Modeling Species' Distributions to Improve Conservation in Semiurban Landscapes: Koala Case Study

Abstract:  Models of species' distributions are commonly used to inform landscape and conservation planning. In urban and semiurban landscapes, the distributions of species are determined by a combination of natural habitat and anthropogenic impacts. Understanding the spatial influence of these two processes is crucial for making spatially explicit decisions about conservation actions. We present a logistic regression model for the distribution of koalas (  Phascolarctos cinereus ) in a semiurban landscape in eastern Australia that explicitly separates the effect of natural habitat quality and anthropogenic impacts on koala distributions. We achieved this by comparing the predicted distributions from the model with what the predicted distributions would have been if anthropogenic variables were at their mean values. Similar approaches have relied on making predictions assuming anthropogenic variables are zero, which will be unreliable if the training data set does not include anthropogenic variables close to zero. Our approach is novel because it can be applied to landscapes where anthropogenic variables are never close to zero. Our model showed that, averaged across the study area, natural habitat was the main determinant of koala presence. At a local scale, however, anthropogenic impacts could be more important, with consequent implications for conservation planning. We demonstrated that this modeling approach, combined with the visual presentation of predictions as a map, provides important information for making decisions on how different conservation actions should be spatially allocated. This method is particularly useful for areas where wildlife and human populations exist in close proximity.     

Species distribution modelling: Does one size fit all? A phytogeographic analysis of Salix in Ontario

Empirical models for predicting the distribution of organisms from environmental data have often focused on principles of ecological niche theory. However, even at large scales, there is little agreement over how to represent the dimensions of a species’ niche. The performance of such models is greatly affected by the nature of species distributional and environmental data. Regional scale distribution models were developed for 30 willow species in Ontario to examine (i) the predictive ability of logistic regression analysis, and (ii) the effects of using different distributional and environmental data sets. Two original measures of model accuracy and over-prediction were employed and evaluated using independent data. Models based on unique combinations of monthly climate data predicted distributions most accurately for all species. Models based on a fixed set of variables, while generating the highest average probabilities of occurrence for certain species with limited ranges, resulted in the greatest under- and over-estimates of willow distributions. Comparisons of models demonstrated climatic patterns among willows of differing habit and habitat. The distribution of dwarf willow species, present only in the Ontario arctic, followed gradients of summer maximum temperatures. The distribution of the tree species in the southerly portions of the province followed gradients of fall and winter minimum temperatures. Regardless of distributional and environmental data input, no algorithm maximized model performance for all species. Individual species models require individual approaches; i.e., the variable selection technique, the set of environmental factors used as predictors, and the nature of species distributional data must be carefully matched to the intended application. An understanding of evolutionary processes enhances the meaningful interpretation of individual species models. Unless sampling bias and species prevalence can be accounted for, models based on collection point data are best used to guide field surveys. While inferred range data may be better suited to determine potential ecological niches, overestimation of species prevalence and environmental tolerance must be recognized. A combination of available distributional data types is recommended to best determine species niches, an important step in developing conservation strategies.     

Minimum viable metapopulation size, extinction debt, and the conservation of a declining species.

A key question facing conservation biologists is whether declines in species' distributions are keeping pace with landscape change, or whether current distributions overestimate probabilities of future persistence. We use metapopulations of the marsh fritillary butterfly Euphydryas aurinia in the United Kingdom as a model system to test for extinction debt in a declining species. We derive parameters for a metapopulation model (incidence function model, IFM) using information from a 625-km2 landscape where habitat patch occupancy, colonization, and extinction rates for E. aurinia depend on patch connectivity, area, and quality. We then show that habitat networks in six extant metapopulations in 16-km2 squares were larger, had longer modeled persistence times (using IFM), and higher metapopulation capacity (lambdaM) than six extinct metapopulations. However, there was a > 99% chance that one or more of the six extant metapopulations would go extinct in 100 years in the absence of further habitat loss. For 11 out of 12 networks, minimum areas of habitat needed for 95% persistence of metapopulation simulations after 100 years ranged from 80 to 142 ha (approximately 5-9% of land area), depending on the spatial location of habitat. The area of habitat exceeded the estimated minimum viable metapopulation size (MVM) in only two of the six extant metapopulations, and even then by only 20%. The remaining four extant networks were expected to suffer extinction in 15-126 years. MVM was consistently estimated as approximately 5% of land area based on a sensitivity analysis of IFM parameters and was reduced only marginally (to approximately 4%) by modeling the potential impact of long-distance colonization over wider landscapes. The results suggest a widespread extinction debt among extant metapopulations of a declining species, necessitating conservation management or reserve designation even in apparent strongholds. For threatened species, metapopulation modeling is a potential means to identify landscapes near to extinction thresholds, to which conservation measures can be targeted for the best chance of success.     

Native ecotypic variation and the role of host identity in the spread of an invasive herbivore, Cactoblastis cactorum

Environmental niche models (ENMs) have gained enormous popularity as tools to investigate potential changes in species distributions resulting from climate change and species introductions. Despite recognition that species interactions can influence the dynamics of invasion spread, most implementations of ENMs focus on abiotic factors as the sole predictors of potential range limits. Implicit in this approach is the assumption that biotic interactions are relatively unimportant, either because of scaling issues, or because fundamental and realized niches are equivalent in a species' native range. When species are introduced into exotic landscapes, changes in biotic interactions relative to the native range can lead to occupation of different regions of niche space and apparent shifts in physiological tolerances. We use an escaped biological control organism, Cactoblastis cactorum (Berg.), to assess the role of the environmental envelope as compared with patterns of host-herbivore associations based on collections made in the native range. Because all nonnative populations are derived from a single C. cactorum ecotype, we hypothesize that biotic interactions associated with this ecotype are driving the species' invasion dynamics. Environmental niche models constructed from known native populations perform poorly in predicting nonnative distributions of this species, except where there is an overlap in niche space. In contrast, genetic isolation in the native range is concordant with the observed pattern of host use, and strong host association has been noted in nonnative landscapes. Our results support the hypothesis that the apparent shift in niche space from the native to the exotic ranges results from a shift in biotic interactions, and demonstrate the importance of considering biotic interactions in assessing the risk of future spread for species whose native range is highly constrained by biotic interactions.     

Forecasting New Zealand Mudsnail invasion range: model comparisons using native and invaded ranges.

Evaluations of the potential distribution of invasive species can increase the efficiency of their management by focusing prevention measures. Generally, ecological models are built using occurrence data from a species' native range to predict the distribution in areas that the species may invade. However, historical and geographical constraints can limit a species' native distribution. Genetic Algorithm for Rule-set Production (GARP), an ecological niche modeling program, was used to predict the potential distribution of the invasive, freshwater New Zealand mudsnail, Potamopyrgus antipodarum, in Australia and North America. We compared the strength of the predictions made by models built with data from the snail's native range in New Zealand to models built with data from the locations invaded by the species. A time-series analysis of the Australian models demonstrated that range-of-invasion data can make better predictions about the potential distribution of invasive species than models built with native range data. Large differences among the model forecasts indicate that uncritical choice of the data set used in training the GARP models can result in misleading predictions. The models predict a large expansion in the range of P. antipodarum in both Australia and North America unless prevention measures are implemented rapidly.     

Niche differentiation and fine-scale projections for Argentine ants based on remotely sensed data.

Modeling ecological niches of species is a promising approach for predicting the geographic potential of invasive species in new environments. Argentine ants (Linepithema humile) rank among the most successful invasive species: native to South America, they have invaded broad areas worldwide. Despite their widespread success, little is known about what makes an area susceptible--or not--to invasion. Here, we use a genetic algorithm approach to ecological niche modeling based on high-resolution remote-sensing data to examine the roles of niche similarity and difference in predicting invasions by this species. Our comparisons support a picture of general conservatism of the species' ecological characteristics, in spite of distinct geographic and community contexts.     

Fine-scale habitat modeling of a top marine predator: do prey data improve predictive capacity?

Predators and prey assort themselves relative to each other, the availability of resources and refuges, and the temporal and spatial scale of their interaction. Predictive models of predator distributions often rely on these relationships by incorporating data on environmental variability and prey availability to determine predator habitat selection patterns. This approach to predictive modeling holds true in marine systems where observations of predators are logistically difficult, emphasizing the need for accurate models. In this paper, we ask whether including prey distribution data in fine-scale predictive models of bottlenose dolphin (Tursiops truncatus) habitat selection in Florida Bay, Florida, U.S.A., improves predictive capacity. Environmental characteristics are often used as predictor variables in habitat models of top marine predators with the assumption that they act as proxies of prey distribution. We examine the validity of this assumption by comparing the response of dolphin distribution and fish catch rates to the same environmental variables. Next, the predictive capacities of four models, with and without prey distribution data, are tested to determine whether dolphin habitat selection can be predicted without recourse to describing the distribution of their prey. The final analysis determines the accuracy of predictive maps of dolphin distribution produced by modeling areas of high fish catch based on significant environmental characteristics. We use spatial analysis and independent data sets to train and test the models. Our results indicate that, due to high habitat heterogeneity and the spatial variability of prey patches, fine-scale models of dolphin habitat selection in coastal habitats will be more successful if environmental variables are used as predictor variables of predator distributions rather than relying on prey data as explanatory variables. However, predictive modeling of prey distribution as the response variable based on environmental variability did produce high predictive performance of dolphin habitat selection, particularly foraging habitat.     

Consequences of niche overlap for ecosystem functioning: an experimental test with pond grazers

While the number of studies investigating the effects of species diversity on ecosystem properties continues to expand, few have explicitly examined how ecosystem functioning depends quantitatively on the degree of niche complementarity among species. We report the results of a microcosm experiment where similarity in habitat use among aquatic snail species was evaluated as a predictor of changes in community and ecosystem properties due to increasing species richness. Replicate microcosms with all possible one- and two-species combinations of a guild of six snail species were stocked with identical initial snail biomass. Microcosms with two species of snails had greater final snail biomass, lower attached algae biomass, and less total organic matter than monocultures. Snail species differed in their use of five distinct habitat types in the microcosms. Similarity in habitat use between a species pair was negatively related to the magnitude of change (e.g., deltaEF [change in ecosystem function]) in dissolved oxygen. periphyton biomass, and accrual of organic matter with a change in diversity. However, using the most stringent criterion for complementarity effects (e.g., Dmax [proportional deviation of the total polyculture yield from the highest yielding monoculture]), a relationship between species' niche similarity and changes in function with increasing species richness was only observed for dissolved oxygen. The identity of snail species present in the microcosms had strong effects on total organic matter, snail biomass, dissolved oxygen, periphyton biomass, and sedimentation rate. In this study, herbivore identity, sampling effects, and niche complementarity all appear to contribute to species richness effects on pond ecosystem properties and community structure. The analytical approach employed here may profitably be used in other systems to quantify the role of niche complementarity in species richness-ecosystem function relationships.     

Predicting the Range of Chinese Mitten Crabs in Europe

Abstract:  Ecological niche modeling provides a means for predicting the potential future distribution of a nonindigenous species based on environmental characteristics of the species' native range. We applied this method to the Chinese mitten crab (Eriocheir sinensis) , a catadromous crustacean with a long history of invasion in Europe. We used genetic algorithm for rule-set prediction to predict the potential European distribution of mitten crab based on its distribution in 42 locations in its native Asia. The climatic variables, air temperature, number of days, amount of precipitation, and wetness index, contributed significantly to predictions of native distribution limits. Although the genetic algorithm for rule-set prediction model was developed for the native range, the species' extensive distribution in Europe ( n = 434) allowed independent validation of the predictions. Application of the model to Europe was successful, with 84% of occurrences in regions predicted to be suitable by >80% of the models and <4% of occurrences in areas predicted suitable by <50% of the models (mainly along the northern range). At the watershed scale, areas with established mitten crab populations had significantly higher habitat matching than sites that were not invaded. The independent validation of the Asian-based model by the European distribution revealed that predictions were highly accurate. The model also identified large areas of Europe, particularly along the Mediterranean coast, as vulnerable to future invasion. These predictions can be used to develop strategies to control the spread of mitten crab by preventing introductions into vulnerable areas.     

Climatic niche divergence or conservatism? Environmental niches and range limits in ecologically similar damselflies

The factors that determine species' range limits are of central interest to biologists. One particularly interesting group comprises odonates (dragonflies and damselflies), which show large differences in secondary sexual traits and respond quickly to climatic factors, but often have minor interspecific niche differences, challenging models of niche-based species coexistence. We quantified the environmental niches at two geographic scales to understand the ecological causes of northern range limits and the coexistence of two congeneric damselflies (Calopteryx splendens and C. virgo). Using environmental niche modeling, we quantified niche divergence first across the whole geographic range in Fennoscandia, and second only in the sympatric part of this range. We found evidence for interspecific divergence along the environmental axes of temperature and precipitation across the northern range in Fennoscandia, suggesting that adaptation to colder and wetter climate might have allowed C. virgo to expand farther north than C. splendens. However, in the sympatric zone in southern Fennoscandia we found only negligible and nonsignificant niche differences. Minor niche differences in sympatry lead to frequent encounters and intense interspecific sexual interactions at the local scale of populations. Nevertheless, niche differences across Fennoscandia suggest that species differences in physiological tolerances limit range expansions northward, and that current and future climate could have large effects on the distributional ranges of these and ecologically similar insects.     

K-select analysis: a new method to analyse habitat selection in radio-tracking studies

Two kinds of wildlife habitat studies can be distinguished in the literature: hindcasting and forecasting studies. Hindcasting studies aim to emphasize among a large set of habitat variables those that are of interest for the focus species, whereas forecasting studies are intended to predict habitat selection according to a small number of habitat variables for a given area. We provide here a new analytical tool which relies on the concept of ecological niche, the K-select analysis, for hindcasting studies of habitat selection by animals using radio-tracking data. Each habitat variable defines one dimension in the ecological space. For each animal, the difference between the vector of average available habitat conditions and the vector of average used conditions defines the marginality vector. Its size is proportional to the importance of habitat selection, and its direction indicates which variables are selected. By performing a non-centered principal component analysis of the table containing the coordinates of the marginality vectors of each animal (row) on the habitat variables (column), the K-select analysis returns a linear combination of habitat variables for which the average marginality is greatest. It is a synthesis of variables which contributes the most to the habitat selection. As with principal component analysis, the biological significance of the factorial axes is deduced from the loading of variables. An example is provided: habitat selection by wild boar is studied in a Mediterranean habitat using the K-select analysis. The numerous advantages of the analysis (a large number of variables that can be included, individual variability in habitat selection taken into account, a lack of too strict underlying hypotheses) make it a powerful approach in radio-tracking studies designed to identify habitat variables that are selected by animals.     

Model of Japanese serow (Capricornis crispus) energetics predicts distribution on Honshu, Japan.

Understanding what determines a species' range is a central objective in ecology and evolutionary biology. It has important applications for predicting species distributions and how they might respond to environmental perturbations. This paper describes a mechanistic approach to predict past and present distribution of the Japanese serow (Capricornis crispus) on Honshu, Japan. We applied state-of-the-art microclimate and animal biophysical/behavioral models coupled with climate and vegetation data to estimate the distribution of potential range expansion under protection. We tested the model results against detailed empirical distribution data from the Ministry of the Environment for a five-prefecture area in central Honshu. We also applied the models to time-series land use/cover maps to investigate the historical transitions in habitat suitability during 1947-1999 in the Arai-Keinan region. This is the first time to our knowledge that mechanistic models have successfully predicted the landscape scale distribution of a mammal species in the absence of other animal species interactions, such as predators. In this case, animal energetics/behavior-plant interactions seem to be critical. Forest cover appears to be important in summer and winter for suitable serow habitats. The energetics model results indicate that the serow can overheat in some open environments in midday hours in summer. In winter, simulation results suggested that forest cover provides effective refuge to avoid increased metabolic demands of cold temperatures and strong winds. The model simulations suggested that land use/cover changes documented during 1947-1999 resulted in increased suitable serow habitat due to expanding forest cover from agricultural marginalization and ecological succession. The models provide a unique tool for estimating species' range expansion under protection or for selecting suitable reintroduction sites.     

Ecological niche modeling of customary medicinal plant species used by Australian Aborigines to identify species-rich and culturally valuable areas for conservation

Customary medicinal plant species used by Australian Aborigines are disappearing rapidly with its associated knowledge, due to the loss of habitats. Conservation and protection of these species is important as they represent sources of novel therapeutic phytochemical compounds and are culturally valuable. Information on the spatial distribution and use of customary medicinal plants is often inadequate and fragmented, posing limitations on the identification and conservation of species-rich areas and culturally valuable habitats.In this study, the habitat suitability modeling program, MaxEnt, was used to predict the potential ecological niches of 431 customary medicinal plant species, based on bioclimatic variables. Specimen locality records were obtained from the Global Biodiversity Information Facility (GBIF) data portal and from Australia's Virtual Herbarium (AVH).Ecological niche models of 414 predicted species, which had 30 or more occurrence points, were used to produce maps indicating areas that were ecologically suitable for multiple species (concordance of high predicted ecological suitability) and having cultural values. For the concordance map, individual species niche models were thresholded and summed. To derive a map of culturally valuable areas, customary medicinal uses from Customary Medicinal Knowledgebase (CMKb) (www.biolinfo.org/cmkb) were used to weight individual species models, resulting in a value within each grid cell reflecting its cultural worth.Even though the available information is scarce and fragmented, our approach provides an opportunity to infer areas predicted to be suitable for multiple species (i.e. concordance hotspots) and to estimate the cultural value of a particular geographical area. Our results also indicate that to conserve bio-cultural diversity, comprehensive information and active participation of Aboriginal communities is indispensable.     

Predictive vegetation modeling for conservation: impact of error propagation from digital elevation data.

The effect of digital elevation model (DEM) error on environmental variables, and subsequently on predictive habitat models, has not been explored. Based on an error analysis of a DEM, multiple error realizations of the DEM were created and used to develop both direct and indirect environmental variables for input to predictive habitat models. The study explores the effects of DEM error and the resultant uncertainty of results on typical steps in the modeling procedure for prediction of vegetation species presence/absence. Results indicate that all of these steps and results, including the statistical significance of environmental variables, shapes of species response curves in generalized additive models (GAMs), stepwise model selection, coefficients and standard errors for generalized linear models (GLMs), prediction accuracy (Cohen's kappa and AUC), and spatial extent of predictions, were greatly affected by this type of error. Error in the DEM can affect the reliability of interpretations of model results and level of accuracy in predictions, as well as the spatial extent of the predictions. We suggest that the sensitivity of DEM-derived environmental variables to error in the DEM should be considered before including them in the modeling processes.     

Interspecific competition between native Eurasian red squirrels and alien grey squirrels: does resource partitioning occur?

In heterogeneous environments, differential niche selection by two competing species will result in niche partitioning so that individuals of each species can maximise their fitness under different sets of environmental variables. Thus, niche partitioning is considered essential to allow co-existence of ecologically related species. To assess whether niche partitioning was occurring between native red squirrels and alien grey squirrels living together in a 13-ha high-quality mixed deciduous woodland in north Italy, we investigated temporal and spatial patterns in their activity and foraging behaviour between 1996 and 1998. We used live trapping and radio-tracking to study numbers, distribution and behaviour of squirrels. Daily and seasonal temporal activity patterns, and activity on the ground and in the trees, were similar in the two species. However, grey squirrels were more tree specialists and had a narrower tree-species niche width than red squirrels, in particular making greater use of oak. Other studies of red and grey squirrels in allopatry show that the two species differ in the extent they utilise oak. Overall, tree-species niche overlap was about 70%. Grey squirrels had larger home ranges than red squirrels. Home ranges and core areas of both species were larger in males than females. Also, intraspecific home range and core-area overlap patterns were similar to those found in allopatric populations of these species. Overall, there was no evidence that the use of space of one species was affected by the other. Our results show that there was no niche partitioning of activity or foraging behaviour in time or space during the study. This suggests that, at moderate grey-squirrel densities, red squirrels are unable to avoid competition with grey squirrels, and that competition for food and/or space will occur when these resources become limiting.     

Habitat Loss and Changes in the Species-Area Relationship

Abstract: The species-area relationship (SAR) has been used successfully to predict extinction from extent of habitat reduction. These extinction estimates assume that species have uniformly distributed range requirements and a minimum abundance level required for persistence; how many species are lost depends solely on how much habitat is removed, not on where it is removed. We consider another limiting case in which range requirements, rather than abundances, determine extinctions. We used a new method for constructing SARs based on assumptions about geographic ranges of species. Our results show that habitat destruction can change the SAR and consequently the number of species predicted to be lost due to habitat destruction. Our method generates SARs that vary in shape according to the specific distributions of geographic range and occupancy but that have the common feature of being described by a power law with an exponent of <1. When the geographic range of species was included in the SAR, the way habitat was lost became important. Although the SAR before habitat destruction is often used to predict species loss after habitat destruction, assumptions must be clearly stated. To predict the damage caused by habitat loss with our model, it is necessary to know the fraction of aggregated species, the distribution of geographic ranges, the form of habitat destruction, and the sampling protocol. The remaining theoretical challenge is to develop a full theory that links abundance and range.     

Evaluating the sources of potential migrant species: implications under climate change

As changes in climate become more apparent, ecologists face the challenge of predicting species responses to the new conditions. Most forecasts are based on climate envelopes (CE), correlative approaches that project future distributions on the basis of the current climate often assuming some dispersal lag. One major caveat with this approach is that it ignores the complexity of factors other than climate that contribute to a species' distributional range. To overcome this limitation and to complement predictions based on CE modeling we carried out a transplant experiment of resident and potential-migrant species. Tree seedlings of 18 species were planted side by side from 2001 to 2004 at several locations in the Southern Appalachians and in the North Carolina Piedmont (U.S.A.). Growing seedlings under a large array of environmental conditions, including those forecasted for the next decades, allowed us to model seedling survival as a function of variables characteristic of each site, and from here we were able to make predictions on future seedling recruitment. In general, almost all species showed decreased survival in plots and years with lower soil moisture, including both residents and potential migrants, and in both locations, the Southern Appalachians and the Piedmont. The detrimental effects that anticipated arid conditions could have on seedling recruitment contradict some of the projections made by CE modeling, where many of the species tested are expected to increase in abundance or to expand their ranges. These results point out the importance of evaluating the potential sources of migrant species when modeling vegetation response to climate change, and considering that species adapted to the new climate and the local conditions may not be available in the surrounding regions.     

Community occupancy responses of small mammals to restoration treatments in ponderosa pine forests, northern Arizona, USA

In western North American conifer forests, wildfires are increasing in frequency and severity due to heavy fuel loads that have accumulated after a century of fire suppression. Forest restoration treatments (e.g., thinning and/or burning) are being designed and implemented at large spatial and temporal scales in an effort to reduce fire risk and restore forest structure and function. In ponderosa pine (Pinus ponderosa) forests, predominantly open forest structure and a frequent, low-severity fire regime constituted the evolutionary environment for wildlife that persisted for thousands of years. Small mammals are important in forest ecosystems as prey and in affecting primary production and decomposition. During 2006-2009, we trapped eight species of small mammals at 294 sites in northern Arizona and used occupancy modeling to determine community responses to thinning and habitat features. The most important covariates in predicting small mammal occupancy were understory vegetation cover, large snags, and treatment. Our analysis identified two generalist species found at relatively high occupancy rates across all sites, four open-forest species that responded positively to treatment, and two dense-forest species that responded negatively to treatment unless specific habitat features were retained. Our results indicate that all eight small mammal species can benefit from restoration treatments, particularly if aspects of their evolutionary environment (e.g., large trees, snags, woody debris) are restored. The occupancy modeling approach we used resulted in precise species-level estimates of occupancy in response to habitat attributes for a greater number of small mammal species than in other comparable studies. We recommend our approach for other studies faced with high variability and broad spatial and temporal scales in assessing impacts of treatments or habitat alteration on wildlife species. Moreover, since forest planning efforts are increasingly focusing on progressively larger treatment implementation, better and more efficiently obtained ecological information is needed to inform these efforts.     

Analysis and prediction of ecological and ecosystem dynamics with response structure models

A method of directly using data concerning species' responses to various combinations of levels of environmental factors is outlined. Examples of its utilization to analyze (and predict) the dynamics of the ecological processes of population growth, interspecific competition, and predation are given for simple systems; a simulation is presented wherein all of these processes interact within a simple ecosystem. A technique of summarizing the pattern and detail of variation of levels of combination of important environmental factors is also introduced. Together these methods permit application of real world information about the components of the biological community and the environment toward the modeling of ecosystems and the making of predictions concerning them to whatever degree of accuracy which may be useful.There are several major advantages of this approach: (1) It makes maximum use of real world data; the limits of its precision are determined only by the adequacy of the data available (hypothetical values also may be used where real data are unavailable and/or where such hypothetical values may be useful in model and theory development). (2) It is powerful (and applicable to the dynamics of diverse kinds of systems), mathematically very simple, and has important advantages in that non-linearities, thresholds, etc. are automatically and accurately taken into account. (3) It can be expressed numerically or graphically, and therefore is especially helpful in permitting visualization of ecological processes and the results of their action.