An Interface Between Ecological Theory and Statistical Modelling

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Ecological Modelling 157 (2002) 101Á/118 www.elsevier.com/locate/ecolmodel

Spatial prediction of species distribution: an interface between ecological theory and statistical modelling

M.P. Austin

CSIRO Sustainable Ecosystems, GPO Box 284, Canberra City, ACT 2601, Australia

Abstract

Neglect of ecological knowledge is a limiting factor in the use of statistical modelling to predict species distribution. Three components are needed for statistical modelling, an ecological model concerning the ecological theory to be used or tested, a data model concerning the collection and measurement of the data, and a statistical model concerning the statistical theory and methods used. This component framework is reviewed with emphasis on ecological theory. The expected shape of a species response curvetoanenvironmental gradient is a central assumption on which agreement has yet to be reached. The nature of the environmental predictors whether indirect variables, e.g. latitude that haveno physiological impact on plants, or direct variables, e.g. temperature also influence the type of response expected. Straight-line relationships between organisms and environment are often used uncritically. Many users of canonical correlation analysis use linear (straight-line) functions to relate ordination axes to variables such as slope and aspect though this is not a necessary part of the method. Some statisticians have used straight lines for species/environment relationships without testing, when evaluating new statistical procedures. Assumptions used in one component often conflict with those in another component. Statistical models can be used to explore ecological theory. Skewed species response curves predominate contrary to the symmetric unimodal curves assumed by some statistical methods. Improvements in statistical modelling can be achieved based on ecological concepts. Examples include incorporating interspecific competition from dominant species; more proximal predictors based on water balance models and spatial autocorrelation procedures to accommodate non-equilibrium vegetation. # 2002 Elsevier Science B.V. All rights reserved.

Keywords: Vegetation models; Species models; Continuum; Niche; Generalised linear models; GLM; Generalised additive models; GAM; Canonical correspondence analysis; CCA; Spatial prediction; Regression; Ordination; Species response curves

1. Introduction Zimmermann, 2000; Elith and Burgman 2002; Scott et al., 2002). A wide variety of statistical The spatial prediction of species distributions and machine-learning methods have been intro- from survey data has recently been recognised as a duced, often in conjunction with geographic in- significant component of conservation planning formation systems (GIS) and remote-sensing (Franklin, 1995; Austin, 1998, 2002; Guisan and (Fitzgerald and Lees, 1992; Aspinall and Veitch, 1993; Pereira and Itami, 1991; Franklin et al., 2000). Guisan and Zimmermann (2000) provide an E-mail address: [email protected] (M.P. Austin). extensivereview of these developments, identifying

0304-3800/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 0 2 ) 0 0 2 0 5 - 3 102 M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 many of the alternative statistical approaches that experiment, by theoretical analysis or by repeating may be used. Various forms of regression analysis the study at a different location. Another use of predominate in the literature. Generalised linear statistical models is calibration, where biotic data models (GLM, McCullagh and Nelder, 1989; are used to estimate past environmental condi- Austin and Cunningham, 1981) and generalised tions, an approach of increasing importance in additive models (GAM, Hastie and Tibshirani, palaeoecology. Birks (1998) states ‘‘The most 1990; Yee and Mitchell, 1991) with logistic regres- significant development in quantitative palaeolim- sion using presenceÁ/absence survey data appear to nology has been the creation of many modern be increasingly popular as the statistical model to calibration data-sets of biotic assemblages and be used (Franklin, 1995; Guisan and Zimmer- associated environmental data. Such calibration mann, 2000; Scott et al., 2002). sets, when analysed by appropriate procedures, In the majority of cases, the purpose of the have the potential to transform fossil biostratigra- statistical modelling is the prediction of species phical data into quantitative estimates of the past distribution. The detection of functional relation- environment.’’ These calibration methods (ter ships between species and environment and the Braak and Verdonschot, 1995) are not considered testing of ecological theory tend to be secondary further except where they relate to ecological considerations (Guisan and Zimmermann, 2000). assumptions. It will be argued in this paper that neglect of There are three major components in any ecological knowledge is a limiting factor in the framework for statistical modelling in plant ecol- application of statistical modelling in ecology and ogy. There needs to be an ecological model, a data conservation planning. At the interface between model, and a statistical model. The ecological ecology and statistics, it is possible for statisticians model consists of the ecological knowledge and to assume inadequate ecological models that may theory to be used or tested in the study. The data confound their evaluation of new statistical meth- model consists of the decisions made regarding ods. On the other hand, ecologists may construct how the data are collected and how the data will simpler statistical models than they believe are be measured or estimated. The statistical model necessary because they are unaware of the power involves the choice of statistical method, error of modern statistical methods, A synthesis of function and significance tests. Each model inter- current ecological theory and modern statistical acts in both obvious and subtle ways with the models is badly needed. This paper reviews some other models to determine the success of any of the current problems and tries to assemble some statistical modelling exercise. of the components necessary for a synthesis to be The ecological model may contain assumptions produced. The review is mainly concerned with to be incorporated into the analysis or hypotheses plant community ecology, though similar argu- to be tested. It may, for example be assumed that ments can be applied to other fields of ecology vegetation and species composition are primarily (Austin, 1999a,b). determined by environment rather than by succes- sional status or by time since last disturbance (Franklin et al., 2000). The functional link between 2. A framework for statistical modelling in plant plants and environment may be assumed to be all community ecology straight-line relationships (Austin, 1999a). A hypothesis might test whether species respond to Statistical models are based on correlation and environmental variables or gradients with a unim- often have as their purpose prediction. It is not odal, symmetric, response function. possible to determine causation from correlation, An important example of the data model is that but a description of functional relationships can be used by classical phytosociologists (Braun-Blan- achieved (Sokal and Rohlf, 1981; Box, 1966). The quet, 1964; Westhoff and van der Maarel, 1978). reality of such relationships and the causal me- Samples (plots) are selected subjectively by the chanisms responsible should then be pursued by ecologist without any statistical sampling design M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 103 considerations. Species abundances are estimated for the selection of resources by animals but is very using a mixed cover/abundance scale with unequal similar to estimating plant species response curves class intervals, the statistical implications of which (Boyce et al., 2002). The ecological model and the have not been considered until recently. Guisan data model for plants and animals differ. An and Harrell (2000, Guisan 2002) point out that this important difference is the difficulty of estimating cover/abundance measurement is an ordinal vari- the non-use of a resource by an animal. Even able that requires, therefore, special attention estimating the absence from a habitat or site of a when selecting the statistical model. Franklin et mobile or cryptic animal is a non-trivial problem. al. (2000) consider a number of problems that can Consequently the statistical models may differ. arise with the data model. These include problems However, Boyce et al.’s discussion of evaluation of the appropriate scale of observations and procedures has general applicability. locational accuracy when ecological field data are The focus here is on the many ecological used with GIS and databases. The scale of resolu- assumptions and theories that can impact on tion and types of errors associated with digital spatial modelling of vegetation. Prediction can be terrain models were also discussed. The authors achieved without the regression equation having included satellite imagery derived variables in their any necessary ecological process basis, but the case studies. Such remotely-sensed variables have result is unlikely to be robust. The more that their own data models, which need to be evaluated knowledge of ecological process can be incorpo- for compatibility with ecological models. Franklin rated, the more robust the prediction and the more et al. (2000) provide a useful geographical per- explanatory power the equation is likely to have. spective on issues associated with the choice of However, statistical models currently used are data model for vegetation prediction. static and contain no dynamic elements (Guisan The term ‘conceptual model’ has been used to and Zimmermann, 2000; Guisan and Theurillat, include both the ecological model and the data 2000). It is assumed that vegetation is in equili- model. Separating theory from data availability brium with the environment, or at least a quasi- and model applicability can ensure debates about equilibrium where change is slow relative to the life theory are not confused with differences in data span of the biota. This is a central ecological models. See the papers by Oksanen (1997), Austin assumption of the use of these statistical models at and Nicholls (1997), Oksanen and Minchin (2002) the present time. O’Connor (2001) has also re- for a recent example of a debate over theory and cently made this point. There are, therefore, limits data models concerning the shape of species on the degree of success that can be achieved by response curves. statistical models using environmental predictors. Similar difficulties can arise with the statistical This will vary depending on the degree to which model regarding the error function assumed, history and disturbance are important in the choice of model whether generalised linear model biome under study. Franklin and colleagues (GLM), generalised additive model (GAM) or (Franklin et al., 2000; Meentemeyer et al., 2001) classification and regression tree (CART, Breiman have recently shown the predictive importance of et al., 1984) and methods of validation. Statistical topographically dependent environmental gradi- decisions can change the ecological model. Choice ents in the fire-prone California chaparral. Note of data transformation, e.g. log to stabilise the the recent use of GAM by Fewster et al. (2000) to variance will change the functional relationship examine the population dynamics of birds and the from an additive to a multiplicative form. Boyce et use of temporal data by Boyce et al. (2002). al. (2002) review statistical methods for evaluating Current techniques need not be limited to static, resource selection functions (RSF) from the per- equilibrium situations but the nature of ecological, spective of the ecological and data models used. data and statistical models will need to be re- RSF are defined as any function that is propor- examined. tional to the probability of use of a resource or Where it is accepted that it is important to area by an organism. The approach was developed maximise our knowledge of the functional rela- 104 M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 tionships between plants and environment, current any predictive modelling effort. Most ecological ecological concepts need to be carefully consid- textbooks (Begon et al., 1990; Giller, 1984; Krebs, ered. For example, consensus is emerging that the 1994) present the response as a unimodal, sym- continuum concept is a more appropriate model of metric bell-shaped curve. Austin (1999a) drew vegetation organisation than that of the plant attention to the lack of evidence for this assump- community or association and is more suitable tion. Niche theory is based on this assumption and for spatial prediction (Austin and Smith, 1989; additionally makes several others (Fig. 1): Franklin, 1995; Guisan and Zimmermann, 2000). Similarly, it is now well recognised that statistical 1) Both the fundamental and realised niches of a models provide a description of the realised niche species are bell-shaped symmetric curves. of a species but can say little about the funda- 2) Competition restricts niche breadth. mental niche (Austin and Smith, 1989; Austin, 3) Maximum abundance occurs at the optimum 1992; Malanson et al., 1992; Franklin, 1995; for the fundamental niche. Guisan and Zimmermann, 2000). The statistical 4) Species maxima are equally spaced along the model may not represent the realised niche. The gradient. source-sink model (Pulliam, 1988) regards the 5) Species maxima are of equal amplitude. geographical distribution of a species as composed 6) Collective properties of species, e.g. species of two areas. The source areas are where popula- richness, dominance or stand abundance show tion growth is positive. Sink areas are those areas no patterns of response along the gradient. where population growth is below replacement and populations are maintained by dispersal from Mueller-Dombois and Ellenberg (1974) Fig. source areas. Hence statistical models may be 12.3, see also Austin (1999a) put forward a more describing an amalgam of realised niche and sink general theory which relaxes several but not all of areas. Pulliam (1988) comments that ‘... in such the above assumptions. Their theory assumes that cases, it can be said that the fundamental niche is a superior competitor can displace a species from smaller than the realised niche.’ Watkinson and the optimum of its fundamental niche. As a Sutherland (1995) discuss some plant examples, consequence, a species realised niche response Cakite edentula is given as an example where may take a wide variety of shapes from skewed reproduction from populations at one position on to bimodal (Fig. 1). There is no general agreement an environmental gradient maintains sink popula- on species response shape. Many predictive studies tions at other positions on the gradient (Keddy, 1982; Watkinson, 1985). A more inclusive theoretical framework is needed which recognises both niche and source-sink ideas. Few statistical models of distribution have considered reproductive success though see Heegaard (2001) for an example of the occurrence of sink populations in an alpine moss. There is much less consensus regarding other theoretical concepts in plant ecology than the continuum concept and their role in spatial modelling.

2.1. Biotic components Fig. 1. Three alternative models of species response shape. The ﬁrst diagram shows the niche theory model. The second and third diagrams show possibilities when a superior competitor Assumptions about the shape of the response of displaces a species from its optimum while the fourth gives species to an environmental variable (usually details of labels and curves. (Austin and Smith, 1989, with kind termed an environmental gradient) are central to permission from Kluwer Academic Publishing.) M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 105 fail to address this issue when specifying the effect on the types of models needed to predict the mathematical forms to be used in fitting their distribution of shrubs and herbs in forests. statistical models. Austin and Smith (1989) tabu- lated the assumptions adopted by many ecologists 2.2. Environmental components in their conceptual models of vegetation. Many failed to specify the shape of species responses, Theoretical discussion of environmental gradi- changes in total biomass along the gradient or ents does not usually distinguish between types of species richness. Conceptual models of vegetation gradients (ter Braak, 1985, 1986; Austin and organisation are incomplete to judge from these Smith, 1989; Guisan and Zimmermann, 2000). tabulations. This may contribute to why many Two distinct classifications of environmental gra- models of species distribution fail to test theore- dients have been used, idealised types and distal or proximal gradients. tical assumptions (Franklin 1995; Austin, 1999b; Austin (1980) suggested there were three idea- Guisan and Zimmermann, 2000; Scott et al., lised kinds of gradients. These were direct, re- 2002). source and indirect gradients. Indirect gradients While the continuum concept of vegetation may are those where the variable has no physiological be accepted, little attention has been paid to what effect on plant growth or competition. Examples alternative patterns of species distributions might are altitude, latitude or longitude. Their correla- be compatible with the concept (Austin, 1985). tion with species distribution is due to their Gleason’s (1926) concept of species having indivi- location-dependent correlation with variables dualistic distributions along the gradient though such as temperature or rainfall. Direct gradients influenced by competition is only one possible are those that do have a direct physiological realisation. Trees, shrubs and herbs may vary influence on growth but are not consumed by the independently of each other but partition the plants, Examples are temperature and pH. Re- environmental gradient according to conventional source gradients are based on those variables that niche theory (Fig. 2). Other more complex possi- are consumed by plants such as light, water and bilities can be imagined. If the response of shrubs nutrients. These are idealised and not exclusive and herbs is dependent on the composition of the categories. Water can be a resource gradient under canopy layer in forest this could have a profound conditions of low availability and an indirect gradient when it is sufficiently abundant to cause anaerobic conditions due to waterlogging. Environmental predictors (gradients) may be either proximal or distal. Proximal and distal refer to the position of the predictor in the chain of processes that link the predictor to its impact on the plant. The most proximal gradient will be the causal variable determining the plant response. For example, available soluble soil phosphate concentration at the root hair would be a more proximal resource gradient than total soil phos- phorus. Indirect gradients are clearly distal variables. Species distribution models will have only Fig. 2. Four alternative patterns of vegetation organisation local value either for prediction or understanding along an environmental gradient. The a and b represent the when using distal variables. Models based on classical community and continuum models. c Shows the niche proximal resource and direct gradients will be the theory model with species partitioning the resource while d shows a possible pattern where trees, shrubs and grasses most robust and widely applicable. They will also partition the gradient independently. (Austin, 1985, with kind be the least practical in terms of knowing what to permission from Annual Reviews Inc.) measure or in terms of resources and time to 106 M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 measure them. It will be very difficult to provide Schluter, 1993; Rosenzweig, 1995). A case can be GIS coverage for proximal variables. Use of made for more critical use of regression models proximal variables for predictive mapping of when applied to collective properties of vegetation species distribution is, therefore, probably imprac- (Austin, 1999b; Pausas and Austin, 2001). Austin tical. (1999b) comments that quadratic equations have The expected shape of the species response will been fitted where the authors’ theory indicates that vary with the nature of the gradient. Response to a hyperbola should be used (Cornell and Karlson, an indirect gradient could take any shape. The 1997). shape would depend on the nature of the correla- Guisan and Theurillat (2000) modelled species tions between the indirect variable and the causal richness using a Poisson GLM. They then com- gradients (Guisan et al., 1999; Guisan and Zim- pared this pattern with that obtained by accumu- mermann, 2000; Franklin et al., 2000; Meente- lating predictions from individual species. The meyer et al., 2001). Guisan and Zimmermann results for current climatic conditions were similar. (2000) point out that in many cases ‘indirect The patterns differed when different climate variables usually replace a combination of differ- change scenarios were used with the models. ent resources and direct gradients in a simple way’. Assumptions about the patterns of response of This is likely to lead to robust predictive models collective properties are also critical when simu- only where the causal gradients are linearly lated data are used to assess the performance of correlated and those gradients are themselves different types of statistical models (Minchin, linearly correlated with the indirect gradient. 1987; Faith et al., 1987; Palmer, 1993; ter Braak Gradients of altitude at sub-alpine and alpine et al., 1993; Austin et al., 1995). The relationship altitudes may show this characteristic. Topo- between patterns of individual species response graphic position is another indirect factor that curves and response curves of collective properties may behave in this way, but once the lithology such as species richness remains to be elucidated. changes, the correlation structure between the variables will change. Austin and Smith (1989) 2.4. Ecological processes discuss the expected response shape to direct and resource gradients. They proposed certain shapes While the equilibrium assumption is essential but no definitive conclusions have been reached. for static modelling, other assumptions about ecological processes tend to remain unconsidered 2.3. Collective properties (Guisan and Zimmermann, 2000). Models of species distribution usually focus on environmen- When plants grow together, the resulting com- tal predictors. Processes such as dispersal, compe- munity can be described by certain collective tition, succession, fire and grazing pressure need to properties such as species richness, dominance be incorporated. Leathwick, in a series of studies and total biomass. The relationships between these on the distribution of New Zealand forest trees has properties have attracted considerable theoretical incorporated a number of these features (Leath- interest (Cornell and Lawton, 1992; Cornell and wick and Mitchell, 1992; Leathwick, 1995, 1998; Karlson, 1997; Naeem et al., 1994, 1996; Tilman et Leathwick et al., 1996, 1998; Leathwick and al., 1996, 1997; Partel et al., 1996). Frequently, Austin, 2001). Leathwick and Mitchell (1992) these studies use only simple regression models modelled the distribution of forest trees, in the (Austin, 1999b). More complex statistical models central North Island of New Zealand using GLM. have been used in exploratory analysis of species The region had been disturbed by the large Taupo richness patterns in relation to environment (Cur- Pumice eruption (130 AD). Depth of the pumice rie, 1991; Margules et al., 1987; Pausas, 1994; deposits was a significant predictor for several Austin, et al., 1996). Investigations of global species. The positive correlation of rapidly disper- patterns of species richness tend to emphasise sing conifers with the deeper deposits close to the indirect gradients such as latitude (Ricklefs and eruption centre was interpreted as a result of a M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 107

dispersal process rather than an environmental species behave differently in relation to environ- control. All forests close to the eruption centre mental predictors. Leathwick and Austin (2001) would have been destroyed. The slower dispersing introduced competition from the dominant tree Nothofagus species would normally dominate such species Nothofagus in modelling the spatial dis- sites but have yet to reach them. This is an example tribution of the density of these other forest where spatial autocorrelation should be an intrin- species. Prediction was also improved by making sic part of the model. Whenever a statistical model the competitive effect of Nothofagus species a is established with spatial autocorrelation or function of position on the environmental gradi- autologistic components, one of two assumptions ents (i.e. a term describing interaction between is being made (Wu and Huffer, 1997; Augustin et Nothofagus density and the environment value was al., 1996; Gumpertz et al., 1997; Weir and Pettit, used). The GAM results indicate substantial 2000). The model may have been mis-specified and reductions in abundance of other tree species as an important environmental predictor showing Nothofagus abundance increases. The magnitude spatial autocorrelation left out of the model. of the reduction varies with position on the Alternatively, biological processes associated with dominant environmental gradient (Leathwick species dispersal ability, historical disturbance or and Austin, 2001). Leathwick (2002) examines dispersal barriers such as mountain ranges may be further the competitive influence of different responsible. Use of spatial autocorrelation in Nothofagus species. distribution models without considering the pro- There are difficulties associated with introdu- cesses responsible can only lead to less than cing competitive terms into habitat modelling. It is adequate models. Both Gumpertz et al. (1997), difficult to distinguish whether absence and reduced abundance is due to competition or is due to Weir and Pettit (2000) consider covariates but an unidentified environmental variable. The their impact on the methodological comparisons Nothofagus example is exceptional. There is strong being made is not examined in detail. Augustin et circumstantial evidence that the Nothofagus spe- al. (1996) include environmental covariates in their cies are absent due to historical events. The study of red deer distribution but admit that herd probability of competitive interactions between behaviour should perhaps have been included in species is much reduced with plot data unless the the model. The interface between spatial autocor- species are sufficiently large and abundant for relation methods and ecological processes needs them to interact as neighbours. When present the greater attention. Nothofagus species are often the natural domi- Work using GAM by Leathwick (1995, 1998), nants and are the most abundant species on the Leathwick et al. (1996, 1998) on the phenomena of plot. At a plot size of 0.4 ha interactions with other ‘Beech Gaps’ in New Zealand emphasised the need species are probable. The occurrence of density to combine ecological knowledge with statistical levels sufficient for interactions to occur will not modelling. The gaps, which occur on both North be the case with less common species, reducing the and South Islands, do not correspond with any of value of this approach. Models based on pre- the environmental factors currently known to senceÁ/absence data are unlikely to detect such influence tree species distribution in New Zealand phenomena. The potential of simultaneous regres- (Leathwick, 1998). Nothofagus displacement is sion models for ecology (Brzeziecki, 1987) is likely thought to have occurred due to major historical to be restricted to very special circumstances for disturbance events, e.g. glacial advances, volcanic this reason. See discussion below of Austin (1971) eruptions and plate tectonics (McGlone et al., for an alternative approach to incorporating 1996). Their slow reinvasion is probably due to competition. Biotic processes as distinct from their limited seed dispersal and obligate ectomy- environmental processes are rare in statistical corrhizal association (Bayliss, 1980). In the ‘Beech models of species distributions and need greater Gaps’ where Nothofagus is absent, other tree consideration. 108 M.P. Austin / Ecological Modelling 157 (2002) 101Á/118

3. Use of ecological theory in plant community regression model y/a/b/x where x is annual 2 modelling rainfall achieves these outcomes (r /0.46). The number of degrees of freedom is the same and the How is ecological theory being used in statistical outlier is now incorporated in the model. An models of plant species distribution? There are a ecological interpretation of the model is that there number of problems at the interface with statistics is minimum rainfall below which no flowers are that need to be considered more carefully by the produced. Above this threshold, flowering in- analyst and perhaps applied more generally in creases with rainfall until a limit is reached above ecology. which rainfall has little or no effect. Statisticians will regard the example as an 3.1. Straight line models example of poor data analysis. If examined, the pattern of deviations in the residuals would have Regression models fitting straight-line relation- exposed the limitations of the straight line. Yet ships between plant responses and environmental statisticians when evaluating new approaches such variables are frequent in the ecological literature. as the Markov Chain Monte Carlo method on real Statements that curvilinear relationships were data for species distributions do the same. Wu and tested are usually absent from such cases, even Huffer (1997) model plant species distributions in though a curvilinear relationship would make Florida with straight-line relationships for climatic ecological sense. variables using logistic regression. They ascribe the An example typical of many in the literature pattern of residuals obtained to spatial autocorre- might concern the number of flowering stems lation in the species data. No mention is made of produced in a given year as a function of rainfall. probable mis-specification of the model. Three This example may seem trivial to many ecologists. methods of autologistic regression are then com- However, many multiple regressions consist solely pared using the same data. Gumpertz et al. (1997) of straight-line relationships without any concern fit straight-line relationships for the effect of soil for the ecological rationale of the relationship. variables on spatial pattern of a disease. They Statistical significance is sufficient. A straight line mention the possibility of other relationships but was fitted to the data. The relationship was do not pursue them. Weir and Pettit (2000) in 2 methodological study of spatial autocorrelation significant with r /0.30 (Fig. 3). There is one noticeable and influential outlier. An alternative using the common toad suggest the use of envir- model, however, exists which is equally parsimo- onmental covariates but only examine linear and nious, explains more of the variance and is quadratic spatial trend surface models. Latitude ecologically more rational (Austin 1991). The appears to be the indirect environmental gradient operating in this data set from Finland at the northern limits of the distribution of the common toad. Ecologically the model appears poorly specified. Temperature might be a more appropriate variable. What are the consequences of potentially mis-specifying the model on the evaluation of different statistical methods? One of the most popular statistical methods in plant ecology at the present time is canonical correspondence analysis (CCA, ter Braak, 1986). The method combines a multivariate ordination of species occurrence data with a constrained regres- Fig. 3. An example of straight-line regression where an equally sion maximising the correlation between the spe- parsimonious and ecologically more rational model is possible. cies ordination axes and selected environmental (From Austin, 1991, with permission of CSIRO Publishing.) variables. The species are assumed to have unim- M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 109 odal responses to the underlying environmental need not be the case. ter Braak (1986, 1987) makes gradients as specified by the ordination axes. clear that various transformations and scalars can Current practice is to estimate the underlying be used as the environmental predictors. Spline gradient as a linear (straight-line) combination of functions can be incorporated (W. Venables pers. environmental variables. An arbitrary selection of com.). A failure in communication between statis- 17 references from Birks and Austin (1992) ticians and ecologists is occurring. bibliography of papers using CCA all used linear A detailed review of the impact of straight-line combinations. Many of the variables would be regressions on ecological interpretation is re- classified as distal indirect variables, e.g. latitude, quired. altitude and aspect. A review of papers using CCA in volumes 10Á/12 of the Journal of Vegetation 3.2. Assumed ecological theory Science (1999Á/2001) gave 11 that did not mention the linear basis of the correlation. Twenty out of Statistical regression models are not normally the 22 papers do not consider the possibility of associated with an explicit ecological theory. One curvilinear relationships or that distal indirect exception is CCA. ter Braak (1985, 1986) makes variables may have very different relationships the ecological assumptions very clear: with the ordination gradients than direct or resource variables. Leps et al. (2000) explicitly 1) the species tolerances (niche breadth) are use interactions between variables in their con- equal; strained regression. However, they used principal 2) the species maxima (amplitudes) are equal; components analysis (PCA) rather than CCA 3) the species optima are homogeneously distrib- because the ordination axes were short as deter- uted along a gradient that is long compared mined by detrended correspondence analysis with species niche breadth. (DCA). Brunet et al. (2000) is one of the few papers reviewed to provide graphical evidence of ‘Homogeneously distributed’ refers to either the relationship between the correspondence ana- equally spaced optima or optima distributed lysis (CA) axes and the environmental predictor randomly from an uniform distribution. This variables. The main predictor was distance from implies a particular species-packing model (ter the woodland edge; their figure one shows that for Braak, 1986). It is a one-dimensional model. several sites the relationship between CA axis one Further axes are assumed to be independent of and distance would be better described by a each other. The model is very similar to the curvilinear function. Ejrnaes and Bruun (2000) standard niche theory discussed above. The as- using DCA found reciprocal and quadratic rela- sumptions define symmetric bell-shaped responses tionships for the DCA axes with pH and max- for the species. The influence of changes in species imum summer irradiation, respectively. richness along gradients is not addressed; see There is no ecological reason to expect these Pausas (1994), Austin et al. (1995) for examples variables to have linear relationships with the of well-defined patterns of species richness along underlying direct and resource gradients that environmental gradients. Changes in patterns of determine species patterns. There is no reason to dominance, another collective property of vegeta- expect that linear combinations of these variables tion, with associated changes in species amplitudes will provide good estimates of the gradients either. along gradients are not considered. The packing The model of species response used in CCA is clear model must be considered incomplete ecologically and theoretically acceptable to a degree, and the in the sense of Austin and Smith (1989). linkage with environment is explicit. In practice CCA is considered robust to changes in the the measurement of relevant environmental vari- assumptions (Palmer, 1993; ter Braak, 1987; ter ables is inadequate. The results will be subject to Braak et al., 1993; ter Braak and Juggins, 1993). the same failure of interpretation as in the one Palmer (1993) showed that CCA recovers the variable straight-line example given above. This distribution of vegetation plots in relation to the 110 M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 true underlying gradients. To do this, the ordina- tempted to test the propositions of Gauch and tion axes were regressed on predictors consisting Whittaker (1972) about species patterns along of the true gradients plus random variables. The gradients. He found that positive-skewed unim- outcome is not surprising; random variables are odal response curves were typical of eucalypt unlikely to be a match for the original generating forest in southeastern Australia. However, tree variables. The success of CCA was not examined species richness was found to increase along a when indirect predictors with complex functional temperature gradient (Austin, 1987). This was in relationships with causal proximal variables are contrast to Gauch and Whittaker (1972) who included. Ter Braak and colleagues (ter Braak et thought there was no consistent relationship of al., 1993; ter Braak and Juggins, 1993) used similar species richness with gradients from a subjective data generated by the same program COMPAS evaluation of their data. Other theoretical propo- (Minchin, 1987), to examine the performance of sitions could not he tested due to the confounding CCA and various statistical procedures ill-adapted effect of the gradient in species richness. Note to deal with unimodal species responses. CCA was running means were used to describe the species assessed as being robust to the relaxation of the response curves. Minchin (1989), in a more assumptions and as being better than the statistical extensive analysis using 100 vascular plant species alternatives. In neither the work of Palmer nor ter and statistical modelling, tested several hypoth- Braak were alternative methods better adapted to eses. He found that only 45% of species had a more complete ecological model of species unimodal, symmetric response curves. The modes responses such as non-metric multidimensional of the ‘major’ species were randomly distributed scaling using the BrayÁ/Curtis coefficient (Faith rather than evenly distributed. Other hypotheses et al., 1987) tested. gave inconsistent results between different struc- Guisan et al. (1999) provide a comparison of tural (growth form) species groups. Total species GLM with CCA. The GLM modelling included richness (alpha diversity) had a complex trend cubic polynomials and interaction terms. The surface pattern in relation to the two complex authors conclude that the spatial distribution of indirect environmental gradients studied. How- individual species is better modelled by GLM than ever, each structural group of species had a CCA. However, CCA may have some advantages response pattern in accord with the hypothesis for rare species with few positive observations. The that species richness shows a unimodal response combined use of information from other species in along environmental gradients. defining the underlying gradient helps to define the Two conclusions in addition to rejection of the behaviour of the rare species. Ejrnaes (2000) also bell-shaped curve hypothesis can be drawn from considers that DCA may have some advantages these studies. Changes in species richness along a over CCA. gradient are inconsistent with assumptions about The simulated studies (Palmer, 1993; ter Braak uniform distributions of species modes (ter Braak, et al., 1993) used reasonable vegetation data but 1986). Species richness is a collective property of the environmental data were unrealistic. The use vegetation, therefore, that needs to be incorpo- of real data always runs the risk that none of the rated into models (Margules et al., 1987; Huston, methods compared recovers the full truth. ‘Truth’ 1994; Pausas, 1994; Austin et al., 1996; Pausas and in real data sets must always remain uncertain Austin, 2001). Secondly, an adequate model of (Austin et al., 1995). In any research some community variation along environmental gradi- assumptions must be taken as given but should ents must take into account differences in response be questioned as soon as possible. patterns between species groups with different growth forms (Minchin, 1989). 3.3. Ecological theory tested Recently, others with more powerful statistical models have tested hypotheses about species Relatively few papers have attempted to test response curves and the appropriateness of the existing vegetation theory. Austin (1987) at- CCA model. Bio et al. (1998) tested the species M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 111 response shapes of 156 species in relation to six onmental gradients using GLM, GAM or CART environmental variables using GAM. Twenty- to describe the individual responses have not been three percent of the 156 models contained exclu- made. Species packing models have been neglected sively sigmoid (linear) and Gaussian response in hypothesis testing so far. One exception is the curves. (Note linear responses arise when species work of Austin et al. (1994), Austin and Gaywood distributions extend beyond the extremities of the (1994), Austin (1999b). It was hypothesised that gradient sampled.) The remaining 77% of models species responses along an environmental gradient had at least one response function that was would have a skewed shape with tails towards the smooth and more complex. Ejrnaes (2000) exam- mesic central portion of the gradient and modes ined also with GAM, the response functions of 146 with steep declines towards the extremes of the species in relation to a single pH gradient. He gradient. The process basis of this was that species found 20% of species with symmetric unimodal limits towards the extremes were determined by curves. In addition, Ejrnaes (2000) compared the physiological tolerances while internal limits were performance of modelling with environmental determined by competition (Austin, 1990). This predictors as opposed to using DCA axes as was confirmed by GLM models applied to eu- predictors. He avoided circularity by deriving calypt species in forests of southeastern Australia DCA ordinations minus the species being mod- (Austin et al., 1994; Austin and Gaywood, 1994). elled. Twenty-two species were modelled. Better The generality of the result requires testing. predictive models were obtained with DCA axes Eucalypt forests in Australia are a unique vegeta- for 19 of the species. The conclusion is drawn that tion type. All the canopy species belong to one the species are responding to the same fundamen- genus (sensu lato). Although several environmental environmental gradients. One may conclude tal predictors were used in the successful models, other species are better estimators of the gradients the pattern was only found for the temperature than an arbitrary selection of environmental vari- gradient. The pattern applied to species with ables. Ejrnaes (2000) suggests that this may be a temperature optima within the range studied, i.e. more important effect than deviations from the temperate species; those species with optima at symmetric unimodal response (Austin and Gay- higher temperature were not included, i.e. sub- wood, 1994). Vayssieres et al. (2000) also examined tropical species with truncated responses in the the shape of species responses during a compar- region. ison of GLM and CART methods using three Oak This work has led to controversy. This con- species. Two of the three species were fitted with cerned the estimation of the coefficients of the b- a g models including cubic polynomials and interac- function ((x/a) ,(b/x) ) when it is used to fit tion terms. species response curves, particularly the depen- Guisan et al. (1999) in a comparison of GLM dence of the a and g values on a and b values and CCA found polynomials and interaction (Oksanen, 1997). Austin and Nicholls (1997) terms were useful in modelling species distribu- responded demonstrating that the results de- tions. They concluded that GLM gave better pended critically on how data limits (i.e. a and b models for individual species but that CCA had values) were set for modelling individual species. some positive features. All the studies agree that Oksanen and Minchin (2002) have now examined symmetric unimodal responses are rare and a wider range of possible models for the shape of skewed curves predominate. More detailed com- species response curves. A significant issue re- parison is difficult because no two studies have mains. Should species whose response curves are used the same approach or similar types of clearly truncated, i.e. their range extends beyond vegetation data. Ejrnaes (2000) and Guisan et al. the limits of the gradient sampled, be used in (1999) agree that CCA types of constrained theoretical studies of response shape? There is then regression haveadvantages but there is no con- no data on which to base an estimate of the species sensus (Austin et al., 1994; Bio et al., 1998). limit. Failure to have an agreed data model is Detailed analyses of species patterns along envir- confusing tests of theory. 112 M.P. Austin / Ecological Modelling 157 (2002) 101Á/118

Progress in testing hypotheses concerning vege- while proximal variables assumed more impor- tation patterns has been slow. Progress in statis- tance as predictors under more optimal conditions tical methods, remote-sensing and GIS has for the species. The regression models and the outstripped that in ecological theory (Franklin et importance of competition effects varied with al., 2000). Even when the purpose is prediction, a plant community the species was occurring in. better synthesis of theory will inform modelling Initial regression modelling of the stem density per and less biased predictions will result. A systematic plot of Eucalyptus roysii in relation to simple approach to testing the ecological theories implicit indirect environmental predictors yielded negligi- in many of the current predictive modelling ble results (Fig. 4). Inclusion of curvilinear func- approaches is needed. tions made little difference. A series of strategies were then introduced into the regression-building process based on ecological ideas. The data were 4. Role of descriptive analytical ecology stratified into different plant communities based on a numerical classification of the 261 sites using Since a comprehensive theory of vegetation will 352 species. E. rossii occurred in only four of the not result from correlative studies of vegetation/ eight communities recognised. Limits were set on environment relationships alone, what role is there the data to exclude those sites clearly beyond the for such descriptive studies? Any mechanistic realised niche of the species (mean annual rain- process model of ecosystem dynamics should be fall/810 mm and altitude above 1100 m). More consistent with a static, quantitative and rigorous proximal and direct variables were estimated from description of the same ecosystem. Statistical the available environmental variables. A solar habitat models offer a means of achieving that radiation index was calculated from slope, aspect description. Discrepancies between process model and latitude, termed a derived variable in Fig. 4. and statistical model will demonstrate our lack of Moisture stress indices were estimated from these knowledge and may also indicate the way forward. derived variables using a water balance model. The Before progress can be made we must recognise most abundant eucalypt species in each commu- current limitations. nity were then added to the regression models as There are too few studies to evaluate how best potential competition variables. Finally, as the to measure either the vegetation or the environ- variance was found to be increasing with the mean, mental determinants of vegetation (Guisan, 2002; a weighted regression procedure was introduced. Guisan and Harrell, 2000; Guisan and Zimmer- The increase in ecological realism increased the mann, 2000; Franklin et al., 2000). We know little predictive success of the models, at least in terms about the relative importance of different environ- of the adjusted r2 (Fig. 4, Austin, 1979). The mental and biotic variables in different biomes importance of different steps varied with the (Meentemeyer et al., 2001). Austin and Smith community. An ecological interpretation was also (1989) summarised the incompleteness of many possible for the models obtained for the individual conceptual frameworks for plant community ecol- communities. The model for the western dry ogy. There has been little progress in theoretical sclerophyll forest dominated by E. rossii, con- plant community ecology since, which is relevant tained curvilinear terms for various components of to predicting the realised niche of a species (Austin the water balance (Austin, 1971). These included and Gaywood, 1994; Austin, 1999a,b; Guisan and moisture stress for the main drought period, Zimmermann, 2000). available soil moisture and mean annual rainfall. A regression study (Austin, 1971) provided No competition terms were significant. The model some insight into the relationships between distal for the eastern dry sclerophyll forest where E. and proximal variables, indirect and direct gradi- rossii was a subordinate species, contained com- ents and plant competition when re-examined petition terms for two eucalypt species. Other (Austin, 1979). Distal variables were shown to be terms were moisture stress index plus simple linear important near the limits of a species distribution terms for rainfall, pH and solar radiation index. M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 113

Fig. 4. Relative success of exploratory regression analysis predicting E. rossii stem density in dry sclerophyll forest in south-eastern Australia. Success as measured by r2 increases after stratiﬁcation and with increasingly relevant ecological and statistical models. (Redrawn from Austin, 1979, with permission of International Co-operative Publishing company.)

Distal variables such as rainfall are likely to be the 5. Conclusion limiting factor in any water balance model when a species is at the limits of its distribution. It is under There is an intimate relationship between theory these same conditions that competition from other and method. In plant community ecology that species is most likely to have an influence. The relationship is often neglected, as it is in other models are consistent with ecological expectations branches of ecology (Austin, 1999a,b; Scott et al., (Austin, 1971). The importance of distal and 2002). When vegetation studies are interfaced with proximal variables varies with the ecological con- statistics, the problems of the relationship of text. Where gradients are steep and environments theory and methodology in two distinct disciplines are extreme, simple distal variables will be as have to be integrated. The studies reviewed here reveal two areas successful in modelling as environmental process where ecological theory is inadequate and methods based models such as moisture stress. Where suspect. These are patterns of species composition environment is changing slowly and a species is and estimation of environmental predictors. occupying its optimal realised niche, then proximal variables will be more successful predictors. The 5.1. Patterns of vegetation composition success of different habitat modelling strategies will be context-sensitive. A similar exercise on the There is no synthesis of theories regarding the same data set with a different species, E. manni- expected pattern of vegetation composition in fera, but using GLM, gave similar results (Austin relation to environment (Gauch and Whittaker, and Cunningham, 1981). A similar modelling 1972; Grime, 1979; Austin and Smith, 1989; strategy needs to be tested on data sets from Keddy, 1989; Huston, 1994). The collective prop- different biomes. erty, species richness is accorded great importance 114 M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 in experimental plant community ecology (Naeem mann, 2000; Iverson and Pasad, 1998; Franklin et et al., 1996; Tilman et al., 1996; Austin, 1999b). al., 2002; Cawsey et al., 2002), and to use either

Yet its impact on vegetation analysis methods such presenceÁ/absence data or cover/abundance values. as CCA remains to be examined. Clear patterns of Guisan and Harrell (2000) indicate that better response to environmental gradients at a regional statistical models (ordinal regression) are required scale have been demonstrated (Margules et al., for use with cover/abundance data. Preferred 1987; Minchin, 1989; Pausas, 1994; Austin et al., statistical models at present are GAM (Yee and 1996; Leathwick et al., 1998). These patterns do Mitchell, 1991; Leathwick, 1995), or CART not relate in any simple fashion to individual (Franklin, 1998). Use of these methods to define species responses, though patterns of richness do the functions to be incorporated into a parametric depend on growth form types (Minchin, 1989). GLM needs to be considered. GAM may produce a complex function where an equivalent para- 5.2. Estimation of environmental predictors metric function may capture most of the same variation and have a more reasonable ecological The second area is in the development of explanation. CART by its nature must produce measures for environmental predictors and the discontinuities when a continuous function may be extent to which knowledge of environmental more realistic. Best practice may be to analyse data process models should be incorporated (Austin, with both GAM and CART using the results to 1979; Guisan and Zimmermann, 2000; Franklin et construct as far as possible a GLM with ecologi- al., 2000). The studies of Austin (1971, 1979), cally rational functions. It is clear, however, that Austin and Cunningham (1981), Leathwick (1995, best practice will need to incorporate spatial 1998), Leathwick et al. (1996, 1998), Leathwick autocorrelation (Leathwick, 1998 for example) and Austin (2001) suggest that process models and probably by means of an autologistic approach even variables for competition from other species (Augustin et al., 1996). Closer collaboration with have a role to play in static statistical models. The statisticians is likely to see many new methods circumstances when they should be used and the being examined (T. Hastie pers. com.). There are strategy for introducing such variables into the further complications from an ecological theory modelling procedure remain uncertain. Statistical viewpoint that need to be examined in future. The theory and methods for spatial autocorrelation role of ecotypic variation within a species influen- and regression smoothing will have a big impact in cing species response curves remains to be exam- how we address the ecological issues. The relative ined. The modelling of species reproductive neglect of interaction terms in recent vegetation potential in response to environment is in its modelling is a current limitation to progressing infancy (Heegaard, 2001). The inconsistency be- beyond a one-dimensional theory of vegetation tween the realised niche and source-sink concepts composition. The apparent power of CART of species distributions (Pulliam, 1988; Heegaard, techniques (Austin et al., 1995; O’Connor et al., 2001) has to be resolved for plants. The upper and 1996; Iverson and Pasad, 1998; Franklin, 1998)to lower limits in relation to temperature will not be detect interactions is only one example of the need determined by the same proximal variables so for new strategies for predictive habitat modelling species statistical response models are unlikely to if they are to address ecological theory as well as have simple unimodal functions. The estimation of predicting species distributions for immediate spatial autocorrelation effects may help in detect- practical conservation purposes. ing this phenomenon and other biotic processes influencing distributions. It remains to be seen to 5.3. Data model and statistical methods what extent vegetation is in equilibrium with environment and the implications of this for static Current practice with respect to the data model analysis. Analytical description of species distribu- is to emphasise the use of stratified sampling tions and the collective properties of vegetation (Austin and Heyligers, 1989; Guisan and Zimmer- can address these issues. Ultimately, theory, phy- M.P. Austin / Ecological Modelling 157 (2002) 101Á/118 115 siological process knowledge, and dynamic mod- Heglund, P.J., Samson, F., Hauﬂer, J., Morrison, M., elling must be tested for explanatory power against Raphael, M., Wall, B. (Eds.), Predicting Species Occur- quantitative description of the natural patterns. rences: Issues of Accuracy and Scale. Island Press, Covelo, CA, pp. 73Á/82. Austin, M.P., Cunningham, R.B., 1981. Observational analysis

of environmental gradients. Proc. Ecol. Soc. Aust. 11, 109Á/ Acknowledgements 119. Austin, M.P., Heyligers, P.C., 1989. Vegetation survey design for conservation: gradsect sampling of forests in North-

I thank J. Reid, A.O. Nicholls, A. Guisan and eastern New South Wales. Biol. Conserv. 50, 13Á/32. R.J. O’Connor for comments on the manuscript Austin, M.P., Gaywood, M.J., 1994. Current problems of and to S. Marsden for help with its preparation. environmental gradients and species response curves in relation to continuum theory. J. Veg. Sci. 5, 473Á/482. Austin, M.P., Nicholls, A.O., 1997. To ﬁx or not to ﬁx the species limits, that is the ecological question: response to

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