# 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.

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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.

MATERIALS AND METHODS

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 ([r.sub.med]), 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.

RESULTS

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).

[FIGURE 1 OMITTED]

DISCUSSION

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.

ACKNOWLEDGMENTS

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; dreed@olemiss.edu

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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.

MATERIALS AND METHODS

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 ([r.sub.med]), 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.

RESULTS

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).

[FIGURE 1 OMITTED]

DISCUSSION

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. Biogeographic 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] N 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 0.02 S[D.sub.r] (N < 0.5K) K [greater than or equal to] 0.35 [+ or -] 0.04 200 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

ACKNOWLEDGMENTS

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; dreed@olemiss.edu

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Author: | Reed, David H. |
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Publication: | Journal of the Mississippi Academy of Sciences |

Geographic Code: | 1U6MS |

Date: | Jul 1, 2005 |

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