Modeling extinction: density-dependent changes in the variance of population growth rates.
The ability to accurately predict the likelihood of extinction extinction, in biology, disappearance of species of living organisms. Extinction occurs as a result of changed conditions to which the species is not suited. in endangered en·dan·ger
tr.v. en·dan·gered, en·dan·ger·ing, en·dan·gers
1. To expose to harm or danger; imperil.
2. To threaten with extinction. populations of animals is crucial to many concerns in conservation biology conservation biology
The branch of biology that deals with the effects of humans on the environment and with the conservation of biological diversity. . 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 [Latin, By the heads or polls.] A term used in the Descent and Distribution of the estate of one who dies without a will. It means to share and share alike according to the number of individuals. growth rate of a population and its density (i.e., density dependence) is one such parameter (1) Any value passed to a program by the user or by another program in order to customize the program for a particular purpose. A parameter may be anything; for example, a file name, a coordinate, a range of values, a money amount or a code of some kind. . However, the extent to which density dependence influences population dynamics Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. , the usual shape(s) of the density-dependent function, and the impact of density dependence on population persistence (1) In a CRT, the time a phosphor dot remains illuminated after being energized. Long-persistence phosphors reduce flicker, but generate ghost-like images that linger on screen for a fraction of a second. , remain controversial. Here we analyze empirical data from 74 populations (40 species) and find evidence for the ubiquity Ubiquity
See also Omnipresence.
their signs seen as “verses of the wayside throughout America.” [Am. Commerce and Folklore: Misc. of density dependent population growth. More importantly, using stochastic By guesswork; by chance; using or containing random values.
stochastic - probabilistic population models, we find that density-specific changes in the variance The discrepancy between what a party to a lawsuit alleges will be proved in pleadings and what the party actually proves at trial.
In Zoning law, an official permit to use property in a manner that departs from the way in which other property in the same locality of population growth rates Growth Rates
The compounded annualized rate of growth of a company's revenues, earnings, dividends, or other figures.
Remember, historically high growth rates don't always mean a high rate of growth looking into the future. 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·def·i·nite
Not definite, especially:
a. Unclear; vague.
b. Lacking precise limits: an indefinite leave of absence.
c. in the face of finite finite - compact 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 Walde is the surname of:
, 1989). Ecologists This is a list of ecologists, in alphabetical order by surname. A-D
n. pl. gen·er·al·i·ties
1. The state or quality of being general.
2. An observation or principle having general application; a generalization.
3. and importance of density-dependent factors (e.g., malnutrition malnutrition, insufficiency of one or more nutritional elements necessary for health and well-being. Primary malnutrition is caused by the lack of essential foodstuffs—usually vitamins, minerals, or proteins—in the diet. , disease epidemics This article is a list of major epidemics. Worldwide Pandemics
Recent advances in statistical techniques and an increase in the number of long-term Long-term
Three or more years. In the context of accounting, more than 1 year.
1. Of or relating to a gain or loss in the value of a security that has been held over a specific length of time. Compare short-term. ecological ecological
emanating from or pertaining to ecology.
the state of balance in an ecosystem when its inhabitants have established their permanent relationships with each 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 Hanski is Polish and Finnish surname. It may refer to Polish Families and Persons
German anatomist whose works, including Handbuch der Rationellen Pathologie (1846-1852), integrated the study of pathology and physiology. 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 Population viability
The ability of a population to persist and to avoid extinction. The viability of a population will increase or decrease in response to changes in the rates of birth, death, and growth of individuals. 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 Bobby Bowden Field at Doak Campbell Stadium (known as "Doak" for short) is the football stadium on the campus of the Florida State University in Tallahassee, Florida. It is the home venue for the university's Seminoles football team. The stadium was named for Doak S. , 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 Hobbs, city (1990 pop. 29,115), Lea co., SE N.Mex.; inc. 1929. With the discovery (c.1928) of oil and natural gas in the area, Hobbs became one of the last great oil boomtowns in the United States. It remains a major shipping and trading center for oil-well supplies. , 2004).
However, competing claims have been made about the utility and reliability of count-based methods (Brook, 1999; Ludwig Ludwig. For German rulers thus named, use Louis. , 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 Having to do with time. Contrast with "spatial," which deals with space. autocorrelation Autocorrelation
The correlation of a variable with itself over successive time intervals. Sometimes called serial correlation. in environmental factors (Pimm and Redfearn 1988; Inchausti and Halley 2001; Reed et al., 2003a), complicates the seemingly seem·ing
Outward appearance; semblance.
seeming·ly adv. 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 Population viability analysis (PVA) is a species-specific method of risk assessment frequently used in conservation biology. It is traditionally defined as the process that determines the probability that a population will go extinct within a given number of years. models need to be in order to accurately convey population dynamics. Modeling population dynamics is crucial to determining minimum viable population Minimum viable population (MVP) is a lower bound on the population of a species, such that it can survive in the wild. This term is used in the fields of biology, ecology and conservation biology. 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 diffusion, in chemistry, the spontaneous migration of substances from regions where their concentration is high to regions where their concentration is low. Diffusion is important in many life processes. approximations and continuous time Markov chain (probability) Markov chain - (Named after Andrei Markov) A model of sequences of events where the probability of an event occurring depends upon the fact that a preceding event occurred.
A Markov process is governed by a Markov chain. models of population dynamics (e.g., Mangel and Tier 1993; Wilcox Wilcox may refer to: Place names in the United States
See Wilcox (surname) Other
MATERIALS AND METHODS
Stochastic, discrete time Discrete time is non-continuous time. Sampling at non-continuous times results in discrete-time samples. For example, a newspaper may report the price of crude oil once every 24 hours. , 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 NERC Natural Environment Research Council (UK)
NERC North American Electric Reliability Corporation (Princeton, New Jersey, USA)
NERC Northeast Recycling Council
NERC National Environment Research Council 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 mul·ti·ply 1
v. mul·ti·plied, mul·ti·ply·ing, mul·ti·plies
1. To increase the amount, number, or degree of.
2. Mathematics To perform multiplication on. by the census period [greater than or equal to] 50. (2) The population was judged to be in a stable equilibrium (Mech.) the kind of equilibrium of a body so placed that if disturbed it returns to its former position, as in the case when the center of gravity is below the point or axis of support; - opposed to
the number of animal units that a farm or area will carry on a year round basis, including that needed for conservation of winter feed. Usually stated as dry cows or dry sheep equivalents per hectare. 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 regression, in psychology: see defense mechanism.
In statistics, a process for determining a line or curve that best represents the general trend of a data set. was allowed to be a first, second, or third order polynomial polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a . We used corrected Akaike Information Criterion Akaike's information criterion, developed by Hirotsugu Akaike under the name of "an information criterion" (AIC) in 1971 and proposed in Akaike (1974), is a measure of the goodness of fit of an estimated statistical model. It is grounded in the concept of entropy. statistics (Burnham Burn·ham , Daniel Hudson 1846-1912.
American architect and city planner. He did his major work in Chicago, including the general design for the Columbian Exposition (1893) and several early skyscrapers.
Noun 1. and Anderson Anderson, river, Canada
Anderson, river, c.465 mi (750 km) long, rising in several lakes in N central Northwest Territories, Canada. It meanders north and west before receiving the Carnwath River and flowing north to Liverpool Bay, an arm of the Arctic , 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 In statistics, the average amount a number varies from the average number in a series of numbers.
(statistics) standard deviation - (SD) A measure of the range of values in a set of numbers. 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 cal·i·brate
tr.v. cal·i·brat·ed, cal·i·brat·ing, cal·i·brates
1. To check, adjust, or determine by comparison with a standard (the graduations of a quantitative measuring instrument): 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 Noun 1. starting point - earliest limiting point
terminus a quo
commencement, get-go, offset, outset, showtime, starting time, beginning, start, kickoff, first - the time at which something is supposed to begin; "they got an early start"; "she knew from the and the was allowed to run stochastically sto·chas·tic
1. Of, relating to, or characterized by conjecture; conjectural.
a. Involving or containing a random variable or variables: stochastic calculus. for a fixed number of time steps (years). The proportion of replicate rep·li·cate
1. To duplicate, copy, reproduce, or repeat.
2. To reproduce or make an exact copy or copies of genetic material, a cell, or an organism.
A repetition of an experiment or a procedure. 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 Linear regression
A statistical technique for fitting a straight line to a set of data points. 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 Noun 1. stochasticity - the quality of lacking any predictable order or plan
haphazardness, randomness, noise
unregularity, irregularity - not characterized by a fixed principle or rate; at irregular intervals , or genetic stochasticity was included in the models. However, it still has much heuristic A method of problem solving using exploration and trial and error methods. Heuristic program design provides a framework for solving the problem in contrast with a fixed set of rules (algorithmic) that cannot vary.
1. 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 stepwise
incremental; additional information is added at each step.
stepwise multiple regression
used when a large number of possible explanatory variables are available and there is difficulty interpreting the partial regression multiple regression Multiple regression
The estimated relationship between a dependent variable and more than one explanatory variable. 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 Coefficient of Variation
A measure of investment risk that defines risk as the standard deviation per unit of expected return. 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 standardized
pertaining to data that have been submitted to standardization procedures.
standardized morbidity rate
see morbidity rate.
standardized mortality rate
see mortality rate. 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 NDD Norton Disk Doctor (Norton Utilities)
NDD National Direct Dialing
NDD Nature Deficit Disorder
NDD Next-Day Delivery
NDD Non-Decision Directed (digital communications)
NDD Narcotics Detection Dog ) and for models where only the mean population growth rate was allowed to change with changing density (SDD (Software Design Description) The architecture of an information system. See IDD. ). 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 according to
1. As stated or indicated by; on the authority of: according to historians.
2. In keeping with: according to instructions.
3. the regression function.
We also used an analysis of covariance Covariance
A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns vary inversely. (using carrying capacity as the covariate covariate
predictors during the allocation of experimental units in a randomized design. ) to examine whether there were broad phylogenetic phy·lo·ge·net·ic
1. Of or relating to phylogeny or phylogenetics.
2. Relating to or based on evolutionary development or history. (Class, Order, Family) or environmental (biogeographic bi·o·ge·og·ra·phy
The study of the geographic distribution of organisms.
bio·ge·og region, global latitude latitude, angular distance of any point on the surface of the earth north or south of the equator. The equator is latitude 0°, and the North Pole and South Pole are latitudes 90°N and 90°S, respectively. ) effects on the mean time to extinction.
A meta-analysis meta-analysis /meta-anal·y·sis/ (met?ah-ah-nal´i-sis) a systematic method that takes data from a number of independent studies and integrates them using statistical 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 Noisy is the name or part of the name of six communes of France:
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 The Allee effect is a phenomenon in biology characterized by a positive correlation between population density and the per capita growth rate. Description
The Allee effect was first written on extensively by its namesake Warder Clyde Allee. . 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 gen·er·al·ized
1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain.
2. Not specifically adapted to a particular environment or function; not specialized.
3. and lack statistical power. However, it is often useful to test for any patterns based on the ecology ecology, study of the relationships of organisms to their physical environment and to one another. The study of an individual organism or a single species is termed autecology; the study of groups of organisms is called synecology. , life history, or evolutionary histories of the organisms Organisms
See also animals; bacteria; biology; plants; zoology.
Biology, Physiology. the synthesis in living organisms of more complex substances from simpler ones. Cf. catabolism. — anabolic, adj. . 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 con·gru·ent
1. Corresponding; congruous.
a. Coinciding exactly when superimposed: congruent triangles.
b. with other studies that have looked for these types of effects (Gaillard Gail`lard´
a. 1. Gay; brisk; merry; galliard. et al., 2000; Inchausti and Halley, 2001; Reed et al., 2003a; Reed and Hobbs, 2004).
[FIGURE 1 OMITTED]
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 Noun 1. simple regression - the relation between selected values of x and observed values of y (from which the most probable value of y can be predicted for any value of x)
regression toward the mean, statistical regression, regression techniques we used, with an assumption of no time lags and that errors are additive additive
In foods, any of various chemical substances added to produce desirable effects. Additives include such substances as artificial or natural colourings and flavourings; stabilizers, emulsifiers, and thickeners; preservatives and humectants (moisture-retainers); and , 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 vi·o·late
tr.v. vi·o·lat·ed, vi·o·lat·ing, vi·o·lates
1. To break or disregard (a law or promise, for example).
2. To assault (a person) sexually.
3. (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 vertebrate, any animal having a backbone or spinal column. Verbrates can be traced back to the Silurian period. In the adults of nearly all forms the backbone consists of a series of vertebrae. All vertebrates belong to the subphylum Vertebrata of the phylum Chordata. 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 Taylor, city (1990 pop. 70,811), Wayne co., SE Mich., a suburb of Detroit adjacent to Dearborn; founded 1847 as a township, inc. as a city 1968. A small rural village until World War II, it developed significantly in the second half of the 20th cent. 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 non·ex·is·tence
1. The condition of not existing.
2. Something that does not exist.
non . 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 For people named O'Grady, see .
O'Grady is an animated television show created by Tom Snyder and Carl Adams and developed for TV by co-star Holly Schlesinger. et al., 2004). The univariate univariate adjective Determined, produced, or caused by only one variable regression gives the following formula for time to extinction at a given carrying capacity:
[log.sub.10] MTE MTE Ministerio do Trabalho e Emprego (Brazilian Ministry of Work)
MTE My Thoughts Exactly
MTE Middleware Technology Evaluation
MTE Multisystem Test Equipment
MTE Moving Target Exploitation
MTE Multiple Tenant Environment = 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 1. (probability) deterministic - Describes a system whose time evolution can be predicted exactly.
2. (algorithm) deterministic - Describes an algorithm in which the correct next step depends only on the current state. 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 intuition, in philosophy, way of knowing directly; immediate apprehension. The Greeks understood intuition to be the grasp of universal principles by the intelligence (nous), as distinguished from the fleeting impressions of the senses. , 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 pes·si·mism
1. A tendency to stress the negative or unfavorable or to take the gloomiest possible view: "We have seen too much defeatism, too much pessimism, too much of a negative approach" , 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 coun·ter·in·tu·i·tive
Contrary to what intuition or common sense would indicate: "Scientists made clear what may at first seem counterintuitive, that the capacity to be pleasant toward a fellow creature is ... 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 equilibrium, state of balance. When a body or a system is in equilibrium, there is no net tendency to change. In mechanics, equilibrium has to do with the forces acting on a body. populations. Simply computing computing - computer 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 See About this product.
We thank the Mississippi Mississippi, state, United States
Mississippi (mĭs'əsĭp`ē), one of the Deep South states of the United States. It is bordered by Alabama (E), the Gulf of Mexico (S), Arkansas and Louisiana, with most of the border formed by Space Grant Consortium and the National Aeronautics aeronautics: see aerodynamics; airplane; aviation. 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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The first public release of a translator to Scheme by Matt Birkholz, Jim Miller, and Ron Weiss, written at Digital Equipment Corporation's Cambridge Research Laboratory runs E. Heering Jr. and David H. Reed (1)
University of Mississippi The University of Mississippi, also known as Ole Miss, is a public, coeducational research university located in Oxford, Mississippi. Founded in 1848, the school is composed of the main campus in Oxford and three branch campuses located in Booneville, Tupelo, and Southaven. , Department of Biology, University, MS, 38677-1848
(1) Author for correspondence. Department of Biology, P. O. Box 1848; firstname.lastname@example.org