Variance and Uncertainty in the Expected Number of Occurrences in Reserve Selection

Abstract:  Reserve selection often concerns the design of reserve networks for the long-term maintenance of biodiversity. We considered uncertainty in the context of three common reserve-selection formulations, the expected number of populations, proportional coverage of land-cover types, and the probability of having at least one population. By uncertainty, we mean variance in the outcome of any probability-based reserve selection formulation. A typical reserve-selection formulation might ask for the least expensive set of sites that contains n populations per species. It is implicit here that this requirement concerns the expected number of populations, which actually is obtained only with a 50% chance. If the requirement is changed to select the least expensive set of sites that gives n populations per species with a 95% probability, the number of sites required in the solution increases and the identity of the sites is changed toward sites that have high probabilities of persistence (or occurrence) and low associated binomial variance. Anthropogenic threat is one factor that may cause probabilistic uncertainty in the context of proportional area coverage.     

Life-history analysis in “physiological” compared with “sidereal” time: An example with an amphipod (Gammarus lawrencianus) in a varying environment

The multivoltine, estuarine amphipodGammarus lawrencianus has four generations per year in an environment where temperatures range seasonally from –1° to 25°C. Temperature-response curves for rates of brood production and development were determined by laboratory experiments and field observation. The life history and population dynamics were observed over a full annual cycle (1981) for a field population located at Rocky Run, Porter's Lake, Nova Scotia, Canada. On a natural (i.e., sidereal) time scale, the generations appear to have very different life histories: the two summer generations have short lives, rapid development and mature at small size (classicr-selection), whereas the overwintering generations have relatively low rates of mortality, slow development and mature at large size (classicK-selection). This pattern (larger size at maturity at lower temperatures) is widespread in aquatic poikilotherms. Similar life-history differences are evident among cohorts of the summer generations that mature at different temperatures. When time is expressed on a physiological scale that removes the effect of temperature on embryonic development and reproductive rate, the variation within and among generations is greatly reduced. In particular, an apparent alternation betweenr- andK-selection largely disappears. Because the generations are temporally isolated, it might be surmised that natural selection acting on the summer generations might antagonize the effects of natural selection acting on the fall and winter generations. However, the scaling of the rates of development, maturation, growth, reproduction and mortality on the physiological time scale derived from the temperature dependence of development and reproductive rate gives a very different and more homogeneous pattern.     

Using expert knowledge to support Endangered Species Act decision-making for data-deficient species

Many questions relevant to conservation decision-making are characterized by extreme uncertainty due to lack of empirical data and complexity of the underlying ecologic processes, leading to a rapid increase in the use of structured protocols to elicit expert knowledge. Published ecologic applications often employ a modified Delphi method, where experts provide judgments anonymously and mathematical aggregation techniques are used to combine judgments. The Sheffield elicitation framework (SHELF) differs in its behavioral approach to synthesizing individual judgments into a fully specified probability distribution for an unknown quantity. We used the SHELF protocol remotely to assess extinction risk of three subterranean aquatic species that are being considered for listing under the U.S. Endangered Species Act. We provided experts an empirical threat assessment for each known locality over a video conference and recorded judgments on the probability of population persistence over four generations with online submission forms and R-shiny apps available through the SHELF package. Despite large uncertainty for all populations, there were key differences between species’ risk of extirpation based on spatial variation in dominant threats, local land use and management practices, and species’ microhabitat. The resulting probability distributions provided decision makers with a full picture of uncertainty that was consistent with the probabilistic nature of risk assessments. Discussion among experts during SHELF's behavioral aggregation stage clearly documented dominant threats (e.g., development, timber harvest, animal agriculture, and cave visitation) and their interactions with local cave geology and species’ habitat. Our virtual implementation of the SHELF protocol demonstrated the flexibility of the approach for conservation applications operating on budgets and time lines that can limit in-person meetings of geographically dispersed experts.     

How certain are we about the certainty-equivalent long term social discount rate?

Theoretical arguments for using a term structure of social discount rates (SDR) that declines with the time horizon have influenced government guidelines in the US and Europe. The certainty equivalent discount rate that often underpins this guidance embodies uncertainty in the primitives of the SDR, such as growth. For distant time horizons the probability distributions of these primitives are ambiguous and the certainty equivalent itself is uncertain. Yet, if a limited set of characteristics of the unknown probability distributions can be agreed upon, ‘sharp’ upper and lower bounds can be defined for the certainty-equivalent SDR. Unfortunately, even with considerable agreement on these features, these bounds are widely spread for horizons beyond 75 years. So while estimates of the present value of intergenerational impacts, including the social cost of carbon, can be bounded in the presence of this ambiguity, they typically remain so imprecise as to provide little practical guidance.     

Spatial selection and inheritance: applying evolutionary concepts to population dynamics in heterogeneous space

Organisms in highly suitable sites generally produce more offspring, and offspring can inherit this suitability by not dispersing far. This combination of spatial selection and spatial inheritance acts to bias the distribution of organisms toward suitable sites and thereby increase mean fitness (i.e., per capita population increase). Thus, population growth rates in heterogeneous space change over time by a process conceptually analogous to evolution by natural selection, opening avenues for theoretical cross-pollination between evolutionary biology and ecology. We operationally define spatial inheritance and spatial selective differential and then combine these two factors in a modification of the breeder's equation, derived from simple models of population growth in heterogeneous space. The modified breeder's equation yields a conservative criterion for persistence in hostile environments estimable from field measurements. We apply this framework for understanding gypsy moth population persistence amidst abundant predators and find that the predictions of the modified breeder's equation match initial changes in population growth rate in independent simulation output. The analogy between spatial dynamics and natural selection conceptually links ecology and evolution, provides a spatially implicit framework for modeling spatial population dynamics, and represents an important null model for studying habitat selection.     

Fitness Decline under Relaxed Selection in Captive Populations

Abstract: We compared life-history schedules among populations of the housefly (  Musca domestica L.) maintained in the laboratory under curtailed life span, such that selection on mutations that affected only late-life fitness traits was reduced. As a result of this regime, late-life ( but not early-life) fecundity declined within a few generations. The results suggest that if captive populations are maintained with minimal selection, either by direct manipulation of the environment or by equalizing family contributions, the increased frequency of potentially deleterious mutations may rapidly lower the ability of these populations to exist under natural conditions. This would be independent of population size, so expanding captive populations would not alleviate potential fitness reductions due to relaxed selection.     

Reproductive plasticity,oviposition site selection,and maternal effects in fragmented landscapes

Traditionally, evolutionary ecology and conservation biology have primarily been concerned with how environmental changes affect population size and genetic diversity. Recently, however, there has been a growing realization that phenotypic plasticity can have important consequences for the probability of population persistence, population growth, and evolution during rapid environmental change. Habitat fragmentation due to human activities is dramatically changing the ecological conditions of life for many organisms. In this review, we use examples from the literature to demonstrate that habitat fragmentation has important consequences on oviposition site selection in insects, with carryover effects on offspring survival and, therefore, population dynamics. We argue that plasticity in oviposition site selection and maternal effects on offspring phenotypes may be an important, yet underexplored, mechanism by which environmental conditions have consequences across generations. Without considering the impact of habitat fragmentation on oviposition site selection, it will be difficult to assess the effect of fragmentation on offspring fitness, and ultimately to understand the impact of anthropogenic-induced environmental change on population viability.     

Context-dependent effects of nestling growth trajectories on recruitment probability in the collared flycatcher

Nestling growth is known as an important determinant of fitness in altricial birds, but its evolutionary potential has been debated, and little is known about detailed patterns of current selection on growth. Relationships are often reported between nestling growth and attributes of nestlings and parents, but the interpretation of these depends on the advantages a given growth difference confers to the chicks. Increased growth may have positive, negative or context-dependent effects on offspring fitness, but these effects are largely unknown in natural populations. We measured growth trajectories of body mass in fostered broods of collared flycatchers (Ficedula albicollis) in 3 years of contrasting food conditions. We examined the growth of young and their recruitment probability to the breeding population in relation to year quality, hatching rank, sex, paternal age and paternal attractiveness. We also looked at the interactive effects of growth and intrinsic offspring attributes on recruitment probability. The predictors of nestling growth and those of recruitment did not agree. Moreover, the recruitment consequences of a given nestling growth rate were significantly influenced by nestling rank and paternal ornamentation. Differential recruitment effects of nestling growth in relation to parental traits and nestling attributes suggest that using growth as a generally applicable measure of nestling quality may not be justified. These findings also have implications for morphological evolution and the indicator value of sexual signals.     

Emerging prion disease drives host selection in a wildlife population

Infectious diseases are increasingly recognized as an important force driving population dynamics, conservation biology, and natural selection in wildlife populations. Infectious agents have been implicated in the decline of small or endangered populations and may act to constrain population size, distribution, growth rates, or migration patterns. Further, diseases may provide selective pressures that shape the genetic diversity of populations or species. Thus, understanding disease dynamics and selective pressures from pathogens is crucial to understanding population processes, managing wildlife diseases, and conserving biological diversity. There is ample evidence that variation in the prion protein gene (PRNP) impacts host susceptibility to prion diseases. Still, little is known about how genetic differences might influence natural selection within wildlife populations. Here we link genetic variation with differential susceptibility of white-tailed deer to chronic wasting disease (CWD), with implications for fitness and disease-driven genetic selection. We developed a single nucleotide polymorphism (SNP) assay to efficiently genotype deer at the locus of interest (in the 96th codon of the PRNP gene). Then, using a Bayesian modeling approach, we found that the more susceptible genotype had over four times greater risk of CWD infection; and, once infected, deer with the resistant genotype survived 49% longer (8.25 more months). We used these epidemiological parameters in a multi-stage population matrix model to evaluate relative fitness based on genotype-specific population growth rates. The differences in disease infection and mortality rates allowed genetically resistant deer to achieve higher population growth and obtain a long-term fitness advantage, which translated into a selection coefficient of over 1% favoring the CWD-resistant genotype. This selective pressure suggests that the resistant allele could become dominant in the population within an evolutionarily short time frame. Our work provides a rare example of a quantifiable disease-driven selection process in a wildlife population, demonstrating the potential for infectious diseases to alter host populations. This will have direct bearing on the epidemiology, dynamics, and future trends in CWD transmission and spread. Understanding genotype-specific epidemiology will improve predictive models and inform management strategies for CWD-affected cervid populations.     

Development of a probabilistic blood lead prediction model

A partially probabilistic blood lead prediction model has been developed, based on the US Environmental Protection Agency integrated exposure-uptake-biokinetic blood lead model (IEUBK model). This study translated the IEUBK model into a spreadsheet format. The uptake submodel incorporates uncertainty distributions for exposure and bioavailability parameters. The biokinetic submodel is duplicated with a table incorporating partitioning and decay of lead levels in the body. As a case study, the probabilistic model is applied to a lead exposure scenario involving a former smelter site in Sandy, Utah. The probabilistic model produces less biased estimates of means and standard deviations than the deterministic model. Parameter uncertainty is propagated in the model by the use of Monte Carlo simulation. Thus, sensitivity analysis is possible, and driving variables can be determined.     

Berücksichtigung von Variabilität und Unsicherheit in quantitativen Risikoabschätzungen (QRA)

Using the example of residential living on a contaminated site, a probabilistic exposure assessment is performed with variability and uncertainty being modelled separately. Probability distributions are used in the exposure model in order to characterize person-related variables (e.g. body weight) only; chemical-specific parameters are being held constant. In addition, uncertainty concerning one selected variable (soil ingestion rate) was modelled. Comparing these results to conventional “worst case” estimates, we find those estimates located in the uppermost range of the probabilistic estimates. The worst case estimates tend to be highly conservative and possibly unrealistic.     

Sex ratio and host size in a parasitoid wasp

Summary Parasitoid wasps often lay male eggs in small hosts and female eggs in larger hosts. The selective advantage of this strategy can be explained by assuming wasp fitness increases with host size and that this fitness increase is greater in females than in males. I conducted experiments to test a model based on this explanation and found the results generally supported the model with one exception; unlike what the model assumed, these wasps were unable to adjust their offspring sex ratios in each generation to different host size distributions. This finding suggests an alternate view as to how selection might operate in the evolution of parasitoid sex ratios.     

Demographic heterogeneity, cohort selection, and population growth

Demographic heterogeneity--variation among individuals in survival and reproduction--is ubiquitous in natural populations. Structured population models address heterogeneity due to age, size, or major developmental stages. However, other important sources of demographic heterogeneity, such as genetic variation, spatial heterogeneity in the environment, maternal effects, and differential exposure to stressors, are often not easily measured and hence are modeled as stochasticity. Recent research has elucidated the role of demographic heterogeneity in changing the magnitude of demographic stochasticity in small populations. Here we demonstrate a previously unrecognized effect: heterogeneous survival in long-lived species can increase the long-term growth rate in populations of any size. We illustrate this result using simple models in which each individual's annual survival rate is independent of age but survival may differ among individuals within a cohort. Similar models, but with nonoverlapping generations, have been extensively studied by demographers, who showed that, because the more "frail" individuals are more likely to die at a young age, the average survival rate of the cohort increases with age. Within ecology and evolution, this phenomenon of "cohort selection" is increasingly appreciated as a confounding factor in studies of senescence. We show that, when placed in a population model with overlapping generations, this heterogeneity also causes the asymptotic population growth rate lambda to increase, relative to a homogeneous population with the same mean survival rate at birth. The increase occurs because, even integrating over all the cohorts in the population, the population becomes increasingly dominated by the more robust individuals. The growth rate increases monotonically with the variance in survival rates, and the effect can be substantial, easily doubling the growth rate of slow-growing populations. Correlations between parent and offspring phenotype change the magnitude of the increase in lambda, but the increase occurs even for negative parent-offspring correlations. The effect of heterogeneity in reproductive rate on lambda is quite different: growth rate increases with reproductive heterogeneity for positive parent-offspring correlation but decreases for negative parent-offspring correlation. These effects of demographic heterogeneity on lambda have important implications for population dynamics, population viability analysis, and evolution.     

Resolving structural uncertainty in natural resources management using POMDP approaches

In recent years there has been a growing focus on the uncertainties of natural resources management, and the importance of accounting for uncertainty in assessing management effectiveness. This paper focuses on uncertainty in resource management in terms of discrete-state Markov decision processes (MDP) under structural uncertainty and partial observability. It describes the treatment of structural uncertainty with approaches developed for partially observable resource systems. In particular, I show how value iteration for partially observable MDPs (POMDP) can be extended to structurally uncertain MDPs. A key difference between these process classes is that structurally uncertain MDPs require the tracking of system state as well as a probability structure for the structure uncertainty, whereas with POMDPs require only a probability structure for the observation uncertainty. The added complexity of the optimization problem under structural uncertainty is compensated by reduced dimensionality in the search for optimal strategy. A solution algorithm for structurally uncertain processes is outlined for a simple example in conservation biology. By building on the conceptual framework developed for POMDPs, natural resource analysts and decision makers who confront structural uncertainties in natural resources can take advantage of the rapid growth in POMDP methods and approaches, and thereby produce better conservation strategies over a larger class of resource problems.     

Relationship between Population Size and Fitness

Abstract:  Long-term effective population size, which determines rates of inbreeding, is correlated with population fitness. Fitness, in turn, influences population persistence. I synthesized data from the literature concerning the effects of population size on population fitness in natural populations of plants to determine how large populations must be to maintain levels of fitness that will provide adequate protection against environmental perturbations that can cause extinction. Integral to this comment on what has been done and what needs to be done, sThe evidence suggests that there is a linear relationship between log population size and population fitness over the range of population sizes examined. More importantly, populations will have to be maintained at sizes of >2000 individuals to maintain population fitness at levels compatible with the conservation goal of long-term persistence. This approach to estimating minimum viable population size provides estimates that are in general agreement with those from numerous other studies and strengthens the argument that conservation efforts should ultimately aim at maintaining populations of several thousand individuals to ensure long-term persistence.     

Prediction of the upper tail of concentration distributions of a continuous point source release in urban environments

The peak values observed in a measured concentration time series of a dispersing gaseous pollutant released continuously from a point source in urban environments, and the hazard level associated with them, demonstrate the necessity of predicting the upper tail of concentration distributions. For the prediction of concentration distributions statistical models are preferably employed which provide information about the probability of occurrence. In this paper a concentration database pertaining to a field experiment is used for the selection of the statistical distribution. The inverses of the gamma cumulative distribution function (cdf) for 75th–99th percentiles of concentration are found to be more consistent with the experimental data than those of the log-normal distribution. The experimental values have been derived from measured high frequency time series by sorting first the concentrations and then finding the concentration which corresponds to each probability. Then the concentration mean and variance that are predicted with Computational Fluid Dynamics-Reynolds Averaged Navier–Stokes (RANS) methodology are used to construct the gamma distribution. The proposed model (“RANS-gamma”) is included in the framework of a computational code (ADREA-HF) suitable for simulating the dispersion of airborne pollutants over complex geometries. The methodology is validated by comparing the inverses of the model cdfs with the observed ones from two wind tunnel experiments. The evaluation is performed in the form of validation metrics such as the fractional bias, the normalized mean square error and the factor-of-two percentage. From the above comparisons it is concluded that the overall model performance for the present cases is satisfactory.     

Building statistical models to analyze species distributions.

Models of the geographic distributions of species have wide application in ecology. But the nonspatial, single-level, regression models that ecologists have often employed do not deal with problems of irregular sampling intensity or spatial dependence, and do not adequately quantify uncertainty. We show here how to build statistical models that can handle these features of spatial prediction and provide richer, more powerful inference about species niche relations, distributions, and the effects of human disturbance. We begin with a familiar generalized linear model and build in additional features, including spatial random effects and hierarchical levels. Since these models are fully specified statistical models, we show that it is possible to add complexity without sacrificing interpretability. This step-by-step approach, together with attached code that implements a simple, spatially explicit, regression model, is structured to facilitate self-teaching. All models are developed in a Bayesian framework. We assess the performance of the models by using them to predict the distributions of two plant species (Proteaceae) from South Africa's Cape Floristic Region. We demonstrate that making distribution models spatially explicit can be essential for accurately characterizing the environmental response of species, predicting their probability of occurrence, and assessing uncertainty in the model results. Adding hierarchical levels to the models has further advantages in allowing human transformation of the landscape to be taken into account, as well as additional features of the sampling process.     

Testing the evolution of increased competitive ability (EICA) hypothesis in a novel framework

The "evolution of increased competitive ability" (EICA) hypothesis proposes that escape from natural enemies, e.g., after transcontinental introductions, alters the selection regime because costly defenses no longer enhance fitness. Such an evolutionary loss of defenses enables resources to be directed toward growth or other traits improving performance. We tested the EICA hypothesis in a novel framework in which the natural enemy is the traveler that follows its widespread host by accidental or deliberate (biocontrol) introductions. In a greenhouse experiment we used populations of Senecio vulgaris from North America, Europe, and Australia that differ in the history of exposure to the rust fungus Puccinia lagenophorae. Contrary to what is predicted by EICA, we found no evidence for increased levels of resistance to the rust fungus in plant populations with a longer history of rust exposure (Australia). Similarly, there was no evidence for reduced fecundity in these populations, although vegetative vigor, measured as secondary branching and growth rate, was lower. The maintenance of high rust resistance in populations with no (North America) or only a short history (Europe) of rust exposure is surprising given that resistance seems to incur considerable fitness costs, as indicated by the negative association between family mean resistance and fitness in the absence of disease observed for all three continents. The comparison of population differentiation in quantitative traits with estimates of differentiation in amplified fragment length polymorphic (AFLP) markers suggests that a number of fitness-related traits are under divergent selection among the studied populations. The proposed framework to test changes in the evolutionary trajectory underlying EICA can be employed in an expanded range of systems. These may include investigations on a cosmopolitan weed or crop when an antagonist is expanding its geographic range (such as our study), studies along a chronosequence of introduction time with expected increasing accumulation of natural enemies over time, or comparisons between introduced plant populations that differ in exposure time to biocontrol organisms.     

Representing the effects of diurnal variations in light on primary production on a seasonal time-scale

Rates of production by primary producers are strongly affected by light. Diurnal variations in irradiance produce characteristic diurnal patterns in primary production and respiration. In cases in which the processes of interest occur on a longer time-scale, it is not uncommon to ignore diurnal variations and use mean daily irradiance to force models of primary producers. This simplification reduces the computer processing time and input data requirements of models. However, this approach results in an error which may be significant if irradiance during part of the day is higher than that needed for growth at the maximum possible rate. Here, an alternative approach is presented and applied to a simulation of phytoplankton in a shallow lake. The model uses a semi-analytical calculation of mean daily growth rates that takes into account periods of supersaturation with respect to light to deliver results that conform closely to those of a model using hourly time-steps and irradiance forcing, but with a substantially smaller computational cost.