The Northern Yellowstone Elk: Density Dependence and Climatic ...
The Northern Yellowstone Elk: Density Dependence and Climatic ... The Northern Yellowstone Elk: Density Dependence and Climatic ...
116 DENSITY DEPENDENCE IN YELLOWSTONE ELK * Taper and GoganJ. Wildl. Manage. 66(1):2002Table 4. Regression equations for a priori models to describe observed trends in population change, northern Yellowston elkherd, 1965-1995, Montana and Wyoming. N is population size, X is In(N), and Z is Gaussian random variable with mean 0 andvariance 1. IE and IL are indicator functions for the early and late periods, respectively. Arguments to the indicator functions areonly expressed for observations from that period.Model nameRegression equationRicker, 2a, & lb,v IE(0.375) + IL(0.483) - (3.009 x 10-5)N + 0.084ZRicker, 1a, b, & v 0.330 - (2.124 x 10-5)N + 0.093ZRicker, 2b, & 1 a,v 0.373 + IE([-2.942 x 10-5] N) + 0.086ZRicker, 2a, b, & lv IE(0.357 - [2.76 x 10-5] N) + IL(1.017 - [7.477 x 10-5] N) + 0.079ZGompertz, 2a, b, & 1vIE(1.790 - 0.186 x 10-5]X) + IL(8.900 - 0.902 x 10-5] X) + 0.082ZGompertz, 2b, & la,v 1.550 + IE(-0.158X) + IL(-0.147X) + 0.102ZGompertz, la, b, & v 1.118 - 0.107X + 0.101ZRicker, 2a, b, &v IE(0.357 - [2.762 x 10-5] + 0.082Z) + IL(1.017 - [7.477 x 10-5] + 0.071Z)Gompertz, 2a, b, &vIE(1.790 - 0.188X + 0.086Z) + IL(8.90 - 0.902X + 0.071Z)Exp single model 0.146 + 0.118ZGompertz, 2a, & lb,vIE(3.051) + IL(3.323) - 0.329X + 0.102ZExp double modelIE(0.155 + 0.119 Z) + IL(0.124 + 0.111Z)Random walk 0.187Z(1989-1992) with predicted growth rates forthose years based on our best model of densityand environmental effects (Fig. 6). In the first 2years following the fire, the observed growthrates are below predicted values, while in thethird and fourth seasons, growth rates are considerablyabove the predicted values. The residualsfor the last 2 years of the time series seemlarger than the pre-fire portion of the time series.It appears that the major effects of the 1988 firesp p2N (0.00) IT p p2N (0.92)p s2p2N (1.06)ps2F2p2N (1.26)Sp p2N (1.44)TSpp2N (1.95)t p p2N (1.98)(AIC) , I , II I I I 1-27.0 -26.5 -26.0 -25.5 -25.0 -24.5SIC values__Fig. 5. Schwarz Information Criteria (SIC) selection for appropriatemodels to describe the impacts of density and weatheron population growth rate, northern Yellowstone elk herd,1965-1995, Montana and Wyoming. Values of SIC for eachmodel are given on the Y-axis. Symbols: t, s, and f are temperature,snow, and forage for the current winter; T, S, and Fare temperature, snow and forage for the previous winter; p isprecipitation for the current spring.on elk population dynamics were brief. It shouldbe noted that although the observed 1989 growthrate is below the predicted growth rate, the residualis quite small. Caution should be taken ininterpreting these post-fire residuals because theanalysis depends heavily on sightability correctionsthat have very large uncertainties.Density Dependence in Component LifehistoryTraitsWe examined the estimated demographic ratesversus density for the early period. Fertility (Fig. 7)was fitted using a log-linear model. Adult sur-vivorship (Fig. 8) is fit with a nonlinear regressionmodel, S(N) = exp(a - bN). The parameter a wasconstrained not to exceed 0 so that survivorshipcannot appear to exceed 1 as population sizegoes to 0. Data points from the late period areincluded in the fit. Our decision to eliminate thelate period from the fit for fertility while includ-aL 0.40 a.(D0.6-0.20.0-0n!+ +? la A+~00!9+ e9 + +A+t-* +M. I . I . I . I . Il . I . .65 70 75 80Year85 90 95Fig. 6. Patterns of observed (circles, squares), predicted(crosses) and immediate post-fire observed (triangles) growthrates, northern Yellowstone elk herd, 1965-1995, Montanaand Wyoming. Single Ricker model with density, spring precipitationand spring precipitation squared as covariates.
J. Wildl. Manage. 66(1):2002DENSITY DEPENDENCE IN YELLOWSTONELK * Taper and Gogan 1170.71.1-0.600Early period1.0-.SEarly period._0.4-0.3 '.0* 4zr.20.S00.200.803,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000Post-hunt density0.7t I3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000tFig. 7. Changes in estimated fertility with population density,northern Yellowstone elk herd, 1965-1980, Montana andWyoming. E(F) = exp(-0.108 - 0.00102*Nt).Post-hunt densityFig. 8. Changes in estimated adult survivorship with populationdensity, northern Yellowstonelk herd, 1965-1980, Montanaand Wyoming. E(S) = exp(-0.219*N3 77).ing the late period in the fit for survivorship isbased on inspection of the data and the observationthat the only environmental effects indicatedto be important by model identification usingthe linear regression were current spring precipitationand current spring precipitation squared.These variables are expected to influence fertilitybut not adult survivorship. Calculation of survivorshipfrom population counts does not allowdistinguishing mortality from emigration.Inserting the above submodels for fertility andsurvivorship into the simple demographic modelof equation 1 produces a model with an SIC ofonly -12.8. Alternatively, one can fit the parametersof this model directly to the population transitiondata using the Nelder/Mead downhill-simplexalgorithm. The fit is similar, although notquite as good (SIC = -11.4). Clearly, the demographicmodel is not supported over the simpleRicker model of population dynamics. Nonetheless,density dependence in both fertility andadult survivorship is strongly suggested (Figs. 7,8). We believe the demographic model is notsupported because it requires 2 extra parameters;this is a heavy penalty to pay with so few datapoints, and the demographic model requires theuse of the herd composition assessments. Thisintroduces new measurement errors and biasesto the analysis and probably degrades the performanceof the model.One of the consequences of density dependencein fertility and adult survivorship (Figs. 7,8) is a biphasic relationship between populationgrowth rate and density (Fig. 9). At low density,population growth rate is dominated by densitydependence in fertility, while high density populationgrowth rate is dominated by density dependencein adult survivorship (Fig. 9). This demonstratesthe biphasic nature to growth rateimposed by separate density dependence in fer-tility and survivorship. Curvature in the relationshipbetween growth rate and density can bemodeled in a simple fashion using a 0-Rickermodel (Gilpin and Ayala 1973, Thomas et al.1980, Hooten 1995). The SIC for this model is-22.58 with AIC = 1.99, indicating this model isnot clearly distinguishable from our best model.Thus, density dependence may be nonlinear inthe northern Yellowstone elk herd.00.4-0.3-0.2? 0.10.0-0.1%5,~ ^ Early period'., I* Observed r ,- - - Predicted r " *- Predicted r with average sex ratio *\I I I j3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,000Post-hunt densityFig. 9. Biphasic density dependence in the northern Yellowstoneelk herd, 1965-1980, Montana and Wyoming. Thedashed line plots the growth rate predicted by the demographicmodel (Equation 1) using the observed sex ratio foreach year. The solid line plots the growth rate derived fromthe model assuming that the sex ratio is constant throughout.
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116 DENSITY DEPENDENCE IN YELLOWSTONE ELK * Taper <strong>and</strong> GoganJ. Wildl. Manage. 66(1):2002Table 4. Regression equations for a priori models to describe observed trends in population change, northern Yellowston elkherd, 1965-1995, Montana <strong>and</strong> Wyoming. N is population size, X is In(N), <strong>and</strong> Z is Gaussian r<strong>and</strong>om variable with mean 0 <strong>and</strong>variance 1. IE <strong>and</strong> IL are indicator functions for the early <strong>and</strong> late periods, respectively. Arguments to the indicator functions areonly expressed for observations from that period.Model nameRegression equationRicker, 2a, & lb,v IE(0.375) + IL(0.483) - (3.009 x 10-5)N + 0.084ZRicker, 1a, b, & v 0.330 - (2.124 x 10-5)N + 0.093ZRicker, 2b, & 1 a,v 0.373 + IE([-2.942 x 10-5] N) + 0.086ZRicker, 2a, b, & lv IE(0.357 - [2.76 x 10-5] N) + IL(1.017 - [7.477 x 10-5] N) + 0.079ZGompertz, 2a, b, & 1vIE(1.790 - 0.186 x 10-5]X) + IL(8.900 - 0.902 x 10-5] X) + 0.082ZGompertz, 2b, & la,v 1.550 + IE(-0.158X) + IL(-0.147X) + 0.102ZGompertz, la, b, & v 1.118 - 0.107X + 0.101ZRicker, 2a, b, &v IE(0.357 - [2.762 x 10-5] + 0.082Z) + IL(1.017 - [7.477 x 10-5] + 0.071Z)Gompertz, 2a, b, &vIE(1.790 - 0.188X + 0.086Z) + IL(8.90 - 0.902X + 0.071Z)Exp single model 0.146 + 0.118ZGompertz, 2a, & lb,vIE(3.051) + IL(3.323) - 0.329X + 0.102ZExp double modelIE(0.155 + 0.119 Z) + IL(0.124 + 0.111Z)R<strong>and</strong>om walk 0.187Z(1989-1992) with predicted growth rates forthose years based on our best model of density<strong>and</strong> environmental effects (Fig. 6). In the first 2years following the fire, the observed growthrates are below predicted values, while in thethird <strong>and</strong> fourth seasons, growth rates are considerablyabove the predicted values. <strong>The</strong> residualsfor the last 2 years of the time series seemlarger than the pre-fire portion of the time series.It appears that the major effects of the 1988 firesp p2N (0.00) IT p p2N (0.92)p s2p2N (1.06)ps2F2p2N (1.26)Sp p2N (1.44)TSpp2N (1.95)t p p2N (1.98)(AIC) , I , II I I I 1-27.0 -26.5 -26.0 -25.5 -25.0 -24.5SIC values__Fig. 5. Schwarz Information Criteria (SIC) selection for appropriatemodels to describe the impacts of density <strong>and</strong> weatheron population growth rate, northern <strong>Yellowstone</strong> elk herd,1965-1995, Montana <strong>and</strong> Wyoming. Values of SIC for eachmodel are given on the Y-axis. Symbols: t, s, <strong>and</strong> f are temperature,snow, <strong>and</strong> forage for the current winter; T, S, <strong>and</strong> Fare temperature, snow <strong>and</strong> forage for the previous winter; p isprecipitation for the current spring.on elk population dynamics were brief. It shouldbe noted that although the observed 1989 growthrate is below the predicted growth rate, the residualis quite small. Caution should be taken ininterpreting these post-fire residuals because theanalysis depends heavily on sightability correctionsthat have very large uncertainties.<strong>Density</strong> <strong>Dependence</strong> in Component LifehistoryTraitsWe examined the estimated demographic ratesversus density for the early period. Fertility (Fig. 7)was fitted using a log-linear model. Adult sur-vivorship (Fig. 8) is fit with a nonlinear regressionmodel, S(N) = exp(a - bN). <strong>The</strong> parameter a wasconstrained not to exceed 0 so that survivorshipcannot appear to exceed 1 as population sizegoes to 0. Data points from the late period areincluded in the fit. Our decision to eliminate thelate period from the fit for fertility while includ-aL 0.40 a.(D0.6-0.20.0-0n!+ +? la A+~00!9+ e9 + +A+t-* +M. I . I . I . I . Il . I . .65 70 75 80Year85 90 95Fig. 6. Patterns of observed (circles, squares), predicted(crosses) <strong>and</strong> immediate post-fire observed (triangles) growthrates, northern <strong>Yellowstone</strong> elk herd, 1965-1995, Montana<strong>and</strong> Wyoming. Single Ricker model with density, spring precipitation<strong>and</strong> spring precipitation squared as covariates.
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