respect to feeding and post-feeding behavior, we hypothesized that seed dispersal by each species will contribute differently to the rate of increase of G. viburnoides, but that seed dispersal by either species will increase population growth when compared to a scenario with no seed dispersal. To examine the effects of individual dispersers on the future population size of G. viburnoides, we projected population growth rate using demographic models for G. viburnoides that explicitly incorporate data on quantitative and qualitative aspects of seed dispersal by each frugivore species. Our model suggests that seed dispersal by C. cyanomelas leads to positive population growth of G. viburnoides, whereas seed dispersal by P. castanotis has a detrimental effect on the population growth of this species. To our knowledge, this is the first study to report negative effects of a "legitimate" seed disperser on the population dynamics of the plant it consumes. Our results stress the importance of incorporating frugivore effects into population projection matrices, to allow a comprehensive analysis of the effectiveness of different dispersers for plant population dynamics.     

Predicting changes in the Antarctic krill, Euphausia superba, population at South Georgia

Variability in the Southern Ocean is frequently reflected in changes in the abundance of Antarctic krill Euphausia superba and subsequent effects on dependent predators. However, the nature and consequences of changes in krill population dynamics that accompany fluctuations in its abundance are essentially unknown. A conceptual model, developed from quantitative measures of krill length in the diet of predators at South Georgia from 1991 to 1997, allowed predictions to be made about the abundance and population structure of krill in 1998 and the consequences for predators. Consistent with model predictions, in 1998 there was a serial change in krill population structure, low krill biomass and low reproductive performance of predators. The change in the modal size of krill, from 56 mm in December to 42 mm in March, was apparently a result of the transport of krill into the region. This is the first occasion when the future status and structure of the krill population at South Georgia has been successfully predicted. By representing local krill population dynamics, which may also reflect large-scale physical and biological processes, predators have a potential key role as indicators of environmental variation in the Southern Ocean at a range of spatial scales. Received: 6 March 1999 / Accepted: 3 September 1999     

A Bayesian approach to line-transect analysis for estimating abundance

Line-transect analysis is a widely used method of estimating plant and animal density and abundance. A Bayesian approach to a basic line-transect analysis is developed for a half-normal detection function. We extend the model of Karunamuni and Quinn [Karunamuni, R.J., Quinn II, T.J., 1995. Bayesian estimation of animal abundance for line-transect sampling. Biometrics 51, 1325–1337] by including a binomial likelihood function for the number of objects detected. The method computes a joint posterior distribution on the effective strip width and the density of objects in the sampled area. Analytical and computational methods for binned and unbinned perpendicular distance data are provided. Existing information about effective strip width and density can be brought into the analysis via prior distributions. The Bayesian approach is compared to a standard line-transect analysis using both real and simulated data. Results of the Bayesian and non-Bayesian analyses are similar when there are no prior data on effective strip width or density, but the Bayesian approach performs better when such data are available from previous or related studies. Practical methods for including prior data on effective strip width and density are suggested. A numerical example shows how the Bayesian approach can provide valid estimates when the sample size is too small for the standard approach to work reliably. The proposed Bayesian approach can form the basis for developing more advanced analyses.     

Fitting animal survival models with temporal random effects

Estimating temporal variance in animal demographic parameters is of particular importance in population biology. We implement the Schall’s algorithm for incorporating temporal random effects in survival models using recovery data. Our frequentist approach is based on a formulation of band-recovery models with random effects as generalized linear mixed models and a linearization of the link function conditional on the random effects. A simulation study shows that our procedure provides unbiased and precise estimates. The method is then implemented on two case studies using recovery data on fish and birds.     

Effects of temperature variation on TSD in turtle (C. picta) populations

We formulate a two-sex model of temperature-dependent sex determination (TSD) for a freshwater turtle (C. picta) population. The aim is to understand how environmental temperature variations and nest heat conduction properties affect the long term dynamics of the population. This is a key to understanding how global temperature changes may affect their survival. With stochastic inputs of ambient temperature and solar radiation, the model uses the heat equation to determine the temperature in the egg layer in the nest; in turn, this determines the sex ratio in the egg clutch using a variable degree-day model. Finally, a nonlinear Leslie type, stage-based, two-sex model, is used to determine the long term male and female populations. A two-sex model is required because of different development rates for males and females. The model is flexible enough to enable other researchers to examine the effects of temperature variation variations on other species with TSD, e.g., crocodilians, reptilians, as well as other turtle species. It can be adapted to study effects of nest location, soil type, rain events, different incubation periods, and density effects, for example, the dependence of the mating function on the ratio of males to females and each’s contribution to the sex of hatchlings. Modifications can be easily made to fit a specific life history traits. The model is a beginning step in understanding the long term, high fitness shown by many reptile species with TSD, and it may suggest to experimentalists what data may be relevant to these issues; it can also be useful to wildlife managers in developing strategies for intervention if needed. Among the principal findings are that temperature variability and detailed nest heat conduction properties may buffer projected negative effects on a population.     

Invasion, Disturbance, and Competition: Modeling the Fate of Coastal Plant Populations

Abstract:  Wetland habitats are besieged by biotic and abiotic disturbances such as invasive species, hurricanes, habitat fragmentation, and salinization. Predicting how these factors will alter local population dynamics and community structure is a monumental challenge. By examining ecologically similar congeners, such as Iris hexagona and I. pseudacorus (which reproduce clonally and sexually and tolerate a wide range of environmental conditions), one can identify life-history traits that are most influential to population growth and viability. We combined empirical data and stage-structured matrix models to investigate the demographic responses of native ( I. hexagona ) and invasive ( I. pseudacorus ) plant populations to hurricanes and salinity stress in freshwater and brackish wetlands. In our models I. hexagona and I. pseudacorus responded differently to salinity stress, and species coexistence was rare. In 82% of computer simulations of freshwater marsh, invasive iris populations excluded the native species within 50 years, whereas native populations excluded the invasive species in 99% of the simulations in brackish marsh. The occurrence of hurricanes allowed the species to coexist, and species persistence was determined by the length of time it took the ecosystem to recover. Rapid recovery (2 years) favored the invasive species, whereas gradual recovery (30 years) favored the native species. Little is known about the effects of hurricanes on competitive interactions between native and invasive plant species in marsh ecosystems. Our models contribute new insight into the relationship between environmental disturbance and invasion and demonstrate how influential abiotic factors such as climate change will be in determining interspecific interactions.