Models for Population Growth Curves

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.

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: 𝑁(𝑡)=𝐾^−𝑧−𝐾^−𝑧−𝑁^0−𝑧𝑒^{−𝑟𝑧𝑡}^−1/𝑧.(1) The corresponding difference equation is𝑁(𝑡)=𝑁(𝑡−1)+𝑟𝑁(𝑡−1)1−𝑁(𝑡−1)𝐾^𝑧.(2) A rearrangement of (2) is designated here as the “Sibly model” [19–22]:pgr=𝑁(𝑡)𝑁(𝑡−1)−1=𝑟1−𝑁(𝐾𝑡−1)^𝑧.(3) The Gompertz curve is𝑁(𝑡)=𝐾explog𝑁⁰𝐾𝑒^−𝑐𝑡.(4) The difference equation in the form used by Dennis et al. [23] is𝑁(𝑡)=𝑁(𝑡−1)exp(𝑟+𝑏log(𝑁(𝑡−1))).(5) The logistic growth curve is obtained by setting 𝑧=1 in (1):𝑁(𝑡)=𝐾1+𝑐𝑒^−𝑟𝑡,𝑐=𝐾𝑁₀−1.(6) The difference equation is𝑁(𝑡)=𝑁(𝑡−1)+𝑟𝑁(𝑡−1)1−𝑁(𝑡−1)𝐾.(7) The exponential model is𝑁(𝑡)=𝑁₀exp(𝑟𝑡).(8) The difference equation is𝑁(𝑡)=𝑁(𝑡−1)+𝑟𝑁(𝑡−1).(9) The equation of Morris and Doak [24] is𝜇=1𝑞^𝑞−1_𝑖=0log𝑁^𝑖+1𝑁_𝑖(10) with variance estimate:var𝜇=1𝑞−1^𝑞−1_𝑖=0log𝑁𝑁^𝑖+1_𝑖−𝜇².(11) 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:𝑁(𝑡)=𝐾⁻²−𝐾⁻²−𝑁⁰⁻²𝑒^−2𝑟𝑡^−1/2.(12) 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]AIC=𝑛log𝜎²+2𝑘,where𝜎²=∑𝑒^2𝑖̂𝑛,(13) the 𝑒_𝑖̂ 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 usingAIC_𝑐=AIC+2𝑘(𝑘+1)𝑛−𝑘−1.(14) Model comparisons are made usingΔ_𝑖=AIC_𝑐−AIC_min,(15) where AIC_min 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 at http://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

Table 1 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.

Table 2 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.

Table 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).

In those cases marked by an asterisk in Table 3, 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.

Table 4 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 AIC_𝑐, 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 parameter z in (1).

Table 5 

Contrasting the 5 integrated models using AIC_𝑐 and Δ_𝑡. Zeros indicate minimum AIC_𝑐 (15).

Figure 1 

Plot of 𝑟 versus 𝑧 from the 27 data sets that fit the generalized logistic model.

Table 6 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.

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]:𝑁(𝑡)=𝑁(𝑡−1)exp𝑟1−𝑁(𝑡−1)𝐾,(16) 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.