Species distribution models and ecological theory: A critical assessment and some possible new approaches

Given the importance of knowledge of species distribution for conservation and climate change management, continuous and progressive evaluation of the statistical models predicting species distributions is necessary. Current models are evaluated in terms of ecological theory used, the data model accepted and the statistical methods applied. Focus is restricted to Generalised Linear Models (GLM) and Generalised Additive Models (GAM). Certain currently unused regression methods are reviewed for their possible application to species modelling.A review of recent papers suggests that ecological theory is rarely explicitly considered. Current theory and results support species responses to environmental variables to be unimodal and often skewed though process-based theory is often lacking. Many studies fail to test for unimodal or skewed responses and straight-line relationships are often fitted without justification.Data resolution (size of sampling unit) determines the nature of the environmental niche models that can be fitted. A synthesis of differing ecophysiological ideas and the use of biophysical processes models could improve the selection of predictor variables. A better conceptual framework is needed for selecting variables.Comparison of statistical methods is difficult. Predictive success is insufficient and a test of ecological realism is also needed. Evaluation of methods needs artificial data, as there is no knowledge about the true relationships between variables for field data. However, use of artificial data is limited by lack of comprehensive theory.Three potentially new methods are reviewed. Quantile regression (QR) has potential and a strong theoretical justification in Liebig's law of the minimum. Structural equation modelling (SEM) has an appealing conceptual framework for testing causality but has problems with curvilinear relationships. Geographically weighted regression (GWR) intended to examine spatial non-stationarity of ecological processes requires further evaluation before being used.Synthesis and applications: explicit theory needs to be incorporated into species response models used in conservation. For example, testing for unimodal skewed responses should be a routine procedure. Clear statements of the ecological theory used, the nature of the data model and sufficient details of the statistical method are needed for current models to be evaluated. New statistical methods need to be evaluated for compatibility with ecological theory before use in applied ecology. Some recent work with artificial data suggests the combination of ecological knowledge and statistical skill is more important than the precise statistical method used. The potential exists for a synthesis of current species modelling approaches based on their differing ecological insights not their methodology.     

Population Variability and Extinction Risk

Abstract: Population models generally predict increased extinction risk (ER) with increased population variability (  PV  ), yet some empirical tests have provided contradictory findings. We resolve this conflict by attributing negative measured relationships to a statistical artifact that arises because PV tends to be underestimated for populations with short persistence. Such populations do not go extinct quickly as a consequence of low intrinsic variability; instead, the measured variability is low because they go extinct so quickly. Consequently, any underlying positive relationship between PV and ER tends to be obscured. We conducted a series of analyses to evaluate this claim. Simulations showed that negative measured relationships are to be expected, despite an underlying positive relationship. Simulations also identified properties of data, minimizing this bias and thereby permitting meaningful analysis. Experimental data on laboratory populations of a bruchid beetle (Callosobruchus maculatus) supported the simulation results. Likewise, with an appropriate statistical approach (Cox regression on untransformed data), reanalysis of a controversial data set on British island bird populations revealed a significant positive association between PV and ER (p = 0.03). Finally, a similar analysis of time series for naturally regulated animal populations revealed a positive association between PV and quasiextinction risk (p < 0.01). Without exception, our simulation results, experimental findings, reanalysis of published data, and analysis of quasiextinction risk all contradict previous reports of negative or equivocal relationships. Valid analysis of meaningful data provides strong evidence that increased population variability leads to increased extinction risk.     

Validating graph-based connectivity models with independent presence–absence and genetic data sets

Habitat connectivity is a key objective of current conservation policies and is commonly modeled by landscape graphs (i.e., sets of habitat patches [nodes] connected by potential dispersal paths [links]). These graphs are often built based on expert opinion or species distribution models (SDMs) and therefore lack empirical validation from data more closely reflecting functional connectivity. Accordingly, we tested whether landscape graphs reflect how habitat connectivity influences gene flow, which is one of the main ecoevolutionary processes. To that purpose, we modeled the habitat network of a forest bird (plumbeous warbler [Setophaga plumbea]) on Guadeloupe with graphs based on expert opinion, Jacobs’ specialization indices, and an SDM. We used genetic data (712 birds from 27 populations) to compute local genetic indices and pairwise genetic distances. Finally, we assessed the relationships between genetic distances or indices and cost distances or connectivity metrics with maximum-likelihood population-effects distance models and Spearman correlations between metrics. Overall, the landscape graphs reliably reflected the influence of connectivity on population genetic structure; validation R² was up to 0.30 and correlation coefficients were up to 0.71. Yet, the relationship among graph ecological relevance, data requirements, and construction and analysis methods was not straightforward because the graph based on the most complex construction method (species distribution modeling) sometimes had less ecological relevance than the others. Cross-validation methods and sensitivity analyzes allowed us to make the advantages and limitations of each construction method spatially explicit. We confirmed the relevance of landscape graphs for conservation modeling but recommend a case-specific consideration of the cost-effectiveness of their construction methods. We hope the replication of independent validation approaches across species and landscapes will strengthen the ecological relevance of connectivity models.     

Bayesian hierarchical mixture models for otolith microchemistry analysis

The field of fisheries research commonly uses classical statistical classification methods to estimate the proportion of fish that return to natal spawning grounds to spawn. With the advent of otolith microchemical analysis, researchers are able to extract information from fish ear stones (otoliths) about the chemical composition of water in which fish have spent distinct periods of their lives. Here we present a method of analysis set in the Bayesian statistical paradigm which enables explicit incorporation of habitat information into the analysis. The ecological system is seen as arising from a mixture of disparate fish populations and information from the biological relationships inherent in otolith formation is exploited through the hierarchical model structure. We present the model and motivation, demonstrate the validity of the model through simulation studies, and conclude with an analysis of a data set from Lake Erie.     

Using circuit theory to model connectivity in ecology, evolution, and conservation

Connectivity among populations and habitats is important for a wide range of ecological processes. Understanding, preserving, and restoring connectivity in complex landscapes requires connectivity models and metrics that are reliable, efficient, and process based. We introduce a new class of ecological connectivity models based in electrical circuit theory. Although they have been applied in other disciplines, circuit-theoretic connectivity models are new to ecology. They offer distinct advantages over common analytic connectivity models, including a theoretical basis in random walk theory and an ability to evaluate contributions of multiple dispersal pathways. Resistance, current, and voltage calculated across graphs or raster grids can be related to ecological processes (such as individual movement and gene flow) that occur across large population networks or landscapes. Efficient algorithms can quickly solve networks with millions of nodes, or landscapes with millions of raster cells. Here we review basic circuit theory, discuss relationships between circuit and random walk theories, and describe applications in ecology, evolution, and conservation. We provide examples of how circuit models can be used to predict movement patterns and fates of random walkers in complex landscapes and to identify important habitat patches and movement corridors for conservation planning.     

Ecological Modelling by ‘Ecological Modelling'

On the basis of a statistical analysis of the papers published in Ecological Modelling from 1975 to 1996, it has been attempted to examine the development of the field of ecological modelling and systems ecology. It was found that while models of aquatic ecosystems and management issues were more in focus in the 1970s, terresterial ecosystems and ecological theory have gained attention during the 1990s. Interest in ecotoxicological models seems to have increased only slightly during the entire period. It is interesting that financial support for the area of ecological modelling and systems ecology is closely related to the number of publications, which can be seen from the number of papers published by various countries in the journal. Hopefully, this clear message can be used to show to politicians that these relationships are real.     

Using the negative binomial distribution to model overdispersion in ecological count data

A Poisson process is a commonly used starting point for modeling stochastic variation of ecological count data around a theoretical expectation. However, data typically show more variation than implied by the Poisson distribution. Such overdispersion is often accounted for by using models with different assumptions about how the variance changes with the expectation. The choice of these assumptions can naturally have apparent consequences for statistical inference. We propose a parameterization of the negative binomial distribution, where two overdispersion parameters are introduced to allow for various quadratic mean-variance relationships, including the ones assumed in the most commonly used approaches. Using bird migration as an example, we present hypothetical scenarios on how overdispersion can arise due to sampling, flocking behavior or aggregation, environmental variability, or combinations of these factors. For all considered scenarios, mean-variance relationships can be appropriately described by the negative binomial distribution with two overdispersion parameters. To illustrate, we apply the model to empirical migration data with a high level of overdispersion, gaining clearly different model fits with different assumptions about mean-variance relationships. The proposed framework can be a useful approximation for modeling marginal distributions of independent count data in likelihood-based analyses.     

New approaches to modelling fish-habitat relationships

Ecologists often develop models that describe the relationship between faunal communities and their habitat. Coral reef fishes have been the focus of numerous such studies, which have used a wide range of statistical tools to answer an equally wide range of questions. Here, we apply a series of both conventional statistical techniques (linear and generalized additive regression models) and novel machine-learning techniques (the support vector machine and three ensemble techniques used with regression trees) to predict fish species richness, biomass, and diversity from a range of habitat variables. We compare the techniques in terms of their predictive performance, and we compare a subset of the models in terms of the influence each habitat variable has for the predictions. Prediction errors are estimated by cross-validation, and variable importance is assessed using permutations of individual variable values. For predictions of species richness and diversity the tree-based models generally and the random forest model specifically are superior (produce the lowest errors). These model types are all able to model both nonlinear and interaction effects. The linear model, unable to model either effect type, performs the worst (produces the highest errors). For predictions of biomass, the generalized additive model is superior, and the support vector machine performs the worst. Depth range, the difference between maximum and minimum water depth at a given site, is identified as the most important variable in the majority of models predicting the three fish community variables. However, variable importance is highly dependent upon model type, which leads to questions regarding the interpretation of variable importance and its proper use as an indicator of causality. The representation of ecological relationships by tree-based ensemble learners will improve predictive performance, and provide a new avenue for exploring ecological relationships, both statistical and causal.     

Cross-correlation bias in lag analysis of aquatic time series

Cross-correlation analysis is the most valuable and widely used statistical tool for evaluating the strength and direction of time-lagged relationships between ecological variables. Although it is well understood that temporal autocorrelation can inflate estimates of cross correlations and cause high rates of incorrectly concluding that lags exist among time series (i.e. type I error), in this study we show that a problem we term intra-multiplicity can cause substantial bias in cross-correlation analysis even in the absence of autocorrelation. Intra-multiplicity refers to the numerous time lags examined and cross-correlation coefficients computed within a pair of time series during cross-correlation analysis. We show using Monte Carlo simulations that intra-multiplicity can spuriously inflate estimates of cross correlations by identifying incorrect time lags. Further, unlike autocorrelation, which generally identifies lags close to the true lag, intra-multiplicity can erroneously identify lags anywhere in the time series and commonly results in a direction change of the correlation (i.e. positive or negative). Using Monte Carlo simulations we develop formulas that quantify the bias introduced by intra-multiplicity as a function of sample size, true cross correlation between the series, and the number of time lags examined. A priori these formulas enable researchers to determine the sample size needed to minimize the biases introduced by intra-multiplicity. A posteriori the formulas can be used to predict the expected bias and type I error rate associated with the data at hand, as well as the maximum number of time lags that can be analyzed to minimize the effects of intra-multiplicity. We examine the relationship between commercial catch of chum salmon and surface temperatures of the North Pacific (1925–1992) to illustrate the problems of intra-multiplicity in fisheries studies and the application of our formulas. These analyses provide a more robust framework to assess the temporal relationships between ecological variables. Received: 28 July 2000 / Accepted: 6 December 2000     

A power-efficiency trade-off in resource use alters epidemiological relationships

Trade-offs play pivotal roles in the ecology and evolution of natural populations. However, trade-offs are probably not static, invariant relationships. Instead, ecological factors can shift, alter, or reverse the relationships underlying trade-offs and create critical genotype x environment (G x E) interactions. But which ecological factors alter trade-offs or create G x E interactions, and why (mechanistically) do they do this? We tackle these questions using resource quality as the central ecological factor and a case study of disease in the plankton. We show that clonal genotypes of a zooplankton host (Daphnia dentifera) exhibit a "power-efficiency" trade-off in resource use, where powerful (fast-feeding) host clones perform well on richer algal resources, but more efficient (slow-feeding) clones perform relatively well on poorer resources. This resource-based trade-off then influences epidemiological relationships due to fundamental connections between resources and fecundity, transmission rate (an index of resistance), and replication of a virulent fungal parasite (Metschnikowia bicuspidata) within hosts. For instance, using experiments and dynamic energy budget models, we show that the power-efficiency trade-off overturned a previously detected trade-off between fecundity and transmission risk of hosts to this parasite. When poorer resources were eaten, transmission risk and fecundity were negatively, not positively, correlated. Additionally, poor resource quality changed positive relationships between yield of infectious stages (spores) and host fecundity: those fecundity-spore relationships with poor food became negative or nonsignificant. Finally, the power-efficiency trade-off set up an interaction between host clone and resource quality for yield of fungal spores: powerful clones yielded relatively more spores on the better resource, while efficient clones yielded relatively more on the poorer resource. Thus, the physiological ecology of resource use can offer potent, mechanistic insight linking environmental factors to epidemiological relationships.     

A new approach to graphical and numerical analysis of links between plant chemotaxonomy and secondary metabolism from HPLC data smoothed by a simplex mixture design

Summary. HPLC analysis of secondary metabolites represents an efficient tool for the studying of plant chemical diversity under different aspects: chemotaxonomy, metabolomics, adaptative responses to ecological factors, etc. Statistical analyses of HPLC databases, e.g. correlation analysis between HPLC peaks, can reliably provide information on the similarity/dissimilarity degrees between the chemical compounds. The similarities, corresponding to positive correlations, can be interpreted in terms of analogies between chemical structures, synchronic metabolisms or co-evolution of two compounds under certain environment conditions, etc. . In terms of metabolism, positive correlations can translate precursor-product relationships between compounds; negative correlations can be indicative of competitive processes between two compounds for a common precursor(s), enzyme(s) or substrate(s). Furthermore, the correlation analysis under a metabolic aspect can help to understand the biochemical origins of an observed polymorphism in a plant species. With the aim of showing this, we present a new approach based on a simplex mixture design, Scheffé matrix, which provides a correlation network making it possible to graphically visualise and to numerically model the metabolic trends between HPLC peaks. The principle of the approach consisted in mixing individual HPLC profiles representative of different phenotypes, then from a complete mixture set, a series of average profiles were calculated to provide a new database with a small variability. Several iterations of the mixture design provided a smoothed final database from which the relationships between the secondary metabolites were graphically and numerically analysed. These relationships were scale-dependent, namely either deterministic or systematic: the first consisted of a monotonic global trend covering the whole variation field of each metabolites’ pair; the second consisted of repetitive monotonic variations which gradually attenuated or intensified along a global trend. This new metabolomic approach was illustrated from 404 individual plants of Astragalus caprinus (Leguminoseae), belonging to four chemical phenotypes (chemotypes) on the basis of flavonoids analysed in their leaves. After smoothing, the relationships between flavonoids were numerically fitted using linear or polynomial models; therefore the co-response coefficients were easily interpreted in terms of metabolic affinities or competitions between flavonoids which would be responsible of the observed chemical polymorphism (the four chemotypes). The statistical validation of the approach was carried out by comparing Pearson correlations to Spearman correlations calculated from the smoothed and the crude HPLC database, respectively. Moreover, the signs of the smoothed relationships were finely supported by analogies and differences between the chemical structures of flavonoids, leading to fluent interpretation in relation to the pathway architecture.     

The evaluation strip: A new and robust method for plotting predicted responses from species distribution models

Increasing use is being made in conservation management of statistical models that couple extensive collections of species and environmental data to make predictions of the geographic distributions of species. While the relationships fitted between a species and its environment are relatively transparent for many of these modeling techniques, others are more ‘black box’ in character, only producing geographic predictions and providing minimal or untraditional summaries of the fitted relationships on which these predictions are based. This in turn prevents robust evaluation of the ecological sensibility of such models, a necessary process if model predictions are to be treated with confidence. Here we propose a new but simple method for visualizing modeled responses that can be implemented with any modeling method, and demonstrate its application using five common methods applied to the prediction of an Australian tree species. This is achieved by insetting an “evaluation strip” into the spatial data layers, which, after predictions have been made, can be clipped out and used for creating plots of the modelled responses. We present findings of the application strip for algorithms GLMs, GAMs, CLIM, DOMAIN and MARS. Evaluation strips can be constructed to investigate either uni-variate responses, or the simultaneous variation in predicted values in relation to two variables. The latter option is particularly useful for evaluating responses in models that allow the fitting of complex interaction terms.     

Hierarchical analysis of species distributions and abundance across environmental gradients

Abiotic and biotic processes operate at multiple spatial and temporal scales to shape many ecological processes, including species distributions and demography. Current debate about the relative roles of niche-based and stochastic processes in shaping species distributions and community composition reflects, in part, the challenge of understanding how these processes interact across scales. Traditional statistical models that ignore autocorrelation and spatial hierarchies can result in misidentification of important ecological covariates. Here, we demonstrate the utility of a hierarchical modeling framework for testing hypotheses about the importance of abiotic factors at different spatial scales and local spatial autocorrelation for shaping species distributions and abundances. For the two orchid species studied, understory light availability and soil moisture helped to explain patterns of presence and abundance at a microsite scale (<4 m2), while soil organic content was important at a population scale (<400 m2). The inclusion of spatial autocorrelation is shown to alter the magnitude and certainty of estimated relationships between abundance and abiotic variables, and we suggest that such analysis be used more often to explore the relationships between species life histories and distributions. The hierarchical modeling framework is shown to have great potential for elucidating ecological relationships involving abiotic and biotic processes simultaneously at multiple scales.     

The evolution of female social relationships in nonhuman primates

Considerable interspecific variation in female social relationships occurs in gregarious primates, particularly with regard to agonism and cooperation between females and to the quality of female relationships with males. This variation exists alongside variation in female philopatry and dispersal. Socioecological theories have tried to explain variation in female-female social relationships from an evolutionary perspective focused on ecological factors, notably predation and food distribution. According to the current “ecological model”, predation risk forces females of most diurnal primate species to live in groups; the strength of the contest component of competition for resources within and between groups then largely determines social relationships between females. Social relationships among gregarious females are here characterized as Dispersal-Egalitarian, Resident-Nepotistic, Resident-Nepotistic-Tolerant, or Resident-Egalitarian. This ecological model has successfully explained differences in the occurrence of formal submission signals, decided dominance relationships, coalitions and female philopatry. Group size and female rank generally affect female reproduction success as the model predicts, and studies of closely related species in different ecological circumstances underscore the importance of the model. Some cases, however, can only be explained when we extend the model to incorporate the effects of infanticide risk and habitat saturation. We review evidence in support of the ecological model and test the power of alternative models that invoke between-group competition, forced female philopatry, demographic female recruitment, male interventions into female aggression, and male harassment. Not one of these models can replace the ecological model, which already encompasses the between-group competition. Currently the best model, which explains several phenomena that the ecological model does not, is a “socioecological model” based on the combined importance of ecological factors, habitat saturation and infanticide avoidance. We note some points of similarity and divergence with other mammalian taxa; these remain to be explored in detail. Received: 30 September 1996 / Accepted after revision: 20 July 1997     

Landscape-level thresholds, and newt conservation.

Ecological thresholds are defined as points or zones at which a rapid change occurs from one ecological condition to another. The existence of thresholds in species-habitat relationships has important implications for management, but the lack of concordance across studies and the wide range of methods used make generalizations difficult. We used two different statistical methods to test for the existence of thresholds for both individual species and the whole community, using three newt species as models. Based on a sample of 371 ponds, we found significant thresholds for both landscape configuration and composition. These were for the relationships between distance to forest and occurrence of Triturus alpestris and T. helveticus, and forest and crop cover and T. helveticus. Variability in the location of thresholds observed for the different species in this study caution against their use at the community level. Future studies should be based on the identification and assessment of thresholds for targeted species. Thresholds can be a useful concept from which tools may be developed to focus conservation effort for threatened species and their habitats.     

Using Scalar Models for Precautionary Assessments of Threatened Species

Abstract:  Scalar population models, commonly referred to as count-based models, are based on time-series data of population sizes and may be useful for screening-level ecological risk assessments when data for more complex models are not available. Appropriate use of such models for management purposes, however, requires understanding inherent biases that may exist in these models. Through a series of simulations, which compared predictions of risk of decline of scalar and matrix-based models, we examined whether discrepancies may arise from different dynamics displayed due to age structure and generation time. We also examined scalar and matrix-based population models of 18 real populations for potential patterns of bias in population viability estimates. In the simulation study, precautionary bias (i.e., overestimating risks of decline) of scalar models increased as a function of generation time. Models of real populations showed poor fit between scalar and matrix-based models, with scalar models predicting significantly higher risks of decline on average. The strength of this bias was not correlated with generation time, suggesting that additional sources of bias may be masking this relationship. Scalar models can be useful for screening-level assessments, which should in general be precautionary, but the potential shortfalls of these models should be considered before using them as a basis for management decisions.     

Invasive species distribution modeling (iSDM): Are absence data and dispersal constraints needed to predict actual distributions?

Species distribution models (SDMs) based on statistical relationships between occurrence data and underlying environmental conditions are increasingly used to predict spatial patterns of biological invasions and prioritize locations for early detection and control of invasion outbreaks. However, invasive species distribution models (iSDMs) face special challenges because (i) they typically violate SDM's assumption that the organism is in equilibrium with its environment, and (ii) species absence data are often unavailable or believed to be too difficult to interpret. This often leads researchers to generate pseudo-absences for model training or utilize presence-only methods, and to confuse the distinction between predictions of potential vs. actual distribution. We examined the hypothesis that true-absence data, when accompanied by dispersal constraints, improve prediction accuracy and ecological understanding of iSDMs that aim to predict the actual distribution of biological invasions. We evaluated the impact of presence-only, true-absence and pseudo-absence data on model accuracy using an extensive dataset on the distribution of the invasive forest pathogen Phytophthora ramorum in California. Two traditional presence/absence models (generalized linear model and classification trees) and two alternative presence-only models (ecological niche factor analysis and maximum entropy) were developed based on 890 field plots of pathogen occurrence and several climatic, topographic, host vegetation and dispersal variables. The effects of all three possible types of occurrence data on model performance were evaluated with receiver operating characteristic (ROC) and omission/commission error rates. Results show that prediction of actual distribution was less accurate when we ignored true-absences and dispersal constraints. Presence-only models and models without dispersal information tended to over-predict the actual range of invasions. Models based on pseudo-absence data exhibited similar accuracies as presence-only models but produced spatially less feasible predictions. We suggest that true-absence data are a critical ingredient not only for accurate calibration but also for ecologically meaningful assessment of iSDMs that focus on predictions of actual distributions.     

Algorithms and biplots for double constrained correspondence analysis

Correspondence analysis with linear external constraints on both the rows and the columns has been mentioned in the ecological literature, but lacks full mathematical treatment and easily available algorithms and software. This paper fills this gap by defining the method as maximizing the fourth-corner correlation between linear combinations, by providing novel algorithms, which demonstrate relationships with related methods, and by making a detailed study of possible biplots and associated approximations. The method is illustrated using ecological data on the abundances of species in sites and where the species are characterized by traits and sites by environmental variables. The trait data and environment data form the external constraints and the question is which traits and environmental variables are associated, how these associations drive species abundances and how they can be displayed in biplots. With microbiome data becoming widely available, these and related multivariate methods deserve more study as they might be routinely used in the future.     

Social dominance,seasonal movements,and spatial segregation in African elephants: a contribution to conservation behavior

The structure of dominance relationships among individuals in a population is known to influence their fitness, access to resources, risk of predation, and even energy budgets. Recent advances in global positioning system radio telemetry provide data to evaluate the influence of social relationships on population spatial structure and ranging tactics. Using current models of socio-ecology as a framework, we explore the spatial behaviors relating to the maintenance of transitive (i.e., linear) dominance hierarchies between elephant social groups despite the infrequent occurrence of contests over resources and lack of territorial behavior. Data collected from seven families of different rank demonstrate that dominant groups disproportionately use preferred habitats, limit their exposure to predation/conflict with humans by avoiding unprotected areas, and expend less energy than subordinate groups during the dry season. Hence, our data provide strong evidence of rank derived spatial partitioning in this migratory species. These behaviors, however, were not found during the wet season, indicating that spatial segregation of elephants is related to resource availability. Our results indicate the importance of protecting preexisting social mechanisms for mitigating the ecological impacts of high density in this species. This analysis provides an exemplar of how behavioral research in a socio-ecological framework can serve to identify factors salient to the persistence and management of at risk species or populations. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Competing probabilistic models for catch-effort relationships in wildlife censuses

Two probabilistic models are presented for describing the chance that an animal is captured during a wildlife census, as a function of trapping effort. The models in turn are used to propose relationships between sampling intensity and catch-per-unit-effort (C.P.U.E.) that were field tested on small mammal populations. Capture data suggests a model of diminishing C.P.U.E. with increasing levels of trapping intensity. The catch-effort model is used to illustrate optimization procedures in the design of mark-recapture experiments for censusing wild populations.