# Models for Population Growth Curves

Authors: L. L. Eberhardt; J. M. Breiwick
Journal: ISRN Ecology
 (2012)

---

## Abstract

We examined models for population growth curves, contrasting integrated versions with various other forms. A sizable number of data sets for birds and mammals were considered, but the main comparisons were based on 27 data sets that could be fit to the generalized logistic curve. Akaike's information criterion was used to rank fits of those data sets to 5 integrated models. We found that the integrated models gave the best fits to the data examined. The difference equations examined gave much poorer fits as judged by AICc and coefficients of variation. We conclude that the integrated models should be used when possible.


---

## Body

## 1. Introduction

Most recent use of population growth curves has focused on difference equation models (also called “finite population models”). Such models may give a somewhat wider scope for applications and for theory, than do the integrated versions of these models. However, the available integrated versions appear to give much better fits to actual growth curve data, raising some questions about the practical utility of the difference equation versions. We thus examine a number of integrated and difference equation models in this paper. The data used are for birds and mammals. Rather different results may apply for insects and some species of fish.


## 2. Methods

### 2.1. Models

The first of the integrated models used here, the generalized logistic reported by Nelder [16] and by Pella and Tomlinson [17], but best known from Ayala et al. [18], and often termed the theta-logistic is:
(1)N(t)=[K-z-(K-z-N0-z)e-rzt]-1/z.
The corresponding difference equation is(2)N(t)=N(t-1)+rN(t-1)[1-(N(t-1)K)z].
A rearrangement of (2) is designated here as the “Sibly model” [19–22]:(3)pgr=[(N(t)N(t-1))-1]=r[1-(N(t-1)K)z].
The Gompertz curve is(4)N(t)=Kexp⁡⁡(log⁡⁡(N0K)e-ct).
The difference equation in the form used by Dennis et al. [23] is(5)N(t)=N(t-1)exp⁡⁡(r+blog⁡⁡(N(t-1))).
The logistic growth curve is obtained by setting z=1 in (1):(6)N(t)=K1+ce-rt,c=KN0-1.
The difference equation is(7)N(t)=N(t-1)+rN(t-1)[1-(N(t-1)K)].
The exponential model is(8)N(t)=N0exp⁡⁡(rt).
The difference equation is(9)N(t)=N(t-1)+rN(t-1).
The equation of Morris and Doak [24] is(10)μ̂=(1q)∑i=0q-1log⁡⁡(Ni+1Ni)
with variance estimate:(11)var⁡μ̂=(1q-1)∑i=0q-1(log⁡⁡(Ni+1Ni)-μ̂)2.
In the notation used here μ=r.  A modification of (1), with z=2, is considered here, as it has significant advantages. It is here denoted as a modified logistic:(12)N(t)=[K-2-(K-2-N0-2)e-2rt]-1/2.
In the aforementioned models, K is the asymptotic value, r is a rate of increase, z is the parameter controlling the inflection point in a growth curve, N0 represents initial population size, and b and c are functions of r and K in the Gompertz equations. The models have been fit by using nonlinear least-squares, [25], as implemented in the R-language [26].


### 2.2. Akaike’s Information Criterion

AIC is calculated as [27](13)AIC=nlog⁡⁡σ̂2+2k,whereσ̂2=∑êi2n,
the êi are deviations (residuals) from the model fit (assumed to be normally distributed), n is sample size, and k is the number of parameters in the model plus 1. Most of the available data sets have small samples, so Burnham and Anderson [27] recommended using(14)AICc=AIC+2k(k+1)n-k-1.
Model comparisons are made using(15)Δi=AICc-AICmin⁡,
where AICmin is the minimum value calculated for a set of models.


### 2.3. Data Sources

Part of the data used here was obtained in 2006 from the Global Population Dynamics Database (GPDD) maintained by the National Environmental Research Council athttp://www.sw.ic.ac.uk/cpb/gpdd.html. ID numbers for data used from the GPDD appear in the tables. Data sets from the GPDD were obtained by searching the entire data set (over 4000 entries) by inspecting plots of each data set and attempting to fit curves. An up-to-date version of the GPDD data was obtained in 2008 and searched for additional examples. Over 100 data sets were examined in detail, but those that could be fit to (1) are the primary data sets used here.


## 2.1. Models

The first of the integrated models used here, the generalized logistic reported by Nelder [16] and by Pella and Tomlinson [17], but best known from Ayala et al. [18], and often termed the theta-logistic is:
(1)N(t)=[K-z-(K-z-N0-z)e-rzt]-1/z.
The corresponding difference equation is(2)N(t)=N(t-1)+rN(t-1)[1-(N(t-1)K)z].
A rearrangement of (2) is designated here as the “Sibly model” [19–22]:(3)pgr=[(N(t)N(t-1))-1]=r[1-(N(t-1)K)z].
The Gompertz curve is(4)N(t)=Kexp⁡⁡(log⁡⁡(N0K)e-ct).
The difference equation in the form used by Dennis et al. [23] is(5)N(t)=N(t-1)exp⁡⁡(r+blog⁡⁡(N(t-1))).
The logistic growth curve is obtained by setting z=1 in (1):(6)N(t)=K1+ce-rt,c=KN0-1.
The difference equation is(7)N(t)=N(t-1)+rN(t-1)[1-(N(t-1)K)].
The exponential model is(8)N(t)=N0exp⁡⁡(rt).
The difference equation is(9)N(t)=N(t-1)+rN(t-1).
The equation of Morris and Doak [24] is(10)μ̂=(1q)∑i=0q-1log⁡⁡(Ni+1Ni)
with variance estimate:(11)var⁡μ̂=(1q-1)∑i=0q-1(log⁡⁡(Ni+1Ni)-μ̂)2.
In the notation used here μ=r.  A modification of (1), with z=2, is considered here, as it has significant advantages. It is here denoted as a modified logistic:(12)N(t)=[K-2-(K-2-N0-2)e-2rt]-1/2.
In the aforementioned models, K is the asymptotic value, r is a rate of increase, z is the parameter controlling the inflection point in a growth curve, N0 represents initial population size, and b and c are functions of r and K in the Gompertz equations. The models have been fit by using nonlinear least-squares, [25], as implemented in the R-language [26].


## 2.2. Akaike’s Information Criterion

AIC is calculated as [27](13)AIC=nlog⁡⁡σ̂2+2k,whereσ̂2=∑êi2n,
the êi are deviations (residuals) from the model fit (assumed to be normally distributed), n is sample size, and k is the number of parameters in the model plus 1. Most of the available data sets have small samples, so Burnham and Anderson [27] recommended using(14)AICc=AIC+2k(k+1)n-k-1.
Model comparisons are made using(15)Δi=AICc-AICmin⁡,
where AICmin is the minimum value calculated for a set of models.


## 2.3. Data Sources

Part of the data used here was obtained in 2006 from the Global Population Dynamics Database (GPDD) maintained by the National Environmental Research Council athttp://www.sw.ic.ac.uk/cpb/gpdd.html. ID numbers for data used from the GPDD appear in the tables. Data sets from the GPDD were obtained by searching the entire data set (over 4000 entries) by inspecting plots of each data set and attempting to fit curves. An up-to-date version of the GPDD data was obtained in 2008 and searched for additional examples. Over 100 data sets were examined in detail, but those that could be fit to (1) are the primary data sets used here.


## 3. Results

### 3.1. Integrated Models versus Difference Equations

Table1 contrasts the generalized logistic (1) with the corresponding difference equation (2), and the ordinary logistic (6) with its corresponding difference equation (7), using residual mean squares in consequence of the experience of Eberhardt et al. [28], who found that residual mean squares performed better on data similar to those used here. We believe that the “adjustment factor” of Burnham and Anderson [27] (in (14)) may discriminate against models with the larger number of parameters. In nearly all instances, the integrated version is to be preferred, as shown by residual mean squares. The two apparent exceptions in Table 1 have highly variable data.Table 1
Comparison of the generalized logistic (1) with the corresponding difference equation (2) and the ordinary logistic (6) compared with the corresponding difference equation (7). Numbers in square brackets refer to references and other numbers are GPDD numbers. Values given are residual mean squares from the fits.
SourceGeneralized logisticDifference equationRatioLogisticDifference equationRatioRed deer 65852.162.471.143.12.80.90Gray fox 3192835912.09378.2566.71.50Gray fox 342131114401.10125413991.12Coyote 263252827291.08245126811.09Gray seals [1]56.8393.21.6465.9390.881.38Bison  [2]5.817.861.355.647.811.38Macaque 60681.111.321.191.451.430.99Beaver 15115960164901.0315850164101.04Beaver 15640960438001.0743400440701.02Elephant seals [3]48.6163.441.3150.0662.791.25Seneca deer [4]29.9350.781.7028.9950.441.74Badger 783754531.21372.5438.81.18Gentoo penguins (Fraser pers. comm.).61.19113.21.8557.88108.61.88Polar bear 14389.968.10.7695.0967.490.71Coyote 2473083801.23335.7362.41.08Gray fox 328549.3901.31.64526.1850.11.62Marten 969517020220101.2916630223901.35Squirrel 96993471003934001.133391003985001.18Gray fox 334338333320.98328031550.96Camargue horses [5]0.5811.721.421.741.23Argentine horses [6]5.018.041.605.387.81.45Przewalski's horses [5, 7]0.891.391.561.231.471.20Red kite [8]3.274.561.393.24.461.39Medians and Mean Ratios89.9113.21.3595.09108.61.20Table2 makes the same contrast for the Gompertz equation (4) with the corresponding difference equation (5). Smaller samples are available due to the fact that the Gompertz did not fit a number of the data sets. That the “Sibly” model gives even more variable results is illustrated in Table 3, where the Sibly model (3) has a larger coefficient of variation in all cases where a comparison could be made, with the exception of the Seneca deer data which has essentially an exponential trend.Table 2
Contrast of fit to Gompertz (4) with difference equation (5) using residual mean squares.
SourceGompertzDifference equationRatioRed deer 65854.013.50.87Gray seals [1]68.5891.911.34Bison [2]6.347.591.20Macaque 60681.571.50.96Beaver 15646930461600.98Elephant seals [3]62.2469.861.12Seneca deer [4]35.453.871.52Badger 78392.74871.24Gray fox 328569.8867.91.52Marten 969516850230601.37Camargue horses [5]1.842.491.35Argentine horses [6]5.678.181.44Przelwalski’s horses [5, 7]1.41.681.20Mean1.24Table 3
Contrasts between the Sibly model (3) and the corresponding difference equation (2). Due to structural differences in the models, coefficients of variation are used for contrasts. An asterisk (*) indicates cases where counts reached or closely approximated an asymptotic value (K).
SourceDifference equationKCV (K)Sibly modelKCV (K)Generalized logisticKRed deer 6585*1660.0201670.028168Gray fox 319*105200.052107600.57410600Gray fox 342*155900.003150600.13915290Coyote 263*193400.063191200.17319110Gray seals* [1]20440.15520160.1672013Bison [2]21201.314no fit1356Macaque 6068*240.026240.05424Beaver 151*1564000.0551585000.057164.500Beaver 156*4227000.0694293000.083421200Elephant seals [3]17520.072no fit1862Seneca deer [4]41340.39626720.3434425Badger 7825500.162no fit3009Gentoo penguins (Fraser pers comm.)9490.534no fit1027Polar bear 1437380.097no fit700Coyote 247*32730.04633100.1033231Gray fox 32892000.296no fit9723Marten 9695983300.062no fit104000Squirrel 9699*2.19E+060.0692.04E+060.1992.17E+06Gray fox 3343.59E+050.0593.54E+050.21722790Camargue horses* [5]750.031770.07478Argentine horses* [6]2090.0512090.069213Przewalski’s horses* [5, 7]440.093430.13144Red kite [8]1397.45no fit182In those cases marked by an asterisk in Table3, the estimates of the asymptotic value (K) are nearly the same for the difference equation (2), the Sibly model (3), and the generalized logistic (1). In all of these cases, the data reach an asymptotic value or closely approximate an asymptote. Coefficients of variation (standard error/K) show that the “Sibly” model (3) gives much more variable estimates than do the other two models considered. These are cases where no gaps existed in the count data. The “Sibly” model does not accommodate isolated data points and has to be fit in segments (yielding more variable estimates), whereas the integrated models can be applied directly in such cases.Table4 gives estimates of the coefficient (z) governing the inflection point in the generalized logistic model, suggesting use of 2.0 in the modified logistic (12). Table 5 contrasts the 5 integrated models using AICc, showing that the modified logistic is generally superior. Clark et al. [29] found, using simulations, that there appeared to be a linear relationship between r and z in the generalized logistic model. Using our data set on birds and mammals, we found a correlation between r and z (−0.51) and a distinctly nonlinear relationship. The data appear in Figure 1.Table 4
Estimates of the parameterz in (1).
SpecieszCVRed deer 65853.890.720Boar 94284.991.138Gray fox 3193.780.407Gray fox 3421.851.151Coyote 2631.931.482Gray seals [1]11.760.882Bison [2]0.750.627Macaque 606816.261.545Beaver 1510.461.739Beaver 1567.431.027Elephant seals [3]1.860.360Seneca deer [4]0.690.652Badger 781.762.290Fur seals [9]4.050.783Blue Springs manatees [10]5.082.108Chinstrap penguins (Fraser, pers. commun.)0.455.778Gentoo penguins (Fraser, pers. commun.)0.692.087Polar bear 1436.131.127Coyote 2476.421.673Gray fox 3281.881.293Boar 94291.660.596Marten 96951.21.892Squirrel 96990.982.920Gray fox 3340.452.480Camargue horses [5]6.060.240Argentine horses [6]7.990.95Przewalski’s horses [5, 7]6.810.690Red kite [8]1.473.380Median1.93Table 5
Contrasting the 5 integrated models using AICc and Δt. Zeros indicate minimum AICc (15).
SpeciesSample sizeGeneralized logisticModified logisticLogisticExponentialGompertzRed deer 6585Cervus elaphus1300.8513.7713.1413.46Gray seals [1]Halichoerus grypus1602.1710.6715.214.62Beaver 156Castor canadensis4600.8410.4152.96.34Polar bear 143Ursus maritimus5106.36.372.75no fitBoar 9428Sus scrufa142.7809.3728.375.2Gray fox 319Urocyon cineroargenteus121.95012.437.79no fitGray fox  342Urocyon cineroargenteus124.8107.227.4no fitCoyote 247Canis latrans114.4601.8422.88no fitCoyote 263Canis latrans173.1035.68165.69no fitElephant seals [3]Mirounga angistirostris261.16010.2561.1613.95Gray fox 328Urocyon cineroargenteus124.8100.3813.752.29Badger 78Taxidea tasus152.97012.378.476.71Fur seals [9]Callorhinus ursinus123.5708.373.473.47Blue springs manatees [10]Trichechus manatus191.607.60.70.7Boar 9429Sus scrufa143.26036.0435.975.09Camargue horses [5]Equs caballus87.3404.5316.978.6Przewalski horses [5, 7]Equs ferus814.0902.14.187.67Marten 9695Martes americana251.2700.9913.370.27Squirrel 9699Sciuris verginianus251.271.27018.230.65Bison [2]Bison bison131.280.26023.290.51Seneca deer [4]Odocoileus virginianus112.585.73040.943.52Gentoo penguins**Pygoscelis papua125.245.77031.144.4Beaver 151Castor canadensis444.730.8013.94no fitChinstrap penguins**Pygoscelis antarcticus222.772.22062.750.05**Fraser, pers. comm.Figure 1
Plot ofr versus z from the 27 data sets that fit the generalized logistic model.Table6 compares the Morris and Doak [24] ((10) and (11)) with exponential models fitted by nonlinear least-squares or a linear fit with log-transformation. The median values indicate that all 3 methods give much the same values, but the Morris and Doak approach is far more variable, as indicated by coefficients of variation.  The coefficients of variation show essentially no difference between the two approaches using exponential models. It should be noted that the Morris and Doak estimator (10) can be used with virtually any data set giving nonsensical results in some cases (only exponential-type data were used in Table 6). Thus Morris and Doak [24] used 39 observations on the Yellowstone grizzly bear data, as shown in their Figure 3.6 and Table 3.1. The problem is that the population was initially decreasing and then, as protective measures began to take effect, started to increase in the early 1980s and continued to do so for the remainder of the series. Consequently, the resulting estimate of r (or μ) is of no practical value. Eberhardt and Breiwick [11] give details and cite 7 published papers that largely ignore the change in trend and thus give largely meaningless results.Table 6
Comparison of Morris and Doak estimates with exponential models.
SpeciesMorris and DoakNonlinear LSLog transformrCVrCVrCVGrizzlies [11]0.0551.4400.0540.1370.0560.126Bison [2]0.2030.080.1880.0370.2010.03Muskox [12]0.1450.1040.1690.0220.1460.027Calif. Sea otters D. Siniff, pers. comm.0.0470.4220.0550.0730.0520.076Gray seals [1]0.0660.2560.0680.0540.0690.041Albatross [13]−0.0073.260−0.0080.290−0.0080.302Sparrow [14]−0.0610.477−0.0550.094−0.0520.088Red kite [8]0.0580.3200.0580.0350.0590.034YNP Bison [15]0.1630.1120.1410.0240.1560.028Seneca deer [4]0.3910.0800.3860.0680.4110.003Medians0.0580.2560.0580.0540.0590.034


### 3.2. Scope of Models

Clark et al. [29] extracted 99 data sets from the 1198 published by Brook and Bradshaw [20] using a set of criteria that included a minimum number (19) of “transitions” (used because they used difference equations), so all of the data they used had at least 20 observations. This data set contained 42 cases involving birds and mammals, which we screened with linear regression, of which 20 could be fit with the modified logistic (12) or with an exponential model (4 cases). Those that did not have significant slopes were examined with (10) and appeared to be essentially stationary series (correlation = 0) with the exception of a few instances (3) with an erratic pattern. There were just 4 cases in the Clark data that also fell in our sample (Fulmar 6527, Blue Wildebeest 7060, Sandhill crane 9990, and Blue tit 6830).


## 3.1. Integrated Models versus Difference Equations

Table1 contrasts the generalized logistic (1) with the corresponding difference equation (2), and the ordinary logistic (6) with its corresponding difference equation (7), using residual mean squares in consequence of the experience of Eberhardt et al. [28], who found that residual mean squares performed better on data similar to those used here. We believe that the “adjustment factor” of Burnham and Anderson [27] (in (14)) may discriminate against models with the larger number of parameters. In nearly all instances, the integrated version is to be preferred, as shown by residual mean squares. The two apparent exceptions in Table 1 have highly variable data.Table 1
Comparison of the generalized logistic (1) with the corresponding difference equation (2) and the ordinary logistic (6) compared with the corresponding difference equation (7). Numbers in square brackets refer to references and other numbers are GPDD numbers. Values given are residual mean squares from the fits.
SourceGeneralized logisticDifference equationRatioLogisticDifference equationRatioRed deer 65852.162.471.143.12.80.90Gray fox 3192835912.09378.2566.71.50Gray fox 342131114401.10125413991.12Coyote 263252827291.08245126811.09Gray seals [1]56.8393.21.6465.9390.881.38Bison  [2]5.817.861.355.647.811.38Macaque 60681.111.321.191.451.430.99Beaver 15115960164901.0315850164101.04Beaver 15640960438001.0743400440701.02Elephant seals [3]48.6163.441.3150.0662.791.25Seneca deer [4]29.9350.781.7028.9950.441.74Badger 783754531.21372.5438.81.18Gentoo penguins (Fraser pers. comm.).61.19113.21.8557.88108.61.88Polar bear 14389.968.10.7695.0967.490.71Coyote 2473083801.23335.7362.41.08Gray fox 328549.3901.31.64526.1850.11.62Marten 969517020220101.2916630223901.35Squirrel 96993471003934001.133391003985001.18Gray fox 334338333320.98328031550.96Camargue horses [5]0.5811.721.421.741.23Argentine horses [6]5.018.041.605.387.81.45Przewalski's horses [5, 7]0.891.391.561.231.471.20Red kite [8]3.274.561.393.24.461.39Medians and Mean Ratios89.9113.21.3595.09108.61.20Table2 makes the same contrast for the Gompertz equation (4) with the corresponding difference equation (5). Smaller samples are available due to the fact that the Gompertz did not fit a number of the data sets. That the “Sibly” model gives even more variable results is illustrated in Table 3, where the Sibly model (3) has a larger coefficient of variation in all cases where a comparison could be made, with the exception of the Seneca deer data which has essentially an exponential trend.Table 2
Contrast of fit to Gompertz (4) with difference equation (5) using residual mean squares.
SourceGompertzDifference equationRatioRed deer 65854.013.50.87Gray seals [1]68.5891.911.34Bison [2]6.347.591.20Macaque 60681.571.50.96Beaver 15646930461600.98Elephant seals [3]62.2469.861.12Seneca deer [4]35.453.871.52Badger 78392.74871.24Gray fox 328569.8867.91.52Marten 969516850230601.37Camargue horses [5]1.842.491.35Argentine horses [6]5.678.181.44Przelwalski’s horses [5, 7]1.41.681.20Mean1.24Table 3
Contrasts between the Sibly model (3) and the corresponding difference equation (2). Due to structural differences in the models, coefficients of variation are used for contrasts. An asterisk (*) indicates cases where counts reached or closely approximated an asymptotic value (K).
SourceDifference equationKCV (K)Sibly modelKCV (K)Generalized logisticKRed deer 6585*1660.0201670.028168Gray fox 319*105200.052107600.57410600Gray fox 342*155900.003150600.13915290Coyote 263*193400.063191200.17319110Gray seals* [1]20440.15520160.1672013Bison [2]21201.314no fit1356Macaque 6068*240.026240.05424Beaver 151*1564000.0551585000.057164.500Beaver 156*4227000.0694293000.083421200Elephant seals [3]17520.072no fit1862Seneca deer [4]41340.39626720.3434425Badger 7825500.162no fit3009Gentoo penguins (Fraser pers comm.)9490.534no fit1027Polar bear 1437380.097no fit700Coyote 247*32730.04633100.1033231Gray fox 32892000.296no fit9723Marten 9695983300.062no fit104000Squirrel 9699*2.19E+060.0692.04E+060.1992.17E+06Gray fox 3343.59E+050.0593.54E+050.21722790Camargue horses* [5]750.031770.07478Argentine horses* [6]2090.0512090.069213Przewalski’s horses* [5, 7]440.093430.13144Red kite [8]1397.45no fit182In those cases marked by an asterisk in Table3, the estimates of the asymptotic value (K) are nearly the same for the difference equation (2), the Sibly model (3), and the generalized logistic (1). In all of these cases, the data reach an asymptotic value or closely approximate an asymptote. Coefficients of variation (standard error/K) show that the “Sibly” model (3) gives much more variable estimates than do the other two models considered. These are cases where no gaps existed in the count data. The “Sibly” model does not accommodate isolated data points and has to be fit in segments (yielding more variable estimates), whereas the integrated models can be applied directly in such cases.Table4 gives estimates of the coefficient (z) governing the inflection point in the generalized logistic model, suggesting use of 2.0 in the modified logistic (12). Table 5 contrasts the 5 integrated models using AICc, showing that the modified logistic is generally superior. Clark et al. [29] found, using simulations, that there appeared to be a linear relationship between r and z in the generalized logistic model. Using our data set on birds and mammals, we found a correlation between r and z (−0.51) and a distinctly nonlinear relationship. The data appear in Figure 1.Table 4
Estimates of the parameterz in (1).
SpecieszCVRed deer 65853.890.720Boar 94284.991.138Gray fox 3193.780.407Gray fox 3421.851.151Coyote 2631.931.482Gray seals [1]11.760.882Bison [2]0.750.627Macaque 606816.261.545Beaver 1510.461.739Beaver 1567.431.027Elephant seals [3]1.860.360Seneca deer [4]0.690.652Badger 781.762.290Fur seals [9]4.050.783Blue Springs manatees [10]5.082.108Chinstrap penguins (Fraser, pers. commun.)0.455.778Gentoo penguins (Fraser, pers. commun.)0.692.087Polar bear 1436.131.127Coyote 2476.421.673Gray fox 3281.881.293Boar 94291.660.596Marten 96951.21.892Squirrel 96990.982.920Gray fox 3340.452.480Camargue horses [5]6.060.240Argentine horses [6]7.990.95Przewalski’s horses [5, 7]6.810.690Red kite [8]1.473.380Median1.93Table 5
Contrasting the 5 integrated models using AICc and Δt. Zeros indicate minimum AICc (15).
SpeciesSample sizeGeneralized logisticModified logisticLogisticExponentialGompertzRed deer 6585Cervus elaphus1300.8513.7713.1413.46Gray seals [1]Halichoerus grypus1602.1710.6715.214.62Beaver 156Castor canadensis4600.8410.4152.96.34Polar bear 143Ursus maritimus5106.36.372.75no fitBoar 9428Sus scrufa142.7809.3728.375.2Gray fox 319Urocyon cineroargenteus121.95012.437.79no fitGray fox  342Urocyon cineroargenteus124.8107.227.4no fitCoyote 247Canis latrans114.4601.8422.88no fitCoyote 263Canis latrans173.1035.68165.69no fitElephant seals [3]Mirounga angistirostris261.16010.2561.1613.95Gray fox 328Urocyon cineroargenteus124.8100.3813.752.29Badger 78Taxidea tasus152.97012.378.476.71Fur seals [9]Callorhinus ursinus123.5708.373.473.47Blue springs manatees [10]Trichechus manatus191.607.60.70.7Boar 9429Sus scrufa143.26036.0435.975.09Camargue horses [5]Equs caballus87.3404.5316.978.6Przewalski horses [5, 7]Equs ferus814.0902.14.187.67Marten 9695Martes americana251.2700.9913.370.27Squirrel 9699Sciuris verginianus251.271.27018.230.65Bison [2]Bison bison131.280.26023.290.51Seneca deer [4]Odocoileus virginianus112.585.73040.943.52Gentoo penguins**Pygoscelis papua125.245.77031.144.4Beaver 151Castor canadensis444.730.8013.94no fitChinstrap penguins**Pygoscelis antarcticus222.772.22062.750.05**Fraser, pers. comm.Figure 1
Plot ofr versus z from the 27 data sets that fit the generalized logistic model.Table6 compares the Morris and Doak [24] ((10) and (11)) with exponential models fitted by nonlinear least-squares or a linear fit with log-transformation. The median values indicate that all 3 methods give much the same values, but the Morris and Doak approach is far more variable, as indicated by coefficients of variation.  The coefficients of variation show essentially no difference between the two approaches using exponential models. It should be noted that the Morris and Doak estimator (10) can be used with virtually any data set giving nonsensical results in some cases (only exponential-type data were used in Table 6). Thus Morris and Doak [24] used 39 observations on the Yellowstone grizzly bear data, as shown in their Figure 3.6 and Table 3.1. The problem is that the population was initially decreasing and then, as protective measures began to take effect, started to increase in the early 1980s and continued to do so for the remainder of the series. Consequently, the resulting estimate of r (or μ) is of no practical value. Eberhardt and Breiwick [11] give details and cite 7 published papers that largely ignore the change in trend and thus give largely meaningless results.Table 6
Comparison of Morris and Doak estimates with exponential models.
SpeciesMorris and DoakNonlinear LSLog transformrCVrCVrCVGrizzlies [11]0.0551.4400.0540.1370.0560.126Bison [2]0.2030.080.1880.0370.2010.03Muskox [12]0.1450.1040.1690.0220.1460.027Calif. Sea otters D. Siniff, pers. comm.0.0470.4220.0550.0730.0520.076Gray seals [1]0.0660.2560.0680.0540.0690.041Albatross [13]−0.0073.260−0.0080.290−0.0080.302Sparrow [14]−0.0610.477−0.0550.094−0.0520.088Red kite [8]0.0580.3200.0580.0350.0590.034YNP Bison [15]0.1630.1120.1410.0240.1560.028Seneca deer [4]0.3910.0800.3860.0680.4110.003Medians0.0580.2560.0580.0540.0590.034


## 3.2. Scope of Models

Clark et al. [29] extracted 99 data sets from the 1198 published by Brook and Bradshaw [20] using a set of criteria that included a minimum number (19) of “transitions” (used because they used difference equations), so all of the data they used had at least 20 observations. This data set contained 42 cases involving birds and mammals, which we screened with linear regression, of which 20 could be fit with the modified logistic (12) or with an exponential model (4 cases). Those that did not have significant slopes were examined with (10) and appeared to be essentially stationary series (correlation = 0) with the exception of a few instances (3) with an erratic pattern. There were just 4 cases in the Clark data that also fell in our sample (Fulmar 6527, Blue Wildebeest 7060, Sandhill crane 9990, and Blue tit 6830).


## 4. Discussion

The results given here indicate that, for any practical purposes, the integrated models should be used for species like those considered here (birds and mammals). The Gompertz model may be preferred for some species of fish and for insects. Our analyses here have largely been restricted to data sets that can be fit by the generalized logistic (1). The modified logistic (12) and the ordinary logistic (6) can be fit to a much wider range of data. In a few cases, the exponential (8) may appear to give a better fit, but these appear to be largely instances where the data are limited to the early stages of population increase.The recent ecological literature contains a wide range of difference equations (finite population growth models). We have studied 5 of these [28], but there are additional examples [20–23, 30–35]. One reason for the popularity of difference equation models is that they provide a wide range of models, some of which have no integrated analogs. A popular example is the Ricker model [36, 37]:(16)N(t)=N(t-1)exp⁡⁡(r[1-N(t-1)K]),
where r is the rate of increase and K the asymptotic value, as before. This model was originally developed for cases where r is quite large and thus may be useful for such situations. If r is not large (as in all of the cases examined here), the model reduces to the difference equation version of the ordinary logistic, so that the ordinary logistic or (preferably the modified logistic) can be used. There may thus be some practical reasons to consider difference equations. In many other cases, it appears that the principal goals are the development of ecological theory.



---

*Source: 815016-2012-02-12.xml*