The Northern Yellowstone Elk: Density Dependence and Climatic ...

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110 DENSITY DEPENDENCE IN YELLOWSTONE ELK * Taper and GoganJ. Wildl. Manage. 66(1):2002(Coughenour and Singer 1996, Lemke et al.1998). These authors estimated that between1980 and 1985 (between our early and late periods),the extent of the winter range increased41%. In analyses that relate to population densities,we adjust the estimated population numbersduring the late period by dividing by 1.41. Thisexpresses all estimates as densities in units of animalsper pre-1980 winter range.Testing for Density DependenceTo legitimately test for density dependence intime series of population estimates, one musthave a true a priori hypothesis that is null andalternative models that are specified beforeinspecting the data. Following Dennis and Taper(1994), we used a null model of exponentialgrowth and an alternative model of Ricker density-dependentgrowth. We used a parametric bootstraplikelihood ratio (PBLR) test (Dennis andTaper 1994). This procedure regresses populationgrowth rate as a response variable against apredictor variable of population size. The distinctionbetween the PBLR test and a standard regressionis that the PBLR develops the distribution ofthe test statistic under the null hypothesis by simulation.Thus, it gives a true size to the test of densitydependence. The test also is robust againstmeasurement error (Dennis and Taper 1994).Identifying the Form of PopulationDynamicsIdentifying Population Dynamic Process withSchwarz's Information Criterion.-Because legitimatehypothesis testing requires a single pair ofnull and alternative hypotheses, it often is quiteconstraining. Model mis-specification is a majorsource of error in data analysis (Chatfield 1995,Buckland et al. 1997, Burnham and Anderson1998). Model identification is an approach thatminimizes the risk of model mis-specification bycomparing the goodness-of-fit of a whole suite ofmodels. A number of possible criteria for comparingmodels exist. We prefer the informationcriteria methods because of their strong statisticalfoundations (Sakamoto et al. 1986, Burnhamand Anderson 1998) and because the validity ofits application to the identification of populationdynamic models in time-series analysis has beeninvestigated through simulation (Hooten 1995).Although model identification formally selectsa single model from a suite of models, as opposedto testing a single null against a single alternative,the process can still be used to compare classes ofmodels. All the population dynamic models weinvestigated were classified as density-independentor density-dependent models. Thus, if themodel identified as the best model is a memberof the set of density-independent models, thenthe population dynamics are considered to bedensity-independent, while, if the identifiedmodel is a member of the density-dependent set,the population dynamics are classified as densitydependent(Hooten 1995, Zeng 1996, Zeng et al.1998). Because of the multiple models involvedand because explicit P-values are not calculated,such a classification by model identification doesnot per se constitute a hypothesis test. Nonetheless,if the compared models have been includedon an a priori basis, then the comparison retainsmuch of the epistemological standing of ahypothesis test (Burnham and Anderson 1998).There are a variety of possible information criteriato choose from (Hooten 1995). All informationcriteria are constructed as -2*(log-likelihood)+ (an adjustment). The adjustment is afunction of the number of parameters or of thenumber of parameters and the number of observations.The adjustment for Akaike's InformationCriterion (AIC; Akaike 1973) is 2*p (p = the numberof parameters including the error variance),while the adjustment for Schwarz's InformationCriterion (SIC; Schwarz 1978) contains a samplesize correction so that the adjustment is the numberof parameters times the natural logarithm ofthe number of observations. The model with thelowest value for the information criterion used isselected as the best-supported model. However,this selection should not be considered absolute.If the difference in information criteria (IC) valueof 2 models is slight, then it is best to consider themodels more or less equally supported and retainboth for consideration. Differences in informationcriterion values (AIC) of >2 are generallyconsidered to indicate that models are statisticallydistinguishable (Sakamoto et al. 1986).It has been shown both theoretically (Stone1979) and through simulation (Hooten 1995)that the SIC estimates the true order of theunderlying model more accurately than the AIC.Furthermore, the SIC tends to choose models oflower order than the AIC (when the selection differs).In contrasting between density-independentand density-dependent classes of models,this feature will make the identification of densi-ty dependence conservative as the density-dependentmodels contain more parameters than thedensity-independent models (Zeng et al. 1998).
J. Wildl. Manage. 66(1):2002DENSITY DEPENDENCE IN YELLOWSTONE ELK * Taper and Gogan 111(AIC)Ricker, 2 a, & lb,v (0.00)Ricker, 1 a, b, & v (0.43)Ricker, 2 b, & 1 a,v (0.75)Ricker, 2 a, b, & 1 v (0.80)Gompertz, 2 a, b, & 1 v (2.00)Gompertz, 2 b, & 1 a,v (2.75)Gompertz, 1 a, b, & v (3.36)Ricker, 2 a, b, & v (3.48)Gompertz, 2 a, b, & v (4.59)Exp single model (5.74)Gompertz, 2 a, & 1b,v (6.60)Exp double model (11.16)Random walk (18.68)Two stationary dist (41.21)Single stationary dist (59.70)-25 -20 -15 -10 -5 -0SIC valuesFig. 2. Schwarz Information Criteria (SIC) selection for appropriatemodels to describe observed trends in populationchange, northern Yellowstonelk herd, 1965-1995, Montanaand Wyoming. Values of SIC for each model are given inparentheses on the Y-axis. Numbers of intercept (a), slope(b), and error (v) terms indicated.On the other hand, selection by minimum AIC isasymptotically equivalent to selection by minimumprediction error (Stone 1977). Becauseour primary intention is to identify the underlyingform of population dynamics of the northernYellowstone elk herd and not to make explicitpredictions, use of the SIC is appropriate.Models Considered.-Density-independent modelclasses are random walk and exponential growthmodels. Included in the density-dependent classare Gompertz (1825) and Ricker (1954) models.We included models that differentiated populationdynamics in the early and late periods fordensity-independent and density-dependentgrowth model classes. For example, we consideredRicker and Gompertz growth models withintercept, slope, and variance parameters commonto the early and late periods, and Ricker andGompertz models that allowed >1 of these parametersto differ between the periods. In Rickermodels, the relationship between growth rateand population density is linear, while for Gomnpertzmodels, growth rate is linearly associatedwith the logarithm of population size (Fig. 2).We also included the stationary statistical distri-bution models. This class of models does notassume any dependence of population size at agiven time on the population size at any previoustimes. Population sizes are simply assumed to bedrawn from some random distribution.Surprisingly, in such a situation, growth rateswill be negatively correlated with population size.This phenomenon was first pointed out by Eberhardt(1970), and has been discussed recently byWolda and Dennis (1993) and by Wolda et al.(1994). Wolda and Dennis (1993) demonstratethat rainfall can yield a significant result in a testof density dependence. This may be disturbing tobiologists. However, stationary statistical distributiondoes not necessarily indicate an absence ofdirect causal density dependence. For example, aGompertz growth model with a growth rate parameterof 1 will generate a log-normal stationarydistribution of population sizes where populationsize at a given time is independent from populationsize at any previous time. We have explicitlyincluded the stationary statistical distributionmodel class to alleviate the uncertainty generatedby this situation. If a model is identified as a density-dependentmodel but is strongly distinguishablefrom a stationary distribution, then a caveatthat could be raised against deciding the populationdynamics represents biological densitydependence has been removed.Recent work has successfully used nonparametriccurve fitting and generalized additive modelsto investigate density dependence in natural populations(Bj0rnstad et al. 1999). These techniquesare data intensive; for example, Bj0rnstadet al. (1999) analyzed a composite dataset with250 observations. Such an approach cannot besuccessfully applied to the northern Yellowstoneelk herd because of the paucity of data.Assessing the Impacts of EnvironmentalFactorsTo determine the impact of weather on thepopulation dynamics of the northern Yellowstoneelk herd, we include weather variables as covariatesin some population dynamic models. Fames'(1996) weather severity index is a weighted sumof 3 normalized components: winter forage avail-ability, snow depth, and winter temperature. Allcomponents are constructed so that higher num-bers indicate milder winters. Thus, a high valuefor the snow component indicates low snow levels.We do not use Fames' full index, but insteaduse his 3 components directly. This enables themost effective weightings to emerge from thedata. Weather variables used (Table 3) comefrom Fames' most recent revision of his compilationof 30 of Yellowstone climate data (P. Fames,Soil Conservation Service, personal communica-tion). We consider these 3 variables, their l-yearlags (previous year's values), and all squares ofthese variables in looking for the effect of weatheron elk population dynamics. We also considerthe current spring's precipitation and its square.
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