AbstractQuestions: Can a statistical model be designed to representmore directly the nature of organismal response to multipleinteracting factors? Can multiplicative kernel smoothers beused for this purpose? What advantages does this approachhave over more traditional habitat modelling methods?Methods: Non-parametric multiplicative regression (NPMR)was developed from the premises that: the response variablehas a minimum of zero and a physiologically-determinedmaximum, species respond simultaneously to multiple eco-logical factors, the response to any one factor is conditionedby the values of other factors, and that if any of the factors isintolerable then the response is zero. Key features of NPMRare interactive effects of predictors, no need to specify anoverall model form in advance, and built-in controls onoverfitting. The effectiveness of the method is demonstratedwith simulated and real data sets.Results: Empirical and theoretical relationships of speciesresponse to multiple interacting predictors can be representedeffectively by multiplicative kernel smoothers. NPMR allowsus to abandon simplistic assumptions about overall modelform, while embracing the ecological truism that habitat fac-tors interact.
Non-parametric habitat models with automatic interactions
McCune, Bruce
Department of Botany and Plant Pathology, Oregon State University, Corvallis, OR 97331 USA;E-mail [email protected]
Introduction
Eugene Odum (1971) stated Shelford’s ‘law’ oftolerance as follows: “The presence and success of anorganism depend upon the completeness of a complexof conditions. Absence or failure of an organism can becontrolled by the qualitative or quantitative deficiencyor excess with respect to any one of several factorswhich may approach the limits of tolerance for thatorganism.”
This straightforward and manifest statement has de-fied a correspondingly simple and general statisticalrepresentation with the traditional tools used by ecolo-gists. These tools are unnecessarily constrained byadditivity of model terms and a limited array of func-tional forms. The problems in applying simple linearand logistic models to species responses to multipleinteracting predictors have been clearly stated (Kaiser etal. 1994; Huston 2002; Cade & Noon 2003). This paperdemonstrates how non-parametric multiplicative regres-sion (NPMR) provides a simple, effective solution tothe problem of representing empirical species responsesurfaces in a multidimensional niche space. Interactionsamong predictors are accommodated automatically andthe overall form of the response surface need not bespecified.
Habitat models describe how variation in speciesperformance relates to one or more predictors. Meas-ures of species performance include presence-absence,abundance, physiological rates, and demographic pa-rameters (e.g. nesting success). Commonly used predic-tors include environmental variables (including bioticvariables), site characteristics, time since disturbance,and other descriptors of disturbance regime.
Habitat models can take both conceptual and statis-tical forms. Conceptual habitat models have been forma-tive in ecological theory, for example, the Hutchinso-nian niche, an n-dimensional hypervolume (Hutchinson1957), and Whittaker’s diagrams of species responsesto environmental gradients (e.g. Whittaker 1956). Statis-tical habitat models have been made for many species ofparticular concern. These models describe the importantfactors underlying a species’ distribution and abundance,
820 McCune, B.
inform management decisions for rare or threatenedspecies, and allow comparison of probable outcomes ofalternative management strategies.
Ecologists readily grant complex species responsefunctions in theory, but often use simplistic forms (lin-ear or logistic; Rushton et al. 2004) that cannot hope tocapture the complexity of a species in relationship to itshabitat (Kaiser et al. 1994; Heglund 2002; Huston 2002).These simple forms do not accommodate the widely-accepted view that species have hump-shaped responsesto environmental gradients. These hump-shaped responsecurves are implicit in Shelford’s law of tolerance. Ofcourse non-linear transformations of environmental gra-dients, such as Gaussian logistic regression (Huisman etal. 1993), can render hump-shaped response functionswithin the framework of linear models, but still con-strain the model to particular functional forms. General-ized additive models (GAMs, e.g. Maravelias & Reid1997; Mysterud et al. 2001; Begg & Marteinsdottir2002) offer a more flexible approach by combiningsmoothing functions, though interactions must still bemodelled explicitly (e.g. Ciannelli et al. 2004). Like-wise, multivariate adaptive regression splines (MARS)provide a flexible way to fit complex surfaces, includingthe possibility of multiplicative basis functions thatshould readily accommodate interactions (Friedman1991; Hastie et al. 2001).
Multiplicative kernel smoothers can also be used torepresent the complexity of species responses to multi-ple interacting predictors. This kind of model providestwo important advantages over other approaches: itautomatically represents predictor interactions by com-bining predictors multiplicatively, such that the effect ofone predictor can covary in a complex way with otherpredictors, and it requires no assumptions about theoverall shape of the response surface. It accepts thatcomplex interactions may result in the response surfacein one part of the predictor space having no simplefunctional relationship to responses elsewhere in thepredictor space. The chief disadvantages of multiplica-tive kernel smoothers are that the response surface mustbe fitted with a computationally intensive trial-and-error method and the results do not include an equationrelating the response to the predictors. Instead, interpre-tation must rely on graphical visualization, measures offit, and sensitivity analysis for individual predictors.Using these models in an explorative way informs, butdoes not preclude, parametric modeling. Having ex-plored the response surface in a multidimensional space,one can then sensibly choose an appropriate functionalform and proceed with non-linear regression or a gener-alized linear model, if desired.
The extension of kernel smoothing multiplicativelyinto many dimensions, and its combination with cross-
validation, provides an easy, intuitive way to fit parsi-moniously a species response surface to multiple pre-dictors. A few papers in the literature have used similarsmoothers, but differ in important ways from the methodproposed here. Gignac et al. (1991a, b) generated 3Dresponse surfaces for species abundance from griddedabundance data along environmental gradients, usingdistance-weighted means. Limitations to that approachincluded an arbitrarily selected (rather than optimized)search radius, arbitrary treatment of zeros and outliers,and no cross-validation. Locally-weighted smoothing(or regression) using the LOWESS method has beenapplied to habitat models in two and three dimensions(Huntley et al. 1989, 1995). These models were effec-tive but had several drawbacks: (1) restriction to two orthree pre-selected predictors instead of a conducting afree search for the best model using an indefinite numberof variables from a pool of available predictors, (2)choosing the smoothing parameter for each predictorarbitrarily, rather than simultaneously optimizing it inall dimensions, and (3) fitting the model at fixed inter-vals within the plane of predictors, with linear interpola-tion between intervals, rather than fitting the model foreach data point.
Multiplicative models
Habitat models and other species response modelsare most often additive, including those created by theusual least-squares multiple linear regression, general-ized linear models (GLMs, including logistic regres-sion), and generalized additive models (GAMs). In thelatter two, using a log link function makes the log(mean)an additive function of the predictors, but the mean is amultiplicative function of the predictors. In GLMs andGAMs, interactions are accommodated by terms includ-ing more than one predictor. This paper proposes a formof non-parametric multiple regression in which the re-sponse is estimated from a multiplicative combinationof all predictors.
Consider an extreme hypothetical example wherewe place an experimental population in the Antarctic,with some amount of food and shelter. Our speciesresponse variable, y, is the reproductive rate, x1 is food,and x2 is shelter. The model y = b1x1 + b2x2 + b0 says thatthe reproductive rate is increased by the availability offood and shelter, and that increasing either of these alonecan increase the reproductive rate. The simple additivemodel comes to the erroneous conclusion that reproductiverates will be high if the population has abundant food butno shelter. Likewise, the model says the population willreproduce if given lots of shelter but no food. Interactionterms in linear models represent the curved responsesurfaces that we expect from interacting predictors, but
- Non-parametric habitat models with automatic interactions - 821
these surfaces are relatively simple functions (in thesimplest case, hyperbolic paraboloids) applied over thewhole predictor space. We have no reason to expect thatsurfaces representing interactions will take this limitedrange in form.
The simplest GAM for this problem would be:g[µ(x)] = α + f1(x1) + f2(x2), where f is an unspecifiedsmooth function and g is a link function of µ (forexample g(µ) = µ is the identity link, or g(µ) = log(µ) isa log link). With the latter, the model is in one sensemultiplicative, in that µ = eα · e f
1(x
1) ∑ e f
2(x
2), but it is still
additive in that the effects of food and shelter are addedindependently to estimate the log of the reproductiverate. In biological terms this means that the effect offood on log(µ(x)) does not depend on shelter and vice-versa. It makes only slightly more sense for the log ofthe mean response to be an additive result of food andshelter than for the mean to be an additive result. On theother hand, if our interest is in µ rather than log(µ), thenin the multiplicative form, if x1 is highly unfavourable,differences in x2 will have little effect on µ. A moredirect GAM for the Antarctic example is g[µ(x)] = f(x1,x2), with the constant set to zero (no reproduction) andusing a non-parametric smoothing function combiningx1 and x2 interactively and multiplicatively. GAM withinteraction terms is challenging (e.g. see Ciannelli et al.2004), out of reach for most of the grassroots practition-ers of habitat modeling. A simpler, more intuitive ap-proach uses multiplicative kernel smoothers to allow theeffect of each predictor to depend on the value of otherpredictors, without needing to specify those interac-tions. This is non-parametric multiplicative regression(NPMR). The method is implemented in easy-to-usesoftware in HyperNiche (McCune & Mefford 2004).
Non-parametric models
Ecological theory does not reliably inform us as tohow a particular species responds to a particular habitatfactor, much less to combinations of interacting factors.In general, however, we expect species responses tomultiple factors to be complex, including non-linear,asymmetric, and multimodal responses. While thesechallenges are readily addressed in models of speciesresponse to a single factor, the problem becomesexponentially more difficult as the dimensionality of thedata increases. This is the ‘curse of dimensionality’(Bellman 1961).
Yet assumptions about the shape of a species re-sponse to environmental variables are central to anypredictive parametric model (Austin 2002). NPMR cir-cumvents this; predictive modeling can be effective with-out making any assumptions about the shapes of speciesresponses to ecological factors or to their interactions.
Non-parametric multiplicative regression
This section derives the logic for using a multiplica-tive kernel smoother from basic biological principles.The goal is to provide an intuitive biological basis forthe statistical approach. The method is then comparedwith additive models in three examples.
We seek to describe or predict an organism’s per-formance in relationship to its environment. An organ-ism’s ‘environment’ includes not just abiotic factors,but also its biotic environment, including competition,predation, and disease. Assume further that we are meas-uring organismal or population performance in relationto environmental variables, and that performance has aminimum of zero and increasing into the positive realnumbers. Some examples are population density, arealcover, rate of reproduction, and rate of respiration. Thefollowing treatment pertains to this class of variablesrepresenting organismal performance.
Axioms
1. Performance of a species has a maximum set byphysiology and morphology. The maximum isfuzzy, rather than a set value, because of geneticvariation among individuals. This axiom refers toshort, ecological time scales, excluding the possi-bility of evolution.
2. As environmental factors weaken performance, apopulation collapses (its organisms die) and per-formance is minimized at zero.
3. Organisms respond simultaneously to many envi-ronmental factors.
4. If any single factor or combination of factors isintolerable, then the organisms in a population dieand performance is zero, regardless of the values ofother environmental variables.
Definitions
Environmental space: A multidimensional space, them coordinates of the space defined by the m measuredenvironmental variables, including both biotic and abi-otic variables. A particular point v in this environmentalspace is thus the vector defining the value of each of them variables: [v1, v2 ... vm], where v can be any realnumber. In habitat modeling, the environmental space isused as the predictor space. It can be used as an opera-tional definition of a niche space.
Species performance (yv): The performance of an indi-vidual or population of individuals at point v in environ-mental space, where yv ≥ 0.
822 McCune, B.
Environmental measurement (xij): The value of predic-tor j for data point i.
Interaction: Response to predictor j depends on valuesof other predictors.
Describing species performance in relationship toenvironment
Usually we are interested in species performanceover a range of multiple environmental factors. Wecollect a sample of species performance, along withmeasured environmental factors (predictors) for n datapoints in the environmental space. We build a responsesurface of y from its relationship to m predictors in X.
y f x x xi m= ( )1 2, ,..., (1)
A multiplicative kernel smoother allows all predic-tors to interact: the effect of each xj can vary with theothers, and the response surface in one region of envi-ronmental space need not bear any relationship to theresponse surface in other regions of that space.
We need to use data from near target point v to helpestimate the response at that point because (1) all of ourmeasurements have error, (2) we need to interpolatebetween data points, and (3) we wish to borrow infor-mation from nearby points to help estimate a response ata particular target point, because in most data sets notwo cases will occupy the same point in environmentalspace.
The tolerance (sj) of a species to a continuouspredictor j defines how broadly we borrow informa-tion from nearby areas in the predictor space, whileattempting to estimate the value at a target point. If aspecies is broadly tolerant to that factor, then we useinformation from a large neighbourhood of data points,while a species with narrow tolerance to that factor isbetter represented by using only data points that areclose to the target point in the predictor space. Asmooth way of representing the neighbourhood of thetarget point is to use a Gaussian weighting functioncentred on the target point v, the function expressingthe weight (w) given to each sample point i in estimat-ing the response at point v, based on the differencebetween xi and v, scaled by the standard deviation(tolerance) to that predictor:
w eij
x v sij j j
=− −( )⎡
⎣⎤⎦
1
2
2
(2)
This univariate weighting function (the kernel; Bow-man & Azzalini 1997; Hastie et al. 2001) specifies theweight (wij) for an observation of a single predictor jat sample unit i, drawn from the matrix X of n obser-vations for m predictors. This weighting can take
various forms, but this Gaussian function is simpleand intuitive. The standard Gaussian probability den-sity function was modified so that the peak height isalways one and the area under the curve varies. Notethat use of Gaussian weights does not limit the globalmodel to Gaussian forms; the weighting function islargely independent of the global model. Otherweightings can be used, for example, a uniform weightof one within an observational window of a specificoptimized width, and zero weight outside the window(McCune et al. 2003), but this gives a relativelyrough response surface.
With categorical predictors a different approachis needed. The simplest method is to apply binaryweights: an observation is given full weight for agiven predictor if xij and the target point were as-signed to the same category (wij = 1 if xij = vj);otherwise, that observation is given zero weight inestimating the response at point v (wij = 0 if xij ≠ vj).An intriguing refinement would be to allow fuzzycategories, such that weights between zero and oneare allowed for categorical predictors.
The multiplicative local mean estimator
We can then estimate the response y at target point vas:
y
y w
wv
ii
n
ijj
m
ijj
m
i
n=
⎛
⎝⎜⎞
⎠⎟
⎛
⎝⎜⎞
⎠⎟
= =
==
∑ ∏
∏
1 1
11∑∑
(3)
This is a local mean estimator extended multipli-catively to m dimensions. In words, the estimate of theresponse is an average of the observed values, eachvalue weighted by its proximity to the target point in thepredictor space, the weights being the product of weightsfor individual predictors. The model allows interac-tions, because weights for individual predictors are com-bined by multiplication rather than addition. A keybiological feature of the model is that failure of a popu-lation with respect to any single dimension j of thepredictor space results in failure at point i, because theproduct of the weights for point i is zero if any wij = 0.
If point v is one of the n sample points, from whichthe response is estimated, we can reduce overfitting byexcluding point i when it is the same as point v:
- Non-parametric habitat models with automatic interactions - 823
ˆ ,
,
y
y w
wv
ii i v
n
ijj
m
i i v
n
ijj
=
⎛
⎝⎜⎞
⎠⎟= ≠ =
= ≠ =
∑ ∏
∑
1 1
1 11
m
∏⎛
⎝⎜⎞
⎠⎟(4)
The notation i ≠ v indicates that if the target point v isone of the calibration data points, then it is excludedfrom the basis for the estimate of yv. This is the funda-mental equation for cross-validated local mean NPMR.Local linear NPMR is the same except that it is based ona locally weighted linear relationship (App. 1), ratherthan a local mean. The local model can take other forms,such as logistic or more complex polynomials.
With multiple categorical predictors, an observationis given full weight only if the target point matches allcategorical predictors, otherwise the observation is givenzero weight. With mixed categorical and quantitativepredictors, the weights are multiplied as usual.
Multiplicative kernel smoothers are not new (e.g.Bowman & Azzalini 1997; Hastie et al. 2001), but thisspecial form is noteworthy for ecologists modeling abun-dance or other performance variables with zero as anatural lower bound. It simply represents the axiom thatorganisms must simultaneously meet all environmentalchallenges or die.
NPMR is, therefore, a particular class of smoothingfunctions, in which an estimate for a particular targetpoint in predictor space is made by combining informa-tion from observations nearby in the predictor space.The closer a data point is to the target point, the moreweight is given to information from that point. Howrapidly weights diminish with distance can be tuned foreach predictor with a smoothing parameter, in this casethe standard deviation of the Gaussian curve (‘toler-ance’ sj to a predictor j). Selecting a large standarddeviation is comparable to having a broad window;conversely a small standard deviation gives appreciableweight only to observations that are very close to thetarget point in the predictor space. The Gaussian func-tion is scaled to the predictor by arbitrarily setting onestandard deviation equal to one sixth of the range of thepredictor, thus representing a Gaussian curve with ± 3standard deviations over the range of the predictor.Then for each predictor, sj is set to maximize fit, subjectto model fitting constraints described below.
For every point estimate of the response variable onecan calculate a neighbourhood size (nv*), the amount ofdata bearing on that particular estimate:
n wv ijj
m
i i v
n* *
,
=⎛
⎝⎜⎞
⎠⎟== ≠∏∑
11(5)
where 0 < n* ≤ n for a Gaussian kernel. If n* = 0, as ispossible with some other kinds of kernels, then noestimate is possible for that point because no data applyto it. Setting a minimum n* required for an estimateprotects against estimating a response in a region of thepredictor space with insufficient data.
Model fitting and evaluation
Fitting an NPMR model requires simultaneous se-lection of predictors and their tolerances from a pool ofavailable predictors, so as to maximize a measure of fitwhile satisfying criteria for parsimony. This demandsan iterative search through a potentially enormousnumber of possible models, which is accomplished withthe software HyperNiche (McCune & Mefford 2004).Variables are added in forward stepwise fashion, at eachstep making a grid search of variables and their toler-ances. Variables already in the model are simultane-ously evaluated for removal or change in toleranceswith the addition of a new variable.
Using leave-one-out cross-validated statistics for fitreduces overfitting and results in more realistic errorestimates. Incorporating the cross-validation into notjust model evaluation but also model fitting expeditesthe search for a parsimonious model. For quantitativedata, model fit can be evaluated by the size of theresidual sum of squares (RSS) in relationship to the totalsum of squares (TSS):
xRRSS
TSS
y y
y y
i ii
n
i ii
n2
2
1
2
1
1 1= − = −−( )
−( )=
=
∑
∑
ˆ
(6)
This ‘cross R2’ differs from the traditional R2 becausedata point i is excluded from the basis for estimating yi .Consequently, with a weak model, it is not uncommonfor RSS > TSS and xR2 < 0. This method is similar to theuse of G-values (Agterberg 1984; Gotway et al. 1996;Guisan & Zimmerman 2000).
For binary response data (presence-absence) a meas-ure of fit was sought that could be applied to any methodof estimating likelihood of occurrence and would avoidthe arbitrary conversion of continuous estimates of prob-ability of occurrence into a statement of ‘present’ or‘absent’ (Fleishman et al. 2003). The xR2 is inappropri-ate in this case because the goal is to estimate probabili-ties from presence-absence data, rather than producing
824 McCune, B.
estimates that exactly match the data. Log likelihoodratios met these criteria, expressing model improvementover a ‘naïve model’. A naïve model is simply that ourbest estimate of the probability of encountering a speciesin a study area is the average frequency of occurrence ofthat species in the data. The ratio of the likelihood of theobserved values (y = y1, y2, … yn) under the fitted model(M1) to the likelihood of the result under the naïvemodel (M2) is given by:
Bp M
p M121
2
=( )( )y
y (7)
where
p M y yiy
i
y
i
ni iy( ) = −( ) −
=∏ ˆ ˆ1
1
1(8)
and yi corresponds to the fitted values for the likelihoodof occurrence under each model, Mj, j = 1,2.
Formal hypothesis testing with log-likelihood ratiosrequires that the parameters for one model be nestedwithin the other and incorporates the difference in de-grees of freedom between the two models. Log10B isapplied here, however, as a descriptive statistic in thesense of ‘weight of evidence’, similar to a Bayes factor(Kass & Raftery 1995), rather than a formal hypothesistest. LogB differs from a Bayes factor in that a Bayesfactor is based on the marginal distribution of y giventhe prior model (the naïve model in this case), whilelogB is a simple log likelihood ratio for the two models,inverted so that as the weight of evidence increases,logB increases. Values of logB reported here from NPMRmodels are based on cross-validated estimates from theM1 using a leave-one-out strategy. LogB can be inter-preted as the ratio of the likelihood of cross-validatedestimates from the fitted model to estimates from thenaïve model expressed in powers of ten. LogB is nega-tive when cross-validated estimates from the fitted modelare worse than the naïve model. The same rationale canbe applied to the difference between logB values calcu-lated for each of two competing models of interest.Because logB is unbounded, it can be quite large when astrong relationship is modelled with a very large dataset. The average contribution of a sample unit to logB,10(logB)/n, can be used to describe the strength of rela-tionship, independent of sample size.
Controlling flexibility and parsimony
Selecting the best model from the multitude of mod-els with many predictors can lead to overfitting the data.Overfitting is particularly problematic with small datasets, a large number of predictors relative to the sample
size, or clumped sampling from the predictor space. TheNPMR models presented here control overfitting inseveral ways simultaneously: built-in cross validationduring model fitting, a flexibility control, and settingparsimony criteria to control inclusion of predictors.Each control restricts a different aspect of overfitting.Flexibility of the response surface can be controlled bysetting a minimum acceptable average neighbourhoodsize, N*, where N* is the average of ni*. Stiff curves(large N*) are needed with small data sets or clumpeddata distribution along important habitat dimensions.More flexibility is allowable with large high-densitydata sets. A reasonable starting point is to set minimumN* = 0.05(n).
Parsimony in number of predictors is partly control-led by cross-validation and partly by the minimum N*.It can be further controlled by setting an improvementcriterion, expressed as a percentage improvement inmodel fit when a new predictor is added. This criterionis particularly important for parsimonious models withlarge data sets. One can also set a minimumdata:predictor ratio, an effective criterion for smalldata sets. For quantitative responses, the data:predictorratio is the number of sample units divided by thenumber of predictors in the model. For binary re-sponses, the data:predictor ratio is the number of obser-vations in the least represented category (presences orabsences) divided by the number of predictors in themodel. Some suggest a minimum ratio of 10 for binarydata (Harrell et al. 1996). Using all four parsimonycriteria simultaneously is effective because each con-trols different aspects of overfitting. With a forwardstepwise search, as soon as any one of the criteria cannotbe met, the search for additional predictors stops.
Sensitivity analysis
Here ‘sensitivity analysis’ evaluates the relative im-portance of particular predictors within a model. This isparticularly important in non-parametric regression, be-cause we have no fixed coefficients or slopes to compare.
A general way to evaluate the importance of indi-vidual predictors is to nudge up and down observedvalues for individual predictors, and measure the result-ing change in the estimated response for that point. Byaccumulating those sensitivities across all data points,one can evaluate the sensitivity of the model to eachpredictor. The greater the sensitivity, the more influencethat variable has in the model.
The change in the response can be measured as afraction of the observed range of the response variable,|ymax –ymin|. Scaling the differences in response anddifferences in predictors to their respective ranges al-lows a sensitivity measure (Qj for predictor j) that is a
- Non-parametric habitat models with automatic interactions - 825
ratio, independent of the units of the variables:
Qy y y y
n y yj
i ii
n
ii
n
=− + −
−
+=
−=
∑ ∑ˆ ˆ ˆ ˆ
max min
1 1
2 ∆(9)
where yi+ and yi− are the estimates of the response
variable for case i, having increased or decreased, re-spectively, the predictor by an arbitrarily small value ∆(here ∆ = 0.05, i.e. 5% of the range of predictor j). Withthe formula above, a Q = 1.0 means that on average,nudging a predictor results in a change in response ofequal magnitude; Q = 0.0 means that nudging a predic-tor has no detectable effect on the response. Sensitivitycan also be calculated in a similar way from the rootmean square of the differences, rather than the absolutedifferences.
Statistical significance
Statistical significance of a model derived by NPMRcan be evaluated by a randomization test obtained whenthe vector of response values is shuffled, randomlyreassigning their relationships to the predictor matrix.This is a simple, readily justified approach for multipleregression, though other kinds of randomizations areneeded for particular experimental designs (Manly 1997,pp. 156-157). Bootstrap methods could also be applied,but more assumptions are made and caution is neededwith small sample sizes (Westfall & Young 1992, p.142-143; Manly 1997). The following procedure teststhe null hypothesis that the fit of the selected model is nobetter than could be obtained by chance alone, given anequal number of predictors selected from the same poolof variables. The relationship between predictors andthe response variable is destroyed by shuffling the val-ues of the response variable, then repeating the samemodel fitting procedure as used with the unshuffleddata, then calculating the fit. The same pool of predic-tors is used, but with the additional constraint that themodel with the randomized data should have no morepredictors than the model being evaluated. After repeat-ing this many times, the proportion of randomizationruns that results in an equal or better fit is used as the p-value for the test.
Example with synthetic data
Fitting models to data sets with known underlyingstructure provides insights into the performance of dif-ferent modeling approaches. The following examplecombines simple responses to two predictors. NPMRmodels were fit with HyperNiche (McCune & Mefford
2004); logistic and linear models were fit with SPSSver. 11.5; GAM with S-PLUS ver. 6.2. The responsefunction simulates the known response of biomass ofnitrogen-fixing lichens (primarily Lobaria oregana) inwestern Oregon to elevation and stand age.
Biomass of Lobaria has a sigmoid response to standage. Slow to establish in clear-cut or burned forests,Lobaria increases on optimum sites to a plateau averag-
Fig. 1.A. Hypothetical response surface from combinedGaussian and sigmoid functions (Eq. 10), representing bio-mass of epiphytic nitrogen-fixing lichens in relation to eleva-tion and stand age on the west slope of the Cascade Range.B. Multiple linear regression (MLR) model fit to a randomsample of the hypothetical response surface. The model hastwo main predictors, a quadratic elevation term, and interac-tions between stand age and the two elevation terms (adj. R2 =0.43). C. Local mean NPMR model fit to the same randomsample (xR2 = 0.97). The small dropout from the surface hadinsufficient local data to fit the model.
826 McCune, B.
ing about 1.5 T/ha by about age 200 years (McCune1993; Berryman 2002). Lobaria has the classic hump-shaped response to elevation, being rare at low eleva-tions and dropping out completely at high elevations,but abundant between 500-1000 m (McCune et al. 2003).
A noiseless response surface was designed that in-corporates these two predictors, elevation and stand age.The surface multiplies a sigmoid function by a Gaussianfunction (Fig. 1A):
ye
ee
a b x
a b x
a x b=+
⎛
⎝⎜⎞
⎠⎟ ( )− +
− +− −( )
1 1 1
1 1 1
2 2 22
1 (10)
The first term is the sigmoid response to stand age (x1)and the second term is the humped response to elevation(x2). I then selected parameters to give a reasonablesurface for elevation in meters and stand age in years (a1= 5, b1 = 0.03, a2 = .00001, b2 = 700), and multiplied by1.52 to set ymax = 1.5 T/ha. This surface was randomlysampled with 200 points and the response at each pointcalculated with this equation. With this model, if eitherstand age or elevation is unfavourable, then the speciesis absent or nearly so.
A series of least-squares multiple linear regression(MLR) models illustrates the difficulty of representingeven this simple two-predictor system with some tradi-tional statistical tools, while the surface is easily fit byGAM and NPMR. The hazards of the more commonlyused linear models are well known by statisticians butoften not appreciated by ecologists. The simplest andmost naïve model relating biomass to elevation and standage is a MLR with two terms and a constant: y = b1x1 +b2x2 + b0. The fit of this model, a tilted plane, to the data isexpectedly poor (adj. R2 = 0.210), yet the terms for bothstand age and elevation differ significantly from zero.With an interaction term, y = b1x1 + b2x2 + b3x1 x2 + b0, butthe fit is still poor (adj. R2 = 0.245). A residual plot revealsthe hump-shaped relationship with elevation that is notbeing fit. Accommodating this with a quadratic term forelevation and the interaction of the quadratic term withstand age, we havey = b1x1 + b2x2 + b3x2
2 + b4x1 x2 + b5x1 x22 + b0,
yielding an improved, yet weak model (adj. R2 = 0.428).The resulting surface (Fig. 1B) starts to resemble ourknown underlying model, but still leaves much to bedesired, as shown by the residuals plotted against standage and elevation (App. 2).
In contrast, both GAM and non-parametric multipli-cative regression (NPMR) with a local mean andGaussian kernel easily captured almost all of the varia-tion in the response variable using the two predictors,elevation and stand age. The GAM (Poisson family, loglink, spline smoother) readily captures the responsesurface (R2 > 0.99), because the log link effectively
decomposes the two multiplied underlying functionsand the smoothing splines capture their shapes. NPMRclosely reproduced the original response surface (Fig.1C; xR2 = 0.975) and had no major problems in theresiduals (App. 2). Using other local models with NPMRmade little difference in fit (local mean with rectangularkernel, xR2= 0.983; local linear with Gaussian kernel,xR2= 0.988). The local linear model resulted in slightlyhigher sensitivity to stand age (Q = 0.37 vs. Q = 0.24)and slightly lower sensitivity to elevation (Q = 1.14 vs.Q = 1.35) than did the local mean.
Example with real data
A second example illustrates a model fitting binary(presence-absence) data for the tree Larix occidentalis(Anon. 1999) to a suite of climate variables (Daly et al.1994): mean January, July, and annual temperatures;mean January, July, and annual precipitation; meanrelative humidity in January and July; and ‘wetdays’,the mean number of wet days in a year. Larix occidentalis,endemic to western North America, has a fairly smallgeographic range (Fig. 2A). Presumably its range wouldbe more vulnerable to climate change than many othertree species, as it appears to have relatively tight cli-matic tolerances. The grids of climatic data and associ-ated distribution data were randomly sampled with 2500points between 43-49 °N and between 112-122 °W,which includes the entire distribution of L. occidentalisin the U.S. Similar climatic data were not available forthe Canadian portion of the range of the species.
NPMR using a local mean, Gaussian weights, and aminimum N* = 100 found a two-predictor model withlogB = 261. The strongest predictors were wetdays (s =9.1 days/year, Q = 1.54) and mean annual temperature (s= 0.67 °C, Q = 1.10). The need for a hump-shapedmodel appears in the distribution of occupied points inthis 2D slice through the sample space (Fig. 2B). Thebest three-predictor model improved the fit to logB =314. The two predictors remained with their previouslystated tolerances and average relative humidity in July(s = 1.9%, Q = 0.65) was added. Including relativehumidity decreased the sensitivity to wetdays and tem-perature to 1.22 and 1.06, respectively.
Because the sample size was so large, the cross-validation penalty to logB was tiny. This resulted in anasymptotic logB as predictors were added, rather thanthe ultimate decline in cross-validated fit expected ofsmall data sets.
The response surface for the two-predictor modelshowed a distinct interaction between wetdays and tem-perature, as indicated by the diagonal ridge in the re-sponse surface (Fig. 2D). The ridge was asymmetric,with slopes varying from steep on the warm-wet side to
- Non-parametric habitat models with automatic interactions - 827
Fig. 2. A. Distribution of Larix occidentalis (green) in western North America and points used in the random sample. B. 2D slicethrough the predictor space; Larix was present at solid points (black), absent at + (red). C. Response surface from two-predictorGAM; gradient from black to green indicates likelihood of occurrence, with the greenest shade indicating the most favorable habitat.D. Response surface from two-predictor NPMR model. E. Estimated probability of occurrence of L. occidentalis superimposed ondistribution map (black lines show range; gray lines are state boundaries). Deeper green indicates a higher probability of occurrence.Blue ellipses indicate areas where L. occidentalis is absent but potentially present, based on recent climate.
828 McCune, B.
gentle on the warm-dry and cool sides.A GAM (binomial family, logit link, spline smoother)
with the two best predictors fit worse than NPMR withthe same predictors (logB for GAM vs. naïve model =241, logB for NPMR vs. GAM = 20). Adding an interac-tion term to the GAM gave only slight improvement.
Logistic regression (LR) of the same data set pro-duced markedly poorer fits for a given number of pre-dictors than did NPMR or GAM. Local mean NPMRwith three predictors yielded logB = 314. In contrast, LRwith forward stepwise selection from a pool of the nineuntransformed climatic variables yielded a three-pre-dictor model with logB = 169 (calculated from omnibusχ2 statistic in SPSS as log10B = χ2 / 4.60517). Includingtwo-way interactions among the four best predictorsimproved this to logB = 210. Adding quadratic terms forall predictors to the pool of predictors improved the bestthree-predictor LR model to logB = 211. Further im-provements might be had with LR, but the point is notthat a parametric model is impossible; rather that NPMRprovides an effective, rapid, model-free assessment ofthe response surface, automatically allowing for inter-actions. NPMR can be used as the final model, or it canhelp to guide the design of an appropriate non-linearmodel, GLM, or GAM by studying 2D or 3D slices ofthe response surface.
Although the estimated likelihood of occurrence fromNPMR and the actual distribution of Larix correspondedwell (Fig. 2E), some differences emerged, suggestingdisequilibrium between its current distribution and mod-ern climates. One can readily identify where Larix ismissing from areas that appear climatically favourablefor it – for example the east slope of the Cascades innorthern Washington. Most likely some factors otherthan modern climate, perhaps historical climates or dis-turbance regimes or both, have excluded the species.
Example with non-linear dynamics
Even a simple deterministic simulation model for twospecies and one environmental factor can produce a re-sponse surface that cannot be easily represented by stand-ard habitat modelling tools. This is illustrated with adynamic model of Picea mariana (Black spruce) andPicea glauca (White spruce) along a moisture gradient(Apps. 3 and 4). In the model, P. mariana and P. glaucaincrease in logistic fashion, following stand-replacingdisturbance. P. mariana has a broader tolerance to mois-ture than does P. glauca, but P. glauca outcompetes P.mariana on mesic sites. This results in a bimodal realizedniche for P. mariana, dominating on very dry and verywet sites (Curtis 1959; Loucks 1962). I used differenceequations (App. 4) to generate a response surface on agrid of 10 dates × 11 steps on a moisture gradient, then
used NPMR and GAM to fit statistical models to thenoiseless response surface.
In this case GAM performs worse than NPMR (de-tails in App. 3) because no single curve shape permeateseither of the dimensions in the predictor space. In otherwords, GAM appears to fall short when parallel slices ofthe response surface along a given predictor have funda-mentally different shapes, for example sigmoid on wetsites and hump-shaped on mesic sites. With NPMR, onthe other hand, the curve shapes in one part of the multi-dimensional response surface need not bear any relation-ship to the shapes in other parts of the response surface.
Extensions to community analysis
So far NPMR has been applied only to problemswith a single response variable. NPMR opens a door,however, to future improvements in multivariate analy-ses of ecological communities. Species abundance datahave three properties in relationship to environmentalgradients that challenge multivariate statistical analysis(Beals 1984; McCune & Grace 2002, pp. 35-43): thezero truncation problem, ‘solid’ response curves, andcomplex response shapes (including polymodality andasymmetry). Together, these properties produce the ‘dustbunny’ distribution of sample units in multidimensionalspecies space (McCune & Grace 2002) rather than amultivariate normal distribution, demanding multivariatetools capable of effectively representing grossly nonlinearrelationships. An ordination axis derived from commu-nity data can be considered a synthetic gradient throughthe dust bunny of sample units in species space. Thecollective relationships of species to these gradients, fitwith NPMR, could be an improved basis for an optimi-zation principle, replacing stress minimization innonmetric multidimensional scaling.
The immediate utility of NPMR, however, will beimproved empirical models of single species in relationto the factors that influence them. NPMR allows us toabandon simplistic assumptions about overall modelform, embracing the ecological truism that habitat fac-tors interact.
Acknowledgements. I thank my colleagues, especially EricB. Peterson for collaborating on early work; Michael J. Meffordfor collaboration with programming; Philip Dixon, EmilyHolt, Erin Martin, Patricia Muir, Sarah Jovan, and NigelYoccoz for suggestions on the manuscript; and Alix Gitelmanfor her advice.
- Non-parametric habitat models with automatic interactions - 829
References
Anon. 1999. Digital representation of “Atlas of United StatesTrees”. by Elbert L. Little, Jr. Digital Version 1.0. USGS.URL http://climchange.cr.usgs.gov/info/veg-clim/
Austin, M.P. 2002. Spatial prediction of species distribution:an interface between ecological theory and statisticalmodeling. Ecol. Model. 157: 101-118.
Beals, E.W. 1984. Bray-Curtis ordination: an effective strat-egy for analysis of multivariate ecological data. Adv. Ecol.Res. 14: 1-55.
Begg, G.A. & Marteinsdottir, G. 1997. Environmental andstock effects on spatial distribution and abundance ofmature cod Gadus morhua. Marine Ecol. Progr. Ser. 229:245-262.
Berryman, S.D. 2002. Epiphytic macrolichens in relation toforest management and topography in a western Oregonwatershed. Ph.D. Dissertation, Oregon State University,Corvallis, OR, US.
Bowman, A.W. & Azzalini, A. 1997. Applied smoothingtechniques for data analysis. Clarendon Press, Oxford,UK.
Cade, B.S. & Noon, B.R. 2003. A gentle introduction toquantile regression for ecologists. Frontiers Ecol. Environ.1: 412-420.
Ciannelli, L., Chan, K.-S., Bailey, K.M. & Stenseth, N.C.2004. Nonadditive effects of the environment on the sur-vival of a large marine fish population. Ecology 85: 3418-3427.
Curtis, J.T. 1959. The vegetation of Wisconsin. University ofWisconsin Press, Madison, WI, US.
Daly, C., Neilson, R.P. & Phillips, D.L. 1994. A statistical-topographic model for mapping climatological precipita-tion over mountainous terrain. J. Appl. Meteorol. 33: 140-158.
Fleishman, E., MacNally, R. & Fay, J.P. 2003. Validation testsof predictive models of butterfly occurrence based onenvironmental variables. Conserv. Biol. 17:806-817.
Gignac, L.D., Vitt, D.H., Zoltai, S.C. & Bayley, S.E. 1991a.Bryophyte response surfaces along climatic, chemical,and physical gradients in peatlands of western Canada.Nova Hedw. 53: 27-71.
Gignac, L.D., Vitt, D.H. & Bayley, S.E. 1991b. Bryophyteresponse surfaces along ecological and climatic gradients.Vegetatio 93: 29-45.
Gotway, C.A., Ferguson, R.B., Hergert, G.W. & Peterson,T.A. 1996. Comparison of kriging and inverse-distancemethods for mapping soil parameters. Soil Sci. Soc. Am. J.60: 1237-1247.
Guisan, A. & Zimmermann, N.E. 2000. Predictive habitatdistribution models in ecology. Ecol. Model. 135: 147-186.
Harrell, F.E., Lee, K.L. & Mark, D.B. 1996. Multivariableprognostic models: Issues in developing models, evaluat-ing assumptions and adequacy, and measuring and reduc-ing errors. Stat. Med. 15: 361-387.
Hastie, T.J. & Tibshirani, R.J. 1990. Generalized AdditiveModels. Chapman and Hall, London, UK.
Hastie, T., Tibshirani, R. & Friedman, J. 2001. The elements ofstatistical learning. Springer-Verlag, New York, NY, US.
Heglund, P.J. 2002. Foundations of species-environment rela-tions. In: Scott, J.M., Heglund, P.J., Morrison, M.L.,Haufler, J.B., Raphael, M.G., Wall, W.A. & Samson, F.B.(eds.) Predicting species occurrences: Issues of accuracyand scale, pp. 35-41. Island Press, Washington, DC, US.
Huisman, J., Olff, H. & Fresco, L.F.M. 1993. A hierarchicalset of models for species response analysis. J. Veg. Sci. 4:37-46.
Huntley, B., Bartlein, P.J. & Prentice, I.C. 1989. Climaticcontrol of the distribution and abundance of beech (FagusL.) in Europe and North America. J. Biogeogr. 16: 551-560.
Huntley, B., Berry, P.M., Cramer, W. & McDonald, A.P.1995. Modelling present and potential future ranges ofsome European higher plants using climate response sur-faces. J. Biogeogr. 22: 967-1001.
Huston, M.A. 2002. Critical issues for improving predictions.In: Scott, J.M., Heglund, P.J., Morrison, M.L., Haufler,J.B., Raphael, M.G., Wall, W.A. &Samson, F.B. (eds.)Predicting species occurrences: Issues of accuracy andscale, pp 7-21. Island Press, Washington, DC, US.
Loucks, O.L. 1962. Ordinating forest communities by meansof environmental scalars and phytosociological indices.Ecol. Monogr. 32: 137-166.
Manly, B.F.J. 1997. Randomization, Bootstrap and MonteCarlo Methods in Biology. 2nd. ed. Chapman and Hall/CRC, Boca Raton, FL, US.
Maravelias, C.D. & Reid, D.G. 1997. Identifying the effects ofoceanographic features and zooplankton on prespawningherring abundance using generalized additive models.Marine Ecol. Progr. Ser. 147: 1-9.
Martinez-Taberner, A., Ruiz-Perez, M., Mestre, I. & Forteza,V. 1992. Prediction of potential submerged vegetation in asilted coastal marsh, Albufera of Majorca, Balearic Is-lands. J. Environ. Manage. 35: 1-12.
McCune, B. 1993. Gradients in epiphyte biomass in threePseudotsuga-Tsuga forests of different ages in westernOregon and Washington. Bryol. 96: 405-411.
McCune, B. & Grace, J.B. 2002. Analysis of ecological com-munities. MjM Software, Gleneden Beach, OR, US.
McCune, B. & Mefford, M.J. 2004. HyperNiche. Non-parametric multiplicative habitat modeling. Version 1.0.MjM Software, Gleneden Beach, OR, US.
2003. Use of a smoother to forecast occurrence of epiphyticlichens under alternative forest management plans. Ecol.Appl. 13: 1110-1123.
Mysterud, A., Stenseth, N.C., Yoccoz, N.G., Langvatn, R. &Steinhelm, G. 2001. Nonlinear effects of large-scale cli-matic variability on wild and domestic herbivores. Nature410: 1096-1099.
Odum, E.P. 1971. Fundamentals of ecology. Saunders, Phila-delphia, PA, US.
Rushton, S.P., Ormerod, S.J. & Kerby, G. 2004. New para-digms for modelling species distributions? J. Appl. Ecol.41: 193-200.
Westfall, P.H. & Young, S.S. 1992. Resampling-based multi-ple testing. John Wiley and Sons, New York, NY, US.
Whittaker, R.H. 1956. Vegetation of the Great Smoky Moun-tains. Ecol. Monogr. 26: 1-80.
Received 27 September 2005;Accepted 25 July 2006;
Co-ordinating Editor: P. Dixon.
For Apps. 1-4, see JVS/AVS Electronic Archives;www.opuluspress.se/
App. 1-4. Internet supplement to: McCune, B. 2006.
Non-parametric habitat models with automatic interactions.
J. Veg. Sci. 17: 819-830.
App. 1. The Local Linear Form of Non-parametric multiplica-tive regression, NPMR.
A fault of the local mean estimator is that estimates nearthe ends of the predictors are biased toward the central ten-dency of the response variable (Bowman & Azzalini 1997, pp.50-51). This occurs because the closer a target point is to theedge of the sample space, the less data are available beyondthe target point. This bias can be removed by using a locallinear estimator rather than a local mean estimator (Bowman& Azzalini 1997; Hastie et al. 2001). The local linear estimatoris a weighted least squares problem, the weights provided bythe kernel function such that points near the target pointreceive more weight than points far from the target point.
Two important characteristics of LLR are: (1) bias isreduced near the edges of the data set and (2) as the kernelfunction becomes broad, the fitted curve will smoothly ap-proaches traditional least squares regression. In contrast, thelocal mean smoothly approaches a horizontal line parallel tothe predictor axis with an intercept equal to the global mean.
The local linear estimator can be represented in matrixnotation if we first create the design matrix Z containing thepredictors plus a first column of 1s. The predictors are trans-formed by subtracting each value for a given variable from thecorresponding value for the target point. Z has n rows m+1columns (variables). The ith row of Z has the elements:
[ 1 (x1i – v1) (x2i –v2) … (xji – vj) ] (A1-1)
We also create a n × n diagonal matrix of weights, W. InNPMR the ith diagonal element of W is the product of theweights for all variables j = 1 to m. For sample unit i, thediagonal element is:
W Wij ij
j
m
= ∗
=
∏1
(A1-2)
Then the local linear estimator is the first element of theweighted least squares solution, b:
b = (Z'WZ)–1 Z'Wy (A1-3)
A separate regression is solved for each target point.Despite the improved fit to responses near the ends of the
ranges of predictors, local linear models are less conservative,because estimates of y can fall outside of its observed range.This behavior can be particularly noticeable with small datasets. In contrast, the local mean can never produce an estimateoutside the observed range of the dependent variable. In prac-tice, the decision between LM- and LLR-NPMR is a tradeoffbetween avoiding the known bias near the edges of the samplespace and avoiding the possibility of wild estimates in thosesame regions. If the former is the more serious risk, then useLLR-NPMR, while if the latter is possible, use LM-NPMR.Small data sets are more safely modeled with LM-NPMR.
The tolerances (smoothing parameters) are related to theimportance of variables, but in different ways depending onthe local model. With local mean models, tolerance is in-versely related to the importance of a variable. With locallinear models, this is not necessarily so, because a largetolerance can be obtained in either of two conditions, a strongglobally linear effect, or a weak effect. On the other hand anarrow tolerance in a locally linear model implies a strongnonlinear global relationship.
App. 2. Illustration of Residuals for Example 1. Unstandardized residuals from regressions of biomass of epiphytic nitrogen-fixinglichens against (A) stand age and (B) elevation, based on a synthetic data set of known underlying structure. Residuals from least-squares multiple linear regression (MLR) are large and strongly patterned, despite the inclusion of a quadratic term for elevation andinteractions with the quadratic and untransformed variables. Residuals from LM-NPMR against (C) stand age and (D) elevation, aresmall and only slightly patterned.
App. 1-4. Internet supplement to: McCune, B. 2006.
Non-parametric habitat models with automatic interactions.
J. Veg. Sci. 17: 819-830.
App. 3. Example with Non-linear Dynamics.
Even a simple deterministic simulation model for two speciesand one environmental factor can produce a response surfacethat cannot be easily represented by standard habitat modelingtools. This is illustrated with a dynamic model of Piceamariana (Black spruce) and Picea glauca (White spruce)along a moisture gradient. In the model, P. mariana and P.glauca increase in logistic fashion, following stand-replacingdisturbance. P. mariana has a broader tolerance to moisturethan does P. glauca, but P. glauca outcompetes P. mariana onmesic sites. This results in a bimodal realized niche for P.mariana, dominating on very dry and very wet sites (Curtis1959; Loucks 1962). Using difference equations in aspreadsheet (App. 4), I generated a hypothetical responsesurface representing cover of P. mariana in relationship tomoisture and time since disturbance (Fig. A3-1). I then usedNPMR and GAM to fit statistical models to the 110 valuesrepresenting response surface on a grid of 10 dates (10, 20, ...100 years) by 11 steps on the moisture gradient.
Local-mean NPMR with minimum N* = 5.5 (5% of thenumber of data points) yielded a reasonable facsimile of theresponse surface with xR2 = 0.90 (Fig. A3-1B). A local linearmodel improved xR2 to 0.93. Decreasing minimum N* to 1.0improved xR2 to 0.96, but at higher risk of overfitting if errorwere present in the data.
GAM (poisson family, log link, spline functions) yielded aresponse surface that captured the behavior of P. mariana onwetter sites, but missed the smaller hump of P. mariana on thedry sites and the temporary rise and fall of P. mariana onmesic sites (Fig. A3-1C). Even without a cross-validationpenalty, the R2 = 0.78 was lower than with NPMR. Adding aninteraction term to the GAM did not improve the fitted surfaceor R2 appreciably (increment was 0.001). Similar results wereobtained using lowess instead of spline functions. In this caseGAM performs worse than NPMR because no single curveshape permeates either of the dimensions in the predictorspace. In other words, GAM appears to fall short when parallelslices of the response surface along a given predictor havefundamentally different shapes, for example sigmoid on wetsites and hump-shaped on mesic sites. With NPMR, on theother hand, the curve shapes in one part of the multidimen-sional response surface need not bear any relationship to theshapes in other parts of the response surface.
Fig. A3-1. A. Dynamics of Picea mariana along a moisturegradient, in competition with P. glauca, based on a determin-istic difference equation model, plotted for ten decades at 11points on the moisture gradient. Each vertical slice representsthe change in P. mariana over time at a point on the moisturegradient. B. Response surface fitted to the data in A using locallinear NPMR and Gaussian kernel, xR2 = 0.93. C. Responsesurface fitted to the data in A using GAM (Poisson family, loglink, spline smoothing, R2 = 0.78).
App. 1-4. Internet supplement to: McCune, B. 2006.
Non-parametric habitat models with automatic interactions.
J. Veg. Sci. 17: 819-830.
App. 4. Formulation of the Black Spruce – White Spruce Simulation Model.
A simple deterministic simulation model for two species and one environmental factor can produce a response surface that is noteasily represented by standard habitat modeling tools. The four basic components of the system are Picea mariana (Black spruce),Picea glauca (White spruce ), a moisture gradient, and time since disturbance.
The basic facts captured in the model are based on Curtis (1959) and Loucks (1962), who noted the bimodality of P. marianaalong moisture gradient in the boreal forest of eastern North America. Assume that P. mariana and P. glauca increase in logisticfashion, following disturbance. P. mariana has a broader tolerance to moisture than does P. glauca, but P. glauca outcompetes P.mariana on mesic sites, resulting in a bimodal realized niche for P. mariana on a moisture gradient, at least in older stands. P. glaucacannot survive on very wet or very dry sites. Of course the system is more complex than this, but even this level of simplicity ischallenging enough for habitat models.
The key response surface to be simulated and used for statistical model tests is cover of P. mariana in relationship to moistureand time since disturbance.
VariablesB = P. mariana cover (0-100%). Starting value = 1 at year 0.W = P. glauca cover (0-100%). Starting value = 1 at year 0.M = position on a moisture gradient (0=driest, 10=wettest)t = time, increment set to one year
Parameters75 Kb = carrying capacity of P. mariana90 Kw = carrying capacity of P. glauca0.15 rb = intrinsic maximum rate of increase in P. mariana, discrete, (%/yr)0.20 rw = intrinsic maximum rate of increase in P. glauca, discrete, (%/yr)-0.02 sb = tolerance of P. mariana to moisture gradient (more negative is steeper)-0.10 sw = tolerance of P. glaucato moisture gradient5 vb = optimum of P. mariana on moisture gradient5 vw = optimum of P. glauca on moisture gradient1.0 cb = competition coefficient for effects of B on W (0 = no effect)1.5 cw = competition coefficient for effects of W on B (0 = no effect)8 a1 = asymmetry constant 1 for moisture response for P. mariana (higher = more asymmetric)3 a2 = asymmetry constant 2 for moisture response for P. mariana (higher = low end higher)
Difference equations
∆
∆
B
tr e
M a
aB
K Bb
s M v bb b= ( ) +⎛
⎝⎜⎞
⎠⎟⎡
⎣⎢
⎤
⎦⎥
− −−( )2 2
1
cc W
Kw
b
⎛
⎝⎜⎞
⎠⎟ (A4-1)
∆
∆
W
tr e W
K W c B
Kws M v w b
w
b b= ( )⎡⎣⎢
⎤⎦⎥
− −⎛
⎝⎜⎞
⎠⎟−( )2
(A4-2)
The first term (in square brackets) is the intrinsic rate of increase, reduced by a Gaussian relationship to moisture (second roundedbrackets), and an asymmetry control for the response of P. mariana to moisture (third rounded brackets). The second term (B or W)is the existing percent cover of the particular spruce species. The last term (in rounded brackets) depresses the rate of increase as thecarrying capacity is approached, in standard logistic fashion, along with a competitive effect from the other species (e.g., cw Wexpresses the resources preempted by W).
