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.
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.
The first of the integrated models used here, the generalized logistic reported by Nelder [
AIC is calculated as [
Part of the data used here was obtained in 2006 from the Global Population Dynamics Database (GPDD) maintained by the National Environmental Research Council at
Table
Comparison of the generalized logistic (
Source | Generalized logistic | Difference equation | Ratio | Logistic | Difference equation | Ratio |
---|---|---|---|---|---|---|
Red deer 6585 | 2.16 | 2.47 | 1.14 | 3.1 | 2.8 | 0.90 |
Gray fox 319 | 283 | 591 | 2.09 | 378.2 | 566.7 | 1.50 |
Gray fox 342 | 1311 | 1440 | 1.10 | 1254 | 1399 | 1.12 |
Coyote 263 | 2528 | 2729 | 1.08 | 2451 | 2681 | 1.09 |
Gray seals [ | 56.83 | 93.2 | 1.64 | 65.93 | 90.88 | 1.38 |
Bison [ | 5.81 | 7.86 | 1.35 | 5.64 | 7.81 | 1.38 |
Macaque 6068 | 1.11 | 1.32 | 1.19 | 1.45 | 1.43 | 0.99 |
Beaver 151 | 15960 | 16490 | 1.03 | 15850 | 16410 | 1.04 |
Beaver 156 | 40960 | 43800 | 1.07 | 43400 | 44070 | 1.02 |
Elephant seals [ | 48.61 | 63.44 | 1.31 | 50.06 | 62.79 | 1.25 |
Seneca deer [ | 29.93 | 50.78 | 1.70 | 28.99 | 50.44 | 1.74 |
Badger 78 | 375 | 453 | 1.21 | 372.5 | 438.8 | 1.18 |
Gentoo penguins (Fraser pers. comm.). | 61.19 | 113.2 | 1.85 | 57.88 | 108.6 | 1.88 |
Polar bear 143 | 89.9 | 68.1 | 0.76 | 95.09 | 67.49 | 0.71 |
Coyote 247 | 308 | 380 | 1.23 | 335.7 | 362.4 | 1.08 |
Gray fox 328 | 549.3 | 901.3 | 1.64 | 526.1 | 850.1 | 1.62 |
Marten 9695 | 17020 | 22010 | 1.29 | 16630 | 22390 | 1.35 |
Squirrel 9699 | 347100 | 393400 | 1.13 | 339100 | 398500 | 1.18 |
Gray fox 334 | 3383 | 3332 | 0.98 | 3280 | 3155 | 0.96 |
Camargue horses [ | 0.58 | 1 | 1.72 | 1.42 | 1.74 | 1.23 |
Argentine horses [ | 5.01 | 8.04 | 1.60 | 5.38 | 7.8 | 1.45 |
Przewalski's horses [ | 0.89 | 1.39 | 1.56 | 1.23 | 1.47 | 1.20 |
Red kite [ | 3.27 | 4.56 | 1.39 | 3.2 | 4.46 | 1.39 |
Medians and Mean Ratios | 89.9 | 113.2 | 1.35 | 95.09 | 108.6 | 1.20 |
Table
Contrast of fit to Gompertz (
Source | Gompertz | Difference equation | Ratio |
---|---|---|---|
Red deer 6585 | 4.01 | 3.5 | 0.87 |
Gray seals [ | 68.58 | 91.91 | 1.34 |
Bison [ | 6.34 | 7.59 | 1.20 |
Macaque 6068 | 1.57 | 1.5 | 0.96 |
Beaver 156 | 46930 | 46160 | 0.98 |
Elephant seals [ | 62.24 | 69.86 | 1.12 |
Seneca deer [ | 35.4 | 53.87 | 1.52 |
Badger 78 | 392.7 | 487 | 1.24 |
Gray fox 328 | 569.8 | 867.9 | 1.52 |
Marten 9695 | 16850 | 23060 | 1.37 |
Camargue horses [ | 1.84 | 2.49 | 1.35 |
Argentine horses [ | 5.67 | 8.18 | 1.44 |
Przelwalski’s horses [ | 1.4 | 1.68 | 1.20 |
Mean | 1.24 |
Contrasts between the Sibly model (
Source | Difference equation | CV (K) | Sibly model | CV (K) | Generalized logistic |
---|---|---|---|---|---|
Red deer 6585* | 166 | 0.020 | 167 | 0.028 | 168 |
Gray fox 319* | 10520 | 0.052 | 10760 | 0.574 | 10600 |
Gray fox 342* | 15590 | 0.003 | 15060 | 0.139 | 15290 |
Coyote 263* | 19340 | 0.063 | 19120 | 0.173 | 19110 |
Gray seals* [ | 2044 | 0.155 | 2016 | 0.167 | 2013 |
Bison [ | 2120 | 1.314 | no fit | 1356 | |
Macaque 6068* | 24 | 0.026 | 24 | 0.054 | 24 |
Beaver 151* | 156400 | 0.055 | 158500 | 0.057 | 164.500 |
Beaver 156* | 422700 | 0.069 | 429300 | 0.083 | 421200 |
Elephant seals [ | 1752 | 0.072 | no fit | 1862 | |
Seneca deer [ | 4134 | 0.396 | 2672 | 0.343 | 4425 |
Badger 78 | 2550 | 0.162 | no fit | 3009 | |
Gentoo penguins (Fraser pers comm.) | 949 | 0.534 | no fit | 1027 | |
Polar bear 143 | 738 | 0.097 | no fit | 700 | |
Coyote 247* | 3273 | 0.046 | 3310 | 0.103 | 3231 |
Gray fox 328 | 9200 | 0.296 | no fit | 9723 | |
Marten 9695 | 98330 | 0.062 | no fit | 104000 | |
Squirrel 9699* | 0.069 | 0.199 | |||
Gray fox 334 | 0.059 | 0.217 | 22790 | ||
Camargue horses* [ | 75 | 0.031 | 77 | 0.074 | 78 |
Argentine horses* [ | 209 | 0.051 | 209 | 0.069 | 213 |
Przewalski’s horses* [ | 44 | 0.093 | 43 | 0.131 | 44 |
Red kite [ | 139 | 7.45 | no fit | 182 |
In those cases marked by an asterisk in Table
Table
Estimates of the parameter
Species | CV | |
---|---|---|
Red deer 6585 | 3.89 | 0.720 |
Boar 9428 | 4.99 | 1.138 |
Gray fox 319 | 3.78 | 0.407 |
Gray fox 342 | 1.85 | 1.151 |
Coyote 263 | 1.93 | 1.482 |
Gray seals [ | 11.76 | 0.882 |
Bison [ | 0.75 | 0.627 |
Macaque 6068 | 16.26 | 1.545 |
Beaver 151 | 0.46 | 1.739 |
Beaver 156 | 7.43 | 1.027 |
Elephant seals [ | 1.86 | 0.360 |
Seneca deer [ | 0.69 | 0.652 |
Badger 78 | 1.76 | 2.290 |
Fur seals [ | 4.05 | 0.783 |
Blue Springs manatees [ | 5.08 | 2.108 |
Chinstrap penguins (Fraser, pers. commun.) | 0.45 | 5.778 |
Gentoo penguins (Fraser, pers. commun.) | 0.69 | 2.087 |
Polar bear 143 | 6.13 | 1.127 |
Coyote 247 | 6.42 | 1.673 |
Gray fox 328 | 1.88 | 1.293 |
Boar 9429 | 1.66 | 0.596 |
Marten 9695 | 1.2 | 1.892 |
Squirrel 9699 | 0.98 | 2.920 |
Gray fox 334 | 0.45 | 2.480 |
Camargue horses [ | 6.06 | 0.240 |
Argentine horses [ | 7.99 | 0.95 |
Przewalski’s horses [ | 6.81 | 0.690 |
Red kite [ | 1.47 | 3.380 |
Median | 1.93 |
Contrasting the 5 integrated models using AI
Species | Sample size | Generalized logistic | Modified logistic | Logistic | Exponential | Gompertz | |
---|---|---|---|---|---|---|---|
Red deer 6585 | 13 | 0 | 0.85 | 13.77 | 13.14 | 13.46 | |
Gray seals [ | 16 | 0 | 2.17 | 10.67 | 15.21 | 4.62 | |
Beaver 156 | 46 | 0 | 0.84 | 10.41 | 52.9 | 6.34 | |
Polar bear 143 | 51 | 0 | 6.3 | 6.3 | 72.75 | no fit | |
Boar 9428 | 14 | 2.78 | 0 | 9.37 | 28.37 | 5.2 | |
Gray fox 319 | 12 | 1.95 | 0 | 12.4 | 37.79 | no fit | |
Gray fox 342 | 12 | 4.81 | 0 | 7.2 | 27.4 | no fit | |
Coyote 247 | 11 | 4.46 | 0 | 1.84 | 22.88 | no fit | |
Coyote 263 | 17 | 3.1 | 0 | 35.68 | 165.69 | no fit | |
Elephant seals [ | 26 | 1.16 | 0 | 10.25 | 61.16 | 13.95 | |
Gray fox 328 | 12 | 4.81 | 0 | 0.38 | 13.75 | 2.29 | |
Badger 78 | 15 | 2.97 | 0 | 12.37 | 8.47 | 6.71 | |
Fur seals [ | 12 | 3.57 | 0 | 8.37 | 3.47 | 3.47 | |
Blue springs manatees [ | 19 | 1.6 | 0 | 7.6 | 0.7 | 0.7 | |
Boar 9429 | 14 | 3.26 | 0 | 36.04 | 35.97 | 5.09 | |
Camargue horses [ | 8 | 7.34 | 0 | 4.53 | 16.97 | 8.6 | |
Przewalski horses [ | 8 | 14.09 | 0 | 2.1 | 4.18 | 7.67 | |
Marten 9695 | 25 | 1.27 | 0 | 0.99 | 13.37 | 0.27 | |
Squirrel 9699 | 25 | 1.27 | 1.27 | 0 | 18.23 | 0.65 | |
Bison [ | 13 | 1.28 | 0.26 | 0 | 23.29 | 0.51 | |
Seneca deer [ | 11 | 2.58 | 5.73 | 0 | 40.94 | 3.52 | |
Gentoo penguins** | 12 | 5.24 | 5.77 | 0 | 31.14 | 4.4 | |
Beaver 151 | 44 | 4.73 | 0.8 | 0 | 13.94 | no fit | |
Chinstrap penguins** | 22 | 2.77 | 2.22 | 0 | 62.75 | 0.05 |
**Fraser, pers. comm.
Plot of
Table
Comparison of Morris and Doak estimates with exponential models.
Species | Morris and Doak | Nonlinear LS | Log transform | |||
CV | CV | CV | ||||
Grizzlies [ | 0.055 | 1.440 | 0.054 | 0.137 | 0.056 | 0.126 |
Bison [ | 0.203 | 0.08 | 0.188 | 0.037 | 0.201 | 0.03 |
Muskox [ | 0.145 | 0.104 | 0.169 | 0.022 | 0.146 | 0.027 |
Calif. Sea otters D. Siniff, pers. comm. | 0.047 | 0.422 | 0.055 | 0.073 | 0.052 | 0.076 |
Gray seals [ | 0.066 | 0.256 | 0.068 | 0.054 | 0.069 | 0.041 |
Albatross [ | −0.007 | 3.260 | −0.008 | 0.290 | −0.008 | 0.302 |
Sparrow [ | −0.061 | 0.477 | −0.055 | 0.094 | −0.052 | 0.088 |
Red kite [ | 0.058 | 0.320 | 0.058 | 0.035 | 0.059 | 0.034 |
YNP Bison [ | 0.163 | 0.112 | 0.141 | 0.024 | 0.156 | 0.028 |
Seneca deer [ | 0.391 | 0.080 | 0.386 | 0.068 | 0.411 | 0.003 |
Medians | 0.058 | 0.256 | 0.058 | 0.054 | 0.059 | 0.034 |
Clark et al. [
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 (
The recent ecological literature contains a wide range of difference equations (finite population growth models). We have studied 5 of these [