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Modeling extinction: density-dependent changes in the variance of population growth rates.

The ability to accurately predict the likelihood of extinction in endangered populations of animals is crucial to many concerns in conservation biology. A number of parameters are generally believed to significantly affect a population's probability of becoming extinct over a given time span. The relationship between the per capita growth rate of a population and its density (i.e., density dependence) is one such parameter. However, the extent to which density dependence influences population dynamics, the usual shape(s) of the density-dependent function, and the impact of density dependence on population persistence, remain controversial. Here we analyze empirical data from 74 populations (40 species) and find evidence for the ubiquity of density dependent population growth. More importantly, using stochastic population models, we find that density-specific changes in the variance of population growth rates have a larger effect on median time until extinction than do changes in the mean population growth rate. Previous studies have focused primarily on density-dependent changes in the mean growth rate. We demonstrate that density-dependent changes in both the mean and the variance of population growth rates can greatly affect the median time to extinction predicted from stochastic population models.


Population growth cannot continue indefinitely in the face of finite resources. As competition for resources increases at higher population densities, the rate of population growth should slow down and eventually stop. Density-dependent population growth is defined as the dependence of the per capita population growth rate on past population densities (Murdoch and Walde, 1989). Ecologists have debated the generality and importance of density-dependent factors (e.g., malnutrition, disease epidemics) to population dynamics for 70 years.

Recent advances in statistical techniques and an increase in the number of long-term ecological studies has led to a growing consensus that density-dependent reproduction and mortality appears to be widespread in natural populations of vertebrates and invertebrates (e.g., Woiwood and Hanski, 1992; Holyoak, 1993; Wolda and Dennis, 1993; Turchin, 1995; Lande et al., 2002), is thought by many to greatly influence the probability of population persistence, and has long been considered important for population dynamics generally (Ferson et al., 1989; Hanski, 1990; Burgman et al., 1993; Dennis and Taper, 1994; Dennis et al., 1995; Hanski et al., 1996; Lande et al., 2002; Saether et al., 2002; Henle et al., 2004). However, whether density dependence increases or decreases the probability of extinction depends on the exact shape of the density dependent function and its interaction with stochastic factors and life history (Lande et al., 2002; Schoener et al., 2003). For example, the inability of individuals to find a mate or engage in group defense at very low densities might create an absorbing boundary that increases the probability of extinction.

Count-based population viability analyses are often used to estimate population persistence, because these types of data are the ones most often available to conservation biologists (Morris and Doak, 2002). Given a starting population size ([N.sub.0]), the probability of extinction might simply be determined by the mean ([mu]) and the variance ([[sigma].sup.2]) of the distribution of population growth rates (see Dennis et al., 1991; Reed and Hobbs, 2004).

However, competing claims have been made about the utility and reliability of count-based methods (Brook, 1999; Ludwig, 1999; Fieberg and Ellner, 2000; Meir and Fagan, 2000; Sabo et al., 2004). One potential problem with count-based models is their failure to take into account the dependence of both the mean and the variance of population growth rates on population density. This density-dependence, along with correlation patterns among time points in population size due to the temporal autocorrelation in environmental factors (Pimm and Redfearn 1988; Inchausti and Halley 2001; Reed et al., 2003a), complicates the seemingly simple relationship between time until extinction and the mean and variance of population growth rates.

At the heart of this debate is the question of how complex population viability analysis models need to be in order to accurately convey population dynamics. Modeling population dynamics is crucial to determining minimum viable population sizes and ranking conservation priorities. Whether estimates of the density-dependence of population growth rates are necessary for accurate and unbiased estimates of extinction risk is a question of current concern among conservation biologists (e.g., Henle et al., 2004; Sabo et al., 2004).

Despite the importance of the variance in population growth rates, and variance in demographic parameters generally, to all forms of population viability analysis, we are not aware of any published empirical results addressing changes in the variance among population growth rates with changes in density. However, variance in growth rates has been included in a very general way in both diffusion approximations and continuous time Markov chain models of population dynamics (e.g., Mangel and Tier 1993; Wilcox and Elderd, 2003). Here, we examine changes in the variance of population growth rates, as well as changes in the mean growth rate, with density. Further, we demonstrate what effect these changes have on the probability of extinction, as compared to models that disregard density-dependence in population growth rates entirely and to those that only consider changes in the mean population growth rate.


Stochastic, discrete time, discrete state population models were built from empirical data on the distribution of population growth rates, at different densities, estimated from census data obtained from the Global Population Dynamics Database (GPDD)(NERC 1999) for 74 populations (40 species). The populations used to build the models were chosen based on the following criteria: (1) The quality of the data as determined by the GPDD and the length of the census period assayed. The reliability of the census data is scored by the GPDD from one to five (with five being the most reliable). Data sets were selected for inclusion in the study when the reliability rating multiplied by the census period [greater than or equal to] 50. (2) The population was judged to be in a stable equilibrium. Thus, the population was not in continuous decline or so far below carrying capacity as to be in a continual growth phase (i.e., the median population growth rate was approximately zero). Extensive analysis of the data sets in the GPDD (Inchausti and Halley, 2001; Reed et al., 2003a,b; Reed, 2004; Reed and Hobbs, 2004) have shown these are the data sets appropriate for answering the questions being addressed in this paper.

Population growth rates (r), for each time step, were calculated using the following formula:

[r.sub.t] = [log.sub.e]([N.sub.t]/[N.sub.t-1]) (1),

where [N.sub.t] is population size at time t. Positive statistical outliers (especially if they did not seem possible given the species life history) were rare and were removed when detected. However, the distribution of growth rates was normally or approximately normally distributed in all but four cases. Once a distribution of r-values had been calculated the distribution was assayed for evidence of density dependence. Thus, [N.sub.t-1] was regressed against [r.sub.t]. This regression was allowed to be a first, second, or third order polynomial. We used corrected Akaike Information Criterion statistics (Burnham and Anderson, 2002) and model averaging to identify the best fit model. No time lags were allowed in the density dependence. Solving for the carrying capacity (K) was carried out by setting [r.sub.t] equal to zero and solving for N. Thus, the carrying capacity is defined as the population size (density) where the mean population growth rate is expected to be zero.

Once values for K were calculated, means and standard deviations for population growth rates were determined for each population at three different density categories: when N [greater than or equal to] K (high density), when 0.5K [less than or equal to] N < K (intermediate density), and when N < 0.5K (low density). Coefficients of variation in the population growth rate (S[D.sub.r]/r) were also calculated, for each population, at each of the three density categories.

In order to examine the effects of both the mean and the standard deviation of the population growth rate changing with density, we developed a set of stochastic discrete time, discrete state, r-models calibrated from empirical data. The basic format of the model is as follows. First, an initial population size ([N.sub.0]) was set equal to the initial population size from the actual time series. [N.sub.0] was compared to K and a population growth rate (r) randomly chosen from a normal distribution with a mean determined from the regression function and a standard deviation estimated from the actual distribution of growth rates from the time series. The growth rate is then used to determine population size ([N.sub.t]) in the next time step. The process is repeated for each time step, with the distribution of possible randomly selected r values for each time step being determined by the ratio of N:K and the regression function. Each model was run for at least 1,000 simulations.

In order to estimate the median time to extinction, we used at least 1,000 simulations of each population. Each simulation had identical starting points and the was allowed to run stochastically for a fixed number of time steps (years). The proportion of replicate populations going extinct within the determined time frame was recorded. The number of time steps was then varied until several estimates of extinction probability above and below 50% were generated. From this data, linear regression was used to estimate the number of time steps sufficient for 50% of the populations to go extinct (median time to extinction). The median time to extinction is certainly not realistic, as no age-structure, explicit demographic stochasticity, or genetic stochasticity was included in the models. However, it still has much heuristic value as concerns the effects of density dependence.

We used corrected Akaike Information Criterion statistics and model averaging (Burnham and Anderson, 2002) to identify what parameters are important with respect to the median time to extinction. Backwards stepwise multiple regression was used to estimate the parameter coefficients from the consensus model. Thirteen variables were initially tested across different model combinations: Initial population size, the carrying capacity (K), the median growth rate for the entire census period ([]), the mean growth rate at high densities, the mean growth rate at intermediate densities, and the mean growth rate at low densities, the maximum value of r during the census period ([r.sub.max]), the standard deviation in the population growth rate at high densities, the standard deviation in the growth rate at intermediate densities, the standard deviation in the population growth rate at low densities, the coefficient of variation in the growth rate at high densities, the coefficient of variation in the growth rate at intermediate densities, and the coefficient of variation in the population growth rate at low densities. Because standard deviations and coefficients of variation are not independent measures, model selection was based on models that included either the standard deviation or the coefficient of variation, not both. Once the important factors were identified, standardized beta values were calculated to rank the significant factors effects on median time to extinction (Neter et al., 1996).

Sensitivity analysis was performed on the density-specific standard deviation in population growth rates by increasing the standard deviation by 10% increments (up to a maximum increase of 50%) for each density category separately and estimating the slope of the best fit linear line using the standard deviation in the population growth rate as the independent variable and median time to extinction as the dependent variable (Morris and Doak, 2002). The slopes were averaged across all 74 populations and then compared for the three different density categories.

To examine the impact of ignoring density dependence altogether or allowing only the mean growth rate to change, relative to the full model where the mean and variance in growth rates were allowed to change with density, we built additional models for 25 randomly chosen species. Thus, two additional models were constructed with either no density dependence or density dependence where only the mean population growth rate changes with density. Median time to extinction was estimated for models with no density dependence (NDD) and for models where only the mean population growth rate was allowed to change with changing density (SDD). For NDD, the mean population growth rate and the variance among growth rates was the same regardless of density and the values were estimated from the entire census period. For SDD, the variance among growth rates was the same regardless of density, but the mean growth rate changed with density according to the regression function.

We also used an analysis of covariance (using carrying capacity as the covariate) to examine whether there were broad phylogenetic (Class, Order, Family) or environmental (biogeographic region, global latitude) effects on the mean time to extinction.


A meta-analysis of 74 populations was conducted with respect to how the mean, standard deviation, and coefficient of variation in population growth rates changes with changes in population density (Table 1). The mean population growth rate is significantly different at all three density categories and decreases with increasing density across all the 74 populations. Likewise, the standard deviation among growth rates at a given density significantly decreases with increasing density. The coefficient of variation is also highly significantly different for each density category, but the maximum coefficient of variation is reached at intermediate densities. Thus, density-dependent effects on population growth rate were consistent and highly significant despite the data being noisy (e.g., differences in species biology and generation length, differences in the quality of the data). Further, these density-dependent changes were not just to the mean population growth, but also to two measures of variation in growth rates.

It is important to be able to separate purely demographic causes of variation in growth rates from those brought about by the effects of density. Table 2 provides a comparison between how the standard deviation among population growth rates, for a given density category, changes in large (K > 200) versus small (K [less than or equal to] 200) populations. Standard deviations significantly increase with decreasing density in both large and small populations, thus there is an effect of density that is independent of demographic stochasticity. However, the standard deviation is consistently larger in smaller populations, indicating that demographic stochasticity plays a significant role in the amount of variance among population growth rates as well.

We performed model selection using the information-theoretic approach of Burnham and Anderson (2002), using independent combinations of 13 different model parameters (Table 3). The consensus model containing five factors (all significant using multiple regression) are listed and ranked according to their standardized beta values. The significant factors, from greatest effect to least effect are: S[D.sub.r] (intermediate densities) (F = 23.48, P < 0.0001, b = -0.431), K (F = 31.69, P < 0.0001, b = 0.398), S[D.sub.r] (low densities) (F = 9.08, P < 0.005, b = -0.242), r (intermediate densities) (F = 8.13, P < 0.005, b = 0.196), and r (low densities) (F = 7.52, P < 0.01, b = 0.186). The overall regression explains 70.0% of the variation in median extinction times (adjusted [R.sup.2]= 0.700).

In the Introduction it was suggested that the shape of the density-dependent function for growth rates was important to whether it decreased the probability of extinction (primarily believed to be true) or increased the probability of extinction as might be true with strong Allee effects. We illustrate the five general forms of density dependence in population growth rate found in this study and give their relative frequencies (Figure 1). A linear model was found to be the best fit function for 45 of 74 populations. However, the statistical power to detect nonlinearities in individual data sets was often low. Thus, the lack of evidence for general nonlinearity in the relationship between density and per capita growth rates in these data sets should not be construed as suggesting that such nonlinearities do not exist. A 3rd degree polynomial with population growth rates increasing at an increasing rate at very low densities, and decreasing at an increasing rate at very high densities, was the best fit model in 19 of 74 populations. A 2nd degree polynomial where population growth rates increase at an increasing rate at low densities was the best fit in 6 of 74 populations. A 2nd degree polynomial where population growth rates decrease at an increasing rate at both high and low densities was the best fit in 3 of 74 populations. A 2nd degree polynomial where the population growth rate decreases at an increasing rate at high densities was the best fit in only 1 of 74 populations.

An important question in conservation biology is how complex population viability models need to be in order to predict the risk of extinction accurately and without bias. Table 4 shows the results of an analysis of variance comparing median time to extinction, for 25 randomly chosen population models, using three different modeling approaches with the models each developed from the same set of observed data. The models contained either no density dependence, density dependent population growth where the mean growth rate changes with density or population growth rates where the mean and variance were allowed to change with respect to density. The median extinction times for the three models are significantly different from each other (P < 0.01), with the models that allow both the mean and variance in population growth rates to change having the longest median times to extinction and those having no density dependence the shortest.

This paper uses the meta-analysis technique to look for factors useful in predicting extinction risk for populations of vertebrates. These techniques are especially important when individual studies cannot be generalized and lack statistical power. However, it is often useful to test for any patterns based on the ecology, life history, or evolutionary histories of the organisms. We find no differences in median time to extinction based on phylogeny (Class, Order, Family) or environmental classification (biogeographic region, global latitude). None of these factors were significant once the effects of carrying capacity were accounted for. These results are congruent with other studies that have looked for these types of effects (Gaillard et al., 2000; Inchausti and Halley, 2001; Reed et al., 2003a; Reed and Hobbs, 2004).



Ubiquity of density dependence. The role of density-dependent mortality and birth rates, in impacting population dynamics, has been a source of controversy for at least 70 years (Turchin, 1995). However, there seems to be a growing consensus that most populations (at least at times) are regulated through density-dependent mechanisms in conjunction with stochastic factors (e.g., Turchin, 1995; Saether et al., 2002; Reed et al., 2003a). There exist far more sophisticated statistical tests for detecting density-dependence than the one we use here (e.g., Dennis and Taper, 1994) and the question of what role density dependence plays in population dynamics will not be answered definitively by this study. However, it is worth noting that the simple regression techniques we used, with an assumption of no time lags and that errors are additive, was significant in 58 of 74 populations examined (78.4%). This is true despite the median time series being only 19 years and the fact that statistical power is reduced if these assumptions are violated (Turchin, 1995). The mean percent of the variance in the per capita population growth rates explained by population size in the previous time step was 29% (SE [+ or -] 1.9%) across all 74 populations. Mean population growth rates clearly, consistently, and significantly declined as densities increased. Thus, though not a robust test of the presence of density dependence, this study suggests that density dependence is widespread across vertebrate populations (Appendix I & II).

Density dependent changes in the variance of population growth rates. In addition to changes in the mean per capita population growth rate, changes in the variance among growth rates at different densities were detected. Lower densities led to larger standard deviations among population growth rates. This is expected for demographic reasons alone, because smaller populations are more variable (Taylor et al., 1980; Reed and Hobbs, 2004). Indeed, the data confirm that smaller populations tend to have greater standard deviations at all three density categories (significantly higher at the highest and lowest densities) than do larger populations. However, there are significant changes in the standard deviation among population growth rates with changes in density for the large populations and the same pattern of increasing variance among growth rates with decreasing densities can be seen. Anecdotally, even populations of tens of thousands of individuals still often showed the characteristic increase in the variance among growth rates at lower densities, despite their being so large (even at their lowest observed densities) as to make demographic stochasticity almost nonexistent. Thus, there seems to be a component of the variance among population growth rates that is driven by population densities and not just population size. This suggests, as one possibility, that widespread Allee effects may impact not just the mean but the variance in population growth rates.

Parameters affecting median time to extinction. Five factors were identified as significantly affecting median time to extinction in our models, in rank order of their standardized beta values they are: the standard deviation among population growth rates at intermediate densities, the carrying capacity, the standard deviation among population growth rates at low densities, the mean population growth rate at intermediate densities, and the mean population growth rate at low densities.

It is certainly not surprising to see that carrying capacity is a major factor affecting median time to extinction. Models of population viability are usually sensitive to changes in carrying capacity and there is plenty of empirical data linking larger population size to a greater probability of population persistence (see review in Reed et al., 2003a; O'Grady et al., 2004). The univariate regression gives the following formula for time to extinction at a given carrying capacity:

[log.sub.10] MTE = 0.9135 + 0.5776 ([log.sub.10]K) (2),

where MTE is the median time to extinction in years and K is the carrying capacity. However, this model undoubtedly overestimates median time to extinction. In fact, these stochastic r models predict median extinction times that are more than five times as long (K = 10), three times as long (K = 100), or 1.5 times as long (K = 10,000) as models that incorporate far greater complexity (Reed et al., 2004).

The standard deviation in the population growth rate is more important than the mean of the population growth rate, in determining median time to extinction. Thus, models that are deterministic or do not carefully consider estimates of the variation in population growth rates, or demographic parameters generally, will not be able to provide accurate information on the probability of extinction.

The median time to extinction was most affected by both the standard deviation in population growth rates and the mean population growth rate at intermediate densities. This is contrary to intuition, as it might be expected that populations would be most vulnerable to extinction when they are at their lowest densities (smallest size). It is possible that this result is simply due to there being so much more variation among models for the parameters at this density. With this hypothesis in mind, we conducted sensitivity analysis on changes in the standard deviation among growth rates for all three densities for 30 randomly chosen models. Using multiple regression, we found that the models were most sensitive to changes in the standard deviation at the lowest densities (data not shown) as expected from theory. The differences in sensitivity at low and intermediate densities was small, but statistically significant.

Importance of including density-dependent changes in the mean and variance. We are not the first to suggest that density dependence is an important component to include in population viability models (see Introduction). In our simple count-based population viability analysis for 25 species, the median time to extinction without density dependence was less than 30% of the median time to extinction when density dependence was included. Thus, the models without density dependence were not just pessimistic, but they were extremely pessimistic relative to the models with density dependence. This suggests that density dependence generally "puts the brakes on" declining populations by creating reflecting rather than absorbing points at low densities.

Including changes in the variance among population growth rates, in addition to changes in the mean growth rate, nearly doubles persistence time over the case where only the mean growth rate is allowed to change with density. This seems counterintuitive at first, given that the variance among growth rates increases as population sizes decline and this is precisely when populations are most vulnerable. However, the reason for this is that the density-specific variance, even at low densities, is less than the variance among population growth rates for the entire census period. Thus, extreme caution must be used in building count-based PVAs from census data even in equilibrium populations. Simply computing the mean and variance of the growth rates over a given number of time steps is not likely to produce the type of dynamics and, therefore, extinction probabilities that exist in natural populations. The reasons why this is true include the autocorrelation structure in environmental variation through time (Pimm and Redfearn, 1988; Reed et al., 2003a), the lack of inclusion of rare catastrophic events that greatly impact population persistence (Reed et al., 2003b), and density dependent changes in the mean and variance of population growth rates.
Appendix I: Class, order, and biogeographic zone for each of the 40
species modeled.

Species Class Order Zone

Vanellus vanellus Aves Charadriiformes Palaearctic
Rissa tridactyla Aves Charadriiformes Nearctic
Zenaida macroura Aves Columbiformes Nearctic
Accipiter nisus Aves Falconiformes Palaearctic
Falco rusticolus Aves Falconiformes Nearctic
Alauda arvensis Aves Passeriformes Palaearctic
Corvus corone Aves Passeriformes Palaearctic
Corvus frugilegus Aves Passeriformes Palaearctic
Cyanocitta cristata Aves Passeriformes Palaearctic
Melospiza melodia Aves Passeriformes Nearctic
Spizella pusilla Aves Passeriformes Nearctic
Fringella coelebs Aves Passeriformes Palaearctic
Fringella montifringella Aves Passeriformes Palaearctic
Anthus pratensis Aves Passeriformes Palaearctic
Ficedula albicollis Aves Passeriformes Palaearctic
Ficedula hypoleuca Aves Passeriformes Palaearctic
Parus atricapillus Aves Passeriformes Nearctic
Parus bicolor Aves Passeriformes Nearctic
Parus caeruleus Aves Passeriformes Palaearctic
Parus major Aves Passeriformes Palaearctic
Sturnus vulgaris Aves Passeriformes Palaearctic
Phylloscopa collybita Aves Passeriformes Palaearctic
Phylloscopa trochilus Aves Passeriformes Palaearctic
Phalacrocorax aristotellis Aves Pelecaniformes Palaearctic
Picoides pubescens Aves Piciformes Nearctic
Ovis canadensis Mammalia Artiodactyla Nearctic
Tragelaphus strepsiceros Mammalia Artiodactyla Ethiopian
Dama dama Mammalia Artiodactyla Palaearctic
Canis lupus Mammalia Carnivora Palaearctic
Lynx canadensis Mammalia Carnivora Nearctic
Enhydra lutris Mammalia Carnivora Nearctic
Gulo gulo Mammalia Carnivora Nearctic
Martes americana Mammalia Carnivora Nearctic
Phoca groenlandica Mammalia Carnivora Nearctic
Phoca vitulina Mammalia Carnivora Palaearctic
Ursus arctos horribilis Mammalia Carnivora Nearctic
Microtus californicus Mammalia Rodentia Nearctic
Merlangus merlangius Osteichthyes Gadiformes Palaearctic
Perca fluviatalis Osteichthyes Perciformes Palaearctic
Esox lucius Osteichthyes Salmoniformes Palaearctic

Appendix II: F(x) = shape of density dependent function (see Figure 1),
[r.sup.2] = proportion of variance in the population growth rate
explained by density in the preceding time step, M[T.sub.E] = median
time to extinction, K = carrying capacity, r([K.sub.n]) = the mean
population growth rate at high, intermediate and low densities
([K.sub.1], [K.sub.2], and [K.sub.3], respectively), and S[D.sub.r]
([K.sub.n]) = the standard deviation among population growth rates at
high, intermediate, and low densities ([K.sub.1], [K.sub.2], and
[K.sub.3], respectively).

Species [F.sub.(x)] [r.sup.2] M[T.sub.E] K

Vanellus vanellus a 0.06 8 13
Vanellus vanellus a 0.12 33 19
Rissa tridactyla b 0.39 919 171
Zenaida macroura d 0.45 33 24
Zenaida macroura a 0.34 21 110
Accipiter nisus a 0.08 329 111
Accipiter nisus a 0.87 710 52
Falco rusticolus b 0.20 144 82
Alauda arvensis c 0.48 213 57
Corvus corone c 0.44 181 17
Corvus frugilegus a 0.24 58 355
Corvus frugilegus a 0.54 315 126
Cyanocitta cristata a 0.24 246 26
Melospiza melodia a 0.57 26 52
Spizella pusilla b 0.29 273 65
Fringella coelebs a 0.09 1581 616
Fringella montifringilla a 0.19 55 71
Anthus pratensis a 0.34 252 88
Ficedula albicollis a 0.36 9 8
Ficedula albicollis a 0.03 834 99
Ficedula hypoleuca a 0.28 10 9
Ficedula hypoleuca d 0.87 444 82
Ficedula hypoleuca c 0.23 551 144
Parus atricapillus b 0.58 498 114
Parus bicolor a 0.32 16 8
Parus bicolor d 0.10 5 12
Parus bicolor a 0.37 7 17
Parus caeruleus b 0.46 46 46
Parus caeruleus a 0.55 96 82
Parus caeruleus a 0.25 745 74
Parus caeruleus b 0.35 81 89
Parus caeruleus a 0.24 53 44
Parus caeruleus b 0.22 151 87
Perca fluviatalis a 0.41 9 18
Perca fluviatalis a 0.14 41 180
Parus major a 0.45 131 126
Parus major a 0.41 293 208
Parus major a 0.25 103 27
Parus major b 0.21 468 94
Sturnus vulgaris c 0.27 188 54
Sturnus vulgaris b 0.19 41 62
Sturnus vulgaris a 0.35 356 61
Phylloscopa collybita a 0.37 92 12
Phylloscopa trochilus b 0.45 5 7
Phylloscopa trochilus a 0.13 17 9
Phalacrocorax aristotellis e 0.22 1258 399
Picoides pubescens a 0.33 54 5
Picoides pubescens a 0.16 39 8
Ovis canadensis b 0.21 1192 185
Tragelaphus strepsiceros a 0.06 92 6327
Tragelaphus strepsiceros a 0.46 1387 58502
Dama dama a 0.38 1987 970
Canis lupus a 0.14 44 399
Lynx canadensis c 0.20 342 3598
Lynx canadensis a 0.19 1557 31915
Lynx canadensis b 0.05 1261 42300
Enhydra lutris a 0.22 1563 1753
Gulo gulo b 0.05 417 682
Gulo gulo b 0.07 1692 799
Martes americana a 0.26 15 73
Martes americana a 0.12 552 44958
Martes americana a 0.29 779 168
Phoca groenlandica c 0.42 108 96
Phoca vitulina b 0.38 2368 1537
Phoca vitulina b 0.56 843 1208
Phoca vitulina b 0.27 538 135
Ursus arctos horribilis a 0.26 1352 81
Microtus californicus a 0.11 3 54
Microtus californicus b 0.14 11 311
Merlangus merlangius a 0.30 211 1619
Esox lucius a 0.33 349 1967
Esox lucius a 0.19 221 2895

Species r([K.sub.1]) r([K.sub.2]) r([K.sub.3])

Vanellus vanellus -0.033 0.078 0.143
Vanellus vanellus -0.249 0.151 0.339
Rissa tridactyla -0.040 0.069 0.181
Zenaida macroura -0.395 0.172 0.619
Zenaida macroura -0.147 0.146 0.321
Accipiter nisus -0.055 0.027 0.033
Accipiter nisus -0.058 0.223 1.035
Falco rusticolus -0.041 0.007 0.297
Alauda arvensis -0.043 0.095 0.353
Corvus corone -0.109 -0.020 0.139
Corvus frugilegus -0.073 -0.044 0.051
Corvus frugilegus -0.121 -0.049 0.249
Cyanocitta cristata -0.090 0.073 0.252
Melospiza melodia -0.200 0.281 0.568
Spizella pusilla -0.107 0.031 0.185
Fringella coelebs -0.028 0.016 0.047
Fringella montifringilla -0.109 -0.030 0.365
Anthus pratensis -0.096 0.017 0.079
Ficedula albicollis -0.139 0.034 1.206
Ficedula albicollis -0.131 0.042 0.184
Ficedula hypoleuca -0.238 0.188 0.495
Ficedula hypoleuca -0.092 0.184 0.758
Ficedula hypoleuca -0.046 -0.022 0.141
Parus atricapillus -0.158 0.340 0.525
Parus bicolor -0.237 0.067 0.625
Parus bicolor -0.294 -0.141 0.116
Parus bicolor -0.210 -0.015 1.333
Parus caeruleus -0.075 0.113 0.359
Parus caeruleus -0.193 0.136 0.378
Parus caeruleus -0.067 0.085 0.436
Parus caeruleus -0.169 0.168 0.259
Parus caeruleus -0.122 0.098 0.367
Parus caeruleus 0.037 0.027 0.199
Perca fluviatalis -0.037 0.031 0.481
Perca fluviatalis -0.184 0.127 0.261
Parus major -0.307 0.353 0.709
Parus major -0.246 0.092 0.366
Parus major -0.087 -0.065 0.282
Parus major -0.015 0.038 0.329
Sturnus vulgaris -0.026 0.103 0.222
Sturnus vulgaris -0.140 0.241 0.402
Sturnus vulgaris -0.269 0.032 0.268
Phylloscopa collybita -0.210 0.091 0.469
Phylloscopa trochilus -0.441 -0.032 0.500
Phylloscopa trochilus -0.059 0.038 0.260
Phalacrocorax aristotellis -0.019 0.002 0.286
Picoides pubescens -0.098 -0.052 0.597
Picoides pubescens -0.080 0.056 0.348
Ovis canadensis -0.033 0.039 0.199
Tragelaphus strepsiceros -0.086 0.003 0.027
Tragelaphus strepsiceros -0.045 -0.006 0.260
Dama dama -0.028 -0.015 0.091
Canis lupus -0.106 0.246 0.193
Lynx canadensis -0.072 0.111 0.355
Lynx canadensis -0.074 0.044 0.462
Lynx canadensis -0.186 0.081 0.235
Enhydra lutris -0.096 0.080 0.072
Gulo gulo -0.033 0.032 0.056
Gulo gulo -0.139 0.041 0.176
Martes americana -0.136 -0.095 0.625
Martes americana -0.117 0.136 0.158
Martes americana -0.130 0.152 0.279
Phoca groenlandica -0.230 0.225 0.756
Phoca vitulina -0.001 0.006 0.061
Phoca vitulina -0.772 0.058 0.070
Phoca vitulina -0.089 -0.034 0.002
Ursus arctos horribilis -0.052 0.007 0.083
Microtus californicus -0.530 0.300 1.070
Microtus californicus -0.044 0.200 0.317
Merlangus merlangius -0.432 0.497 0.324
Esox lucius -0.136 0.046 0.046
Esox lucius -0.044 -0.038 0.336

 S[D.sub.r] S[D.sub.r] S[D.sub.r]
Species ([K.sub.1]) ([K.sub.2]) ([K.sub.3])

Vanellus vanellus 0.382 0.496 0.685
Vanellus vanellus 0.287 0.292 0.659
Rissa tridactyla 0.120 0.112 0.198
Zenaida macroura 0.117 0.527 0.516
Zenaida macroura 0.516 0.415 0.214
Accipiter nisus 0.077 0.197 0.047
Accipiter nisus 0.093 0.055 2.309
Falco rusticolus 0.308 0.323 0.488
Alauda arvensis 0.124 0.223 0.719
Corvus corone 0.160 0.231 0.266
Corvus frugilegus 0.075 0.214 0.249
Corvus frugilegus 0.174 0.179 0.234
Cyanocitta cristata 0.206 0.267 0.372
Melospiza melodia 0.417 0.365 0.096
Spizella pusilla 0.326 0.332 0.287
Fringella coelebs 0.100 0.062 0.147
Fringella montifringilla 0.399 0.371 0.463
Anthus pratensis 0.123 0.142 0.301
Ficedula albicollis 0.440 0.335 1.706
Ficedula albicollis 0.184 0.214 0.060
Ficedula hypoleuca 0.302 0.619 0.746
Ficedula hypoleuca 0.133 0.269 0.719
Ficedula hypoleuca 0.117 0.154 0.146
Parus atricapillus 0.257 0.387 0.526
Parus bicolor 0.275 0.458 0.744
Parus bicolor 0.131 0.516 0.729
Parus bicolor 0.347 0.666 1.282
Parus caeruleus 0.284 0.479 0.535
Parus caeruleus 0.203 0.451 0.365
Parus caeruleus 0.228 0.345 0.302
Parus caeruleus 0.300 0.333 0.517
Parus caeruleus 0.193 0.389 0.621
Parus caeruleus 0.399 0.290 0.254
Perca fluviatalis 0.210 0.211 0.105
Perca fluviatalis 0.294 0.675 0.145
Parus major 0.254 0.469 0.387
Parus major 0.090 0.372 0.295
Parus major 0.249 0.158 0.495
Parus major 0.154 0.230 0.405
Sturnus vulgaris 0.188 0.395 0.431
Sturnus vulgaris 0.434 0.500 0.685
Sturnus vulgaris 0.189 0.233 0.280
Phylloscopa collybita 0.248 0.288 0.460
Phylloscopa trochilus 0.498 0.302 1.389
Phylloscopa trochilus 0.333 0.461 0.522
Phalacrocorax aristotellis 0.046 0.326 0.433
Picoides pubescens 0.494 0.269 0.392
Picoides pubescens 0.273 0.423 0.429
Ovis canadensis 0.235 0.273 0.375
Tragelaphus strepsiceros 0.097 0.153 0.191
Tragelaphus strepsiceros 0.147 0.098 0.405
Dama dama 0.088 0.074 0.065
Canis lupus 0.322 0.526 0.499
Lynx canadensis 0.264 0.386 0.594
Lynx canadensis 0.207 0.507 0.805
Lynx canadensis 0.257 0.172 0.353
Enhydra lutris 0.199 0.195 0.170
Gulo gulo 0.138 0.184 0.308
Gulo gulo 0.167 0.151 0.399
Martes americana 0.558 0.227 0.873
Martes americana 0.264 0.367 0.315
Martes americana 0.158 0.277 0.202
Phoca groenlandica 0.157 0.475 0.076
Phoca vitulina 0.067 0.071 0.257
Phoca vitulina 0.097 0.104 0.061
Phoca vitulina 0.122 0.137 0.075
Ursus arctos horribilis 0.189 0.078 0.264
Microtus californicus 0.396 1.246 2.099
Microtus californicus 0.214 0.702 0.848
Merlangus merlangius 0.237 0.613 0.222
Esox lucius 0.209 0.244 0.154
Esox lucius 0.216 0.287 0.643

Table 1. Means and standard errors are presented for three different
parameters for three different density ranges.

Parameter Mean [+ or -] SE F P

C[V.sub.r] (high) 0.331 [+ or -] 0.045 30.57 < 0.0001
C[V.sub.r] (intermediate) 0.683 [+ or -] 0.045
C[V.sub.r] (low) 0.205 [+ or -] 0.031
r (high) -0.155 [+ or -] 0.021 123.15 < 0.0001
r (intermediate) 0.082 [+ or -] 0.013
r (low) 0.356 [+ or -] 0.035
S[D.sub.r] (high) 0.251 [+ or -] 0.016 11.29 < 0.0001
S[D.sub.r] (intermediate) 0.355 [+ or -] 0.025
S[D.sub.r] (low) 0.515 [+ or -] 0.047

Table 2. Comparison of the mean (with standard error) standard deviation
in population growth rates across the 74 populations, for three
different density categories, divided as to whether the carrying
capacity was less than or greater than 200 individuals.

 S[D.sub.r] (K
 [greater than or equal to]
 S[D.sub.r] [greater than or equal to]
 (N > K) 0.5K)

K [greater than or equal to] 0.18 [+ or -] 0.30 [+ or -] 0.04
 200 0.02
K < 200 0.28 [+ or -] 0.40 [+ or -] 0.03

 S[D.sub.r] (N < 0.5K)

K [greater than or equal to] 0.35 [+ or -] 0.04
K < 200 0.58 [+ or -] 0.06

Table 3. The results of multiple regression analysis examining 13
factors suspected of being important in determining median time to
extinction in 74 population viability models created from census data on
natural populations of animals. The significant parameters from each
model are listed in order of importance as determined by their
standardized beta values (adjusted [R.sup.2] = 0.700, p < 0.0001).

Parameter Probability Std Beta

S[D.sub.r] (medium) < 0.0001 -0.431
log K < 0.0001 0.398
S[D.sub.r] (low) 0.0036 -0.242
r (medium) 0.0036 0.196
r (low) 0.0108 0.186

Table 4. Results from an analysis of variance (randomized block design
and Tukey's HSD test), comparing median extinction times (in years) for
25 species. Models were built with either no density dependence (NDD),
density dependence where only the mean population growth rate changes
with density (DDM), and density dependence where both the mean and
standard deviation of the population growth rate were allowed to change
with changes in density (DDMS). Each of the three model assumptions
leads to significantly different median times to extinction (M[T.sub.E])
(F = 18.95, p < 0.001).

Model M[T.sub.E]

DDM 476.4
DDMS 913.1
NDD 126.2


We thank the Mississippi Space Grant Consortium and the National Aeronautics and Space Agency for their financial support for this project. We also thank three anonymous reviewers for their helpful comments regarding a prior version of this paper.


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Thomas E. Heering Jr. and David H. Reed (1)

University of Mississippi, Department of Biology, University, MS, 38677-1848

(1) Author for correspondence. Department of Biology, P. O. Box 1848;
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