T. G. Benton, Institute of Biological Sciences, University of Stirling, Stirling FK9 4LA, UK. Tel: + 44 (0)1786 467809. Fax: + 44 (0)1786 464994. E-mail: firstname.lastname@example.org
1 Variation in an organism’s environment can influence its life history and therefore its population size. Understanding the interplay between noise and population dynamics is of considerable importance, especially for prescribing management of economically important or threatened species. The impact of noise on population biology has been the subject of frequent theoretical investigations and discussed in posthoc analyses of time-series. However, there is a dearth of experimental investigations.
2 Here we report results from a replicated laboratory study of soil mites, Sancassania berlesei, kept in controlled environments but with food supplied randomly, with different synchrony and variances, while maintaining the same mean rate.
3 Increasing environmental variance increases population variance, but decreases mean population size, therefore changing the observed shape of the distribution of population sizes: the shape becomes more skewed with more environmental variance.
4 The distribution of population sizes are best described by negative binomial or gamma distributions. Log-normal and normal distributions rarely fit and Poisson distributions never fit.
5 The correlation between populations is sensitive to the correlation in environmental noise but insensitive to its variance.
6 Different life stages (eggs, juveniles and adults) respond differently to noise, in that there are significant differences in the way that environmental variation changes the mean, variance, shape of distribution and relationship between environmental and population synchrony.
Stochasticity therefore ‘matters’ in ecology and, generally, stochastic models may behave differently from deterministic models (Dennis & Costantino 1988; Boyce 1992; Taylor 1992). However, understanding the impact of stochasticity on populations is difficult practically. The pattern of environmental forcing can rarely, if ever, be reconstructed from observed time-series of population sizes (Ranta et al. 2000; Kaitala & Ranta 2001; Laakso, Kaitala & Ranta 2001). Thus, population biologists face a conundrum: we know stochasticity is important but have little understanding of its biological properties because knowledge of the forcing is rarely known. Although the theory describing the behaviour of populations in stochastic environments has taken strides forward in the last decade, there is a lack of empirical understanding of how known variability affects population biology in different systems.
It is with this background that we undertook an empirical investigation of the population biological consequences of imposing known variability on replicated populations. The model animal we used was a soil mite, Sancassania berlesei (Michael), and by maintaining cultures under controlled conditions, but varying food supply, cultures were subject to environmental variation. The environmental variation was applied with different variances, and within batches of replicates, correlated by different amounts. The relationship between environmental synchrony and population synchrony is addressed elsewhere (Benton, Lapsley & Beckerman 2001). Here we address three further issues: (1) what is the empirical distribution of population sizes and how is this related to different distribution functions; (2) what happens to population size and variability as environmental variability changes; and (3) how is the population synchrony affected by environmental variance but unchanging synchrony?
Knowing the distribution of organisms in stochastic environments, especially when the distribution of environmental states is known, is important information. Under density-independent conditions, the expectation is that populations will be log-normally distributed because a sequence of good years will lead to very large population sizes, due to the multiplicative nature of population growth. However, in density-dependent situations, the nature of the interaction between the noise and the density-dependence will determine the population size distribution. Dennis & Costantino (1988) show that the gamma distribution is a good model of the stochastic population equilibrium of logistic models incorporating multiplicative noise, and it describes the distribution of Tribolium sp. populations, as well as a number of other organisms (such as grasshoppers: Kemp & Dennis 1993). Alternatively, Renshaw (1991) shows that for a stochastic logistic process (incorporating demographic stochasticity alone) then the population size distribution is best described by a negative binomial distribution, especially for smaller population sizes. As different models may make different predications, because their assumptions differ, their validity can be tested: as Dennis & Costantino (1988, p. 1212) say ‘the concept of stochastic equilibrium becomes a parsimonious hypothesis about population regulation that is vulnerable to empirical testing’. Knowledge of the distribution of population sizes is useful for a number of reasons. For example, in developing theory it is often useful to be able to make assumptions about population sizes. The moment closure techniques advanced by Keeling (2000a,b) to investigate the persistence of metapopulations rely on the assumption of the log-normality of population sizes. We aimed to identify if different distributions could fit the observed population sizes, whether these distributions fitted (or not) generally, or only in specific circumstances (such as high/low variability, or only to specific life stages).
The variance in population sizes, in relation to its mean size, bears an important relationship to persistence probability and, as such, forms the basis of population viability analysis. Given time-series data from a locality it is possible to estimate the variance in population size. However, much recent interest centres around predicting population sizes when habitats change (see review by Bradbury et al. 2001). Predicting how a change in environmental variability (such as may occur under global warming scenarios: Dai et al. 2001) will relate to a change in mean population size and variance is outside the domain of most data, and requires empirical information. ‘To understand the role of stochasticity in population extinction, we must understand how environmental variability affects the organism’ (Boyce 1992; p. 484); we aim here to provide some empirical information from a model organism which will help inform situations where data are unavailable.
Environmental synchrony is important in predicting population synchrony and has been extensively studied (Kendall et al. 2000; Ripa 2000; Greenman & Benton 2001). However, the relationship between environmental variance and correlation has received relatively little attention. As synchrony has important management implications, whether one can manipulate environmental variance and therefore change the population synchrony is an interesting question. A priori, one might expect a positive relationship between correlation and variance. As discussed extensively in recent literature (Grenfell et al. 1998, 2000; Blasius & Stone 2000; Benton et al. 2001; Greenman & Benton 2001) for a given level of environmental synchrony, the population synchrony is usually lower: there is a ‘loss’ of correlation. As a result, increasing variation may decrease synchrony (Greenman & Benton 2001; Fig. 2), although this may depend on the distribution from which noise is drawn. The sensitivity of results to model assumptions requires that we make some efforts to investigate this question empirically in order to ‘ground truth’ modelling approaches.
Materials and methods
Populations of the soil mite, Sancassania berlesei, were collected from an agricultural manure heap (the composted waste from poultry farms) in September 1996. Stock cultures are maintained with population sizes of several thousand. Cultures are kept at a constant 24 °C in three unlit incubators. Food is supplied in the form of granules of yeast, which are sieved to reduce the variance in size. One granule averages 2·07 mg ± SD 0·25 (median = 2·07, Q1 = 1·91, Q3 = 2·24, n = 123). Experimental culture vessels are 20 mm diameter, 50 mm high, flat-bottomed glass test-tubes. These are half-filled filled with plaster of Paris which, when kept damp, maintains the humidity. The top of the tube is sealed by a circle of filter paper (which allows gaseous diffusion) held in place by the tube’s standard plastic cap with ventilation holes cut into it. All tubes receive an average of one granule of yeast per day. A daily granule is sufficient to maintain an average population of 135 ± SE 2 adults, 659 ± 68 juveniles and 427 ± 32 eggs (n = 5 tubes, each counted 28 times).
Populations are censused by counting mites using a Leica MZ8 binocular microscope and a hand counter. In each tube, a sampling grid is scratched on the plaster base. All adult males and all adult females are counted directly. The number of juveniles and eggs are each counted in a quarter of the tube, so analyses are based on the ‘quarter counts’. To reduce the impact of counting error, the analyses described below are based on the counts of a single individual (CTL), although qualitatively similar results occur when TGB’s and APB’s counts are included.
Environmental variability is introduced by varying the food regime, while keeping the mean supply constant (at 1 granule of yeast per day). The food available over a sampling interval (7, 14 or 21 days) is supplied in random amounts and at random days within the interval (where both days and amounts are selected from uniform distributions). Thus, with ‘14-day’ variability, the mites could receive from one extreme of 14 granules on one day, with the other 13 days without food, to the other extreme of a daily granule. Increasing the sampling interval increases the variance in daily food supply (Table 1).
Table 1. Descriptive statistics of the environmental forcing given the mites during the period of analysis. Cultures received an average of one granule of yeast per day, but at random amounts and random times within a sampling interval: as sampling interval increases (from 7 to 21 days) the variance in food supply increases (measured by the mean standard deviation in daily food supply, based on n = 5 cultures per treatment). Food was supplied in identical regimes to tubes within treatments (correlation equals 1·0), or different regimes correlated at 0·5. The observed mean correlations (average of 10 pairwise correlations between food regimes) were not significantly different from the 0·5 correlation from which they were randomly derived. The distribution of food is described in the columns labelled P0, P1.P7. P0 is the proportion of days with no food. P1 is the proportion of days with one granule (7-day variability), 1–2 granules (14-day variability) or 1–3 granules (21-day variability)
Actual mean standard deviation in daily food
Actual mean correlation between daily food within treatments
Distribution of food
7-day variability: food supplied with correlation 0·5 between replicates
14-day variability: food supplied with correlation 0·5 between replicates
21-day variability: food supplied with correlation 0·5 between replicates
7-day variability: identical food supplied between replicates (correlation = 1)
14-day variability: identical food supplied between replicates (correlation = 1)
21-day variability: identical food supplied between replicates (correlation = 1)
Eight treatments were set up, based on the feeding regime, each of five replicates:
(a)‘constant’ treatment: 1 granule of yeast per day;
(b)7-day variability: food supplied with correlation 0·5 between replicates;
(c)14-day variability: correlation in food = 0·5;
(d)21-day variability: correlation in food = 0·5;
(e)7-day variability: identical food supplied between replicates (correlation = 1);
(f)14-day variability: correlation in food = 1·0;
(g)21-day variability: correlation in food = 1·0; and
(h)14-day variability: correlation in food = 1·0, replicates fed a non-random regime – 14 balls on the first day of each 14-day period. This regime mimics regular environmental forcing (such as seasonal forcing, or multiannual forcing, as with the North Atlantic Oscillation).
Replicate tubes were set up with ~100 males and ~100 females in March 2000. Mites were censused every 4th day, starting 17 March 2000 (treatments, a, c and f) or 30 March 2000 (treatments b, d, e, g, h).
Inspection of the time-series indicates non-stationarity (caused principally by the initial cohort of juveniles maturing and overshooting the carrying capacity). As a result, the first 28 counts were discarded (see Benton et al. 2001 for details). This left 36 counts for treatments a, c and f, or 41 counts for the others. Details of the food supply during this period (the ‘environmental variance’) is given in Table 1.
To investigate the shape of the distribution of population sizes, the data from sets of five tubes were pooled (as per Dennis & Costantino 1988). To these distributions of counts were fitted normal, lognormal, Poisson, gamma and negative binomial distributions. The former three distributions used the observed mean and/or variance. The latter two distributions were fitted with Splus routines (gamma fitter was originally written by Peter Perkins and available from the StatLib at http://www.stat.unipg.it/pub/stat/statlib/ and the NB fitter was written by Darren Shaw and is available from email@example.com). The goodness of fit of the observed data to the fitted distributions was assessed using Kolmogorov–Smirnov and chi-square tests (Splus routines ks.gof and chisq.gof).
For other analyses, counts were logged. Analysis of the population size and variance was undertaken on the logged data; analysis of the correlations was undertaken on the differences of the logged data. Within each treatment there are five experimental tubes. Monte Carlo resampling techniques were used to generate the means, variances, correlations between the replicate tubes and associated confidence intervals. There are two strata of sampling error: estimating the parameters for each time-series (the within-tube error) and estimating the statistics for each group of replicated time-series (the within-group error). Resampling was therefore conducted as a two-stage process. The five replicate time-series were used to generate resampled time-series: for each time (1 … 36 or 1 … 41), one of the five values from the replicates was chosen randomly. The mean and variance of each resampled time-series was calculated, and the bootstrap distribution of 1000 samples gave confidence limits for the means and variances of single tubes. To estimate confidence limits for the mean and variances in population size for each group of five tubes, the 1000 resampled time-series were themselves resampled: 1000 resamples of five values were taken. To estimate confidence limits for the average within-treatment correlation, replicate sets of five time-series were created by resampling all five time-series at random times (n = 36 or 41). For each set of five time-series, correlations between the 10 possible pairwise combinations were estimated. The mean of these 10 values was itself bootstrapped by taking 1000 resamples to give the bootstrap distribution of mean values.
The bootstrapping analysis provides bootstrapped means and their 95% confidence intervals for the mean population size, population variance and correlations between tubes within each treatment group. For most analyses in this study, the independent variable is the environmental variance (1 per day, 7 per week, 14 per 2 weeks, 21 per 3 weeks). Standard linear regression and anova was used to analyse the relationship between the environmental variance and the bootstrapped median population size, variance or correlation.
Cultures fed granulated yeast supplied at a constant rate exhibit initial oscillations which rapidly decay (Fig. 1). This initial over-compensatory behaviour is not strongly excited by stochasticity, as accidental perturbations in the dynamics of ‘constant cultures’ lead to a rapid return to equilibrium (see Fig. 1a, points marked by *). There is little qualitative difference between cultures fed food in random sequences or in sequences which show regular variation (Fig. 1c,d).
WHAT MODELS DESCRIBE THE DISTRIBUTION OF POPULATION SIZES AS ENVIRONMENTAL VARIATION INCREASES?
The stationary population counts for tubes within treatments were pooled and five different model-distributions were fitted to the resulting empirical distributions. As may be expected from the relatively small sample sizes (n ranges from 160 to 225), the power of goodness-of-fit models is relatively low, resulting in more than one model fitting the data for any treatment. However, what is evident from the results (Fig. 2, Table 2) is that (a) the shape of the distribution varies with the environmental variance and the life-history stage in question, and (b) different distributions fit better than others: of 16 treatment combinations the negative binomial distribution is not significantly different from the observed data in 15 cases, the gamma distribution in 12 cases, the log-normal distribution in 10 cases and the normal distribution in four cases; the data are always significantly different from a Poisson distribution.
Table 2. Details of models fitted to the empirical distributions of population sizes for mite life-history stages at different environmental variances (all tubes within a treatment received the same regime of food, i.e. correlation = 1, and counts were pooled for the five tubes). The parameters given are those necessary to fit the distribution and are directly estimated from the data. Goodness-of-fit test details also given (χ2 = chi square test statistic; KS = Kolmogorov–Smirnov statistic). Distributions which show a significant lack of fit are italicized, those which do not show a significant lack of fit are shown in bold
Negative binomial distribution
There are clear patterns of change in the shape of distributions as variance in the food supply increases (Fig. 2). Typically, as environmental variance increases, the centre of distributions shift towards zero and skew increases (see below). This is clear from examination of the fitted shape parameter, k, for the negative binomial distribution, which determines the over-dispersion of the distribution. For all life stages, k increases as environmental variance increases (Table 2). As a result, distributions which can be asymmetrical generally fit the data, especially when the environment varies greatly (the only cases when the normal distribution describes population sizes is when environmental variance is low; when environmental variance is high population sizes are fitted by skewed distributions: gamma, negative binomial and log-normal).
In all cases the modal daily food supply was no food, such that the distribution of food supplies was highly skewed and over-dispersed (Table 1). In no cases did any of gamma, negative binomial or log-normal distribution functions fit the observed distributions in food supply, indicating a lack of close correspondence between the noise distribution and population distribution. For all the ‘correlation 1·0’ tubes the skew, kurtosis, CV and range of the distribution of food and each of the life-stages were estimated. Regressing population estimates against food estimates indicates some similarity between the ‘forcing’ and the ‘forced’ distributions. Controlling for differences between stages, there are significant relationships between the CVs of the food and population distributions (GLM, CV of populations ln transformed, F1,14 = 20·8, P < 0·0005, R2 = 60%), the skew (GLM, skew of populations ln transformed, F1,14 = 9·1, P = 0·009, R2 = 39%), kurtosis (F1,14 = 8·7, P = 0·011, R2 = 38%) and range (F1,11 = 7·5, P = 0·02, R2 = 14%). So, the shape of the ‘forcing’ distribution weakly predicts the shape of the population distribution, with the strongest relationship being the CV.
RELATIONSHIP BETWEEN MEAN POPULATION SIZE AND ENVIRONMENTAL VARIATION
As might be expected, as environmental variance increases the mean population size decreases (Figs 2 and 3, Tables 2 and 3). The slope of the regression between population size and variance depends on the life stage: there is a strong and significant reduction in mean number of eggs as variance increases, but juveniles show a much weaker relationship (which is not significant). The correlation in food supply between replicate tubes did not affect mean food supply (Table 1); as a result the relationship between population size and variance in food supply does not differ between treatments where all tubes receive identical food or they receive correlated patterns of food supply.
Table 3. Summary table of statistics for relationship between mean ln population size and environmental variance (var = 0, 1 … 3, where 1 = 7-day variance, 2 = 14-day variance, etc.). Bootstrapped mean population sizes were used as the response variable in linear regressions with environmental variance as predictor variables. Details of the regressions are displayed for (a) each life stage and (b) treatments grouped by the synchrony in food supply. t-tests were conducted on the coefficients of the linear regressions to see if there were differences between the synchrony treatments
Difference between 0·5 and 1·0
y = 4·51(± 0·10) – 0·363(± 0·05) var
y = 4·58(± 0·14) − 0·501(± 0·075) var
intercepts t = 0·23, NS
R2 = 0·962, F1,2 = 51, P = 0·019
R2 = 0·96, F1,2 = 45, P = 0·022
slopes t = 1·52, NS
y = 5·00(± 0·15) – 0·128(± 0·08) var
y = 5·03(± 0·16) − 0·135(± 0·09) var
intercepts t = 0·14, NS
R2 = 0·57, F1,2 = 2·7, P = 0·245
R2 = 0·55, F1,2 = 2·5, P = 0·256
slopes t = 0·08, NS
y = 4·14(± 0·05) – 0·212(± 0·03) var
y = 4·23(± 0·17) − 0·217(± 0·09) var
intercepts t = 0·52, NS
R2 = 0·97, F1,2 = 67·3, P = 0·015
R2 = 0·75, F1,2 = 6·1, P = 0·133
slopes t = 0·05, NS
y = 4·17(± 0·09) – 0·133(± 0·05) var
y = 4·25(± 0·17) − 0·135(± 0·09) var
intercepts t = 0·36, NS
R2 = 0·78, F1,2 = 7·3, P = 0·114
R2 = 0·53, F1,2 = 2·3, P = 0·27
slopes t = 0·20, NS
The environmental variation received by the ‘seasonal tubes’ (a pattern of 14 granules of yeast on one day, followed by 13 days without food) has a higher daily variance in food supply than any of the treatments with randomly supplied food (Table 1). The reduction in mean population size for these seasonal tubes is in line with the relationships between population size and variance (Fig. 3), suggesting that whether the environmental variability is stochastic or deterministic may be relatively unimportant compared to its absolute variance.
RELATIONSHIP BETWEEN POPULATION VARIABILITY AND ENVIRONMENTAL VARIATION
As environmental variation increases, so does the variation in population size (as measured by the coefficient of variation: the ratio of the standard deviation to mean) (Figs 2 and 4; Table 4). The strength of this relationship varies between life stages (as with mean population size, the relationship is strongest with eggs and weakest with juveniles). In only one case, the males, does the slope of the relationship differ between treatments where all tubes receive the same food supply and treatments where tubes receive correlated food supplies. If male population sizes decrease sufficiently juvenile males develop into fighter morphs (Radwan 1995) which kill other males and so lead to smaller numbers of males. Tubes in which fighters develop may therefore vary more in the long term than tubes in which fighters do not (because the lower tail of the population sizes is prolonged). One tube in the 14-day, 0·5 correlations, and two in the 21-day, 0·5 correlations received food supplies which repeatedly caused sufficient reduction in population size to promote fighter development. These individual tubes are responsible for the difference in average variability between the treatments receiving correlated or identical food supplies.
Table 4. Summary table of statistics for relationship between mean coefficient of variation (CV) of ln-transformed population size and environmental variance (var = 0, 1 … 3, where 1 = 7 day variance, 2 = 14 day variance, etc.). Bootstrapped CVs were used as the response variable in linear regressions with environmental variance as predictor variables. See legend to Table 2 for further details
Difference between 0·5 and 1·0
y = 0·071(± 0·018) + 0·101(± 0·009) var
y = 0·63(± 0·025) + 0·118(± 0·013) var
intercepts t = 0·26, NS
R2 = 0·98, F1,2 = 117, P = 0·008
R2 = 0·98, F1,2 = 78, P = 0·013
slopes t = 1·05, NS
y = 0·068(± 0·012) + 0·023(± 0·007) var
y = 0·065(± 0·006) + 0·020(± 0·003) var
intercepts t = 0·22, NS
R2 = 0·86, F1,2 = 12·3, P = 0·07
R2 = 0·95, F1,2 = 40, P = 0·024
slopes t = 0·41, NS
y = 0·067(± 0·008) + 0·062(± 0·004) var
y = 0·072(± 0·005) + 0·042(± 0·003) var
intercepts t = 0·53, NS
R2 = 0·99, F1,2 = 214, P = 0·005
R2 = 0·99, F1,2 = 221, P = 0·005
slopes t = 3·96, P = 0·029
y = 0·060(± 0·016) + 0·045(± 0·009) var
y = 0·069(± 0·012) + 0·024(± 0·007) var
intercepts t = 0·44, NS
R2 = 0·93, F1,2 = 27, P = 0·035
R2 = 0·87, F1,2 = 13·7, P = 0·066
slopes t = 1·934, P = 0·15
As with mean population size, the seasonal supply of food generally creates variation in population size in line with the expectations given the daily variance.
RELATIONSHIP BETWEEN POPULATION SYNCHRONY AND ENVIRONMENTAL VARIATION
Population synchrony is proportional to environmental synchrony: populations that received the same food supply (i.e. correlation = 1·0) are generally more synchronized than populations which received food supplies correlated at 0·5 (Fig. 5). When each stage is analysed separately using analysis of variance, the minimally adequate statistical model in all cases includes only the correlation in food supply and not the variance in food supply (response variable is bootstrapped median correlation for each stage, predictor variable is correlation of food supply, eggs F1,4 = 79·6, P < 0·001; juveniles, F1,4 = 37·7, P = 0·004; females, F1,4 = 8·14, P = 0·046; males, F1,4 = 8·79, P = 0·041). Populations receiving food at a given correlation generally are synchronized to the same degree irrespective of the variance in the food supply: there is no detectable increase in population synchrony with environmental variation.
In all cases, the population synchrony is significantly less than the environmental correlation, and depends on the life stage (Fig. 5, Benton et al. 2001). When the correlation in food supplies is 1·0, the average correlation in numbers of eggs across the three variances is 0·74 ± 0·6, juveniles is 0·45 ± 0·03, males is 0·43 ± 0·08 and females is 0·48 ± 0·08. Similarly, for food correlations of 0·5, the resulting population synchrony for eggs, juveniles, males and females is 0·33 ± 0·05, 0·23 ± 0·06, 0·22 ± 0·1 and 0·22 ± 0·13, respectively.
For each life stage, the mean population correlation for the ‘seasonal tubes’ is lower than the correlation of the 21-day, correlation 1·0 tubes, even though the variance in food supply is higher for the former (Table 1). Although the 95% CIs overlap the binomial probability of this is 0·065, perhaps indicating a weak influence of the temporal pattern of variance on levels of synchrony.
Variation in the environment affects all organisms to some extent, causing variability in life histories and therefore population dynamics. Quantitative, rather than qualitative, predictions require population models that incorporate environmental variation. However, there is a dearth of information about how populations respond to environmental variation: how does environmental noise map onto population noise (Ranta et al. 2000)? Are populations log-normally (Keeling 2000a,b) or gamma distributed (Dennis & Costantino 1988)? Population viability analysis produces estimates based on the variance and mean population size of organisms in their current environment (Boyce 1992); but what would happen to the mean and variance in population sizes as environmental variability increases, such as caused by global warming (Dai et al. 2001) or by populations being concentrated by habitat loss in poor habitats (Bradbury et al. 2001)? The Moran effect tends to synchronize populations sharing similar patterns of environmental variation, but if variance changes does this increase or decrease synchrony? Answers to question such as these can be found by using modelling approaches. However, all modelling involves making assumptions. Often with stochastic models, the details of the answer are sensitive to the assumptions made (Mills et al. 1996; Benton & Grant 1999; Greenman & Benton 2001), hence empirical investigations are needed in order to give theoreticians ‘a steer’. Furthermore, the difficulty of relating noise input to population response (Ranta et al. 2000) suggests experimental investigations are needed to investigate how noise affects ‘real’ organisms.
SHAPES OF DISTRIBUTIONS
What shape might we expect distributions of population sizes to be in stochastic environments? Under density-independent conditions population sizes are expected to be log-normally distributed (Tuljapurkar 1990). However, density dependence will prevent extremely large values of population sizes which are likely to occur from a run of ‘good years’ and geometric growth. Density dependence is likely to be ubiquitous in organisms (at least occasionally) and occurs in Sancassania living in culture (Fig. 1). Block & Allen (2000) derive structured models, based on the branching process, from which the quasi-stationary distribution of population sizes can be estimated. These models are stochastic in that they incorporate demographic stochasticity and they predict, for large population sizes, a distribution which is approximately normal (Block & Allen 2000). Similarly, Nåssel (1998) showed that models based on a stochastic version of the logistic gave rise to approximately normal distributions at large K. Renshaw (1991) also using a stochastic version of the logistic shows that the normal approximation works for large K, but that a negative binomial distribution may fit better, especially when K decreases. Finally, Dennis & Constantino (1988), use a gamma distribution also derived from a logistic model, and show that it is a good fit to a range of empirical datasets. The stochastic model they used to derive the gamma distribution incorporates stochasticity as a multiplicative noise term that is more analogous to environmental stochasticity rather than demographic stochasticity.
The gamma distribution and negative binomial distributions are related. The relationship between them is seen most clearly when they are used to model ‘waiting times’: the distribution of times between events. The negative binomial (or ‘binomial waiting time distribution’) was derived for the discrete time Bernoulli process (e.g. coin flipping) and models the probability that one waits through × independent trials to gain k successes (Wilks 1962; Johnson & Kotz 1969). The gamma is most familiar as a model of the quadratic forms of sums of normally distributed random variables (the chi-square distribution is an example), but can also model the waiting times to obtain k successes for the continuous time Poisson process (e.g. number of decays of radioactive particles per unit time) (Wilks 1962; Johnson & Kotz 1970). The gamma can be derived from the negative binomial (Wilks 1962), and the similarity between the continuous gamma distribution and discrete negative binomial is indicated by both models having the same shape parameter, k. Table 2 gives the maximum likelihood estimates of k for both distributions and they are close, with differences due perhaps to differences in the way the algorithms bin the data – the goodness-of-fit tests for the negative binomial used ~2× more bins (and so d.f.) than the algorithm used for the gamma.
The gamma and negative binomial distributions share two important properties. First, the shape parameter describes the skew in the distribution (if k is large, skew is small). Hence k can be thought to be related inversely to the ‘stochastic force’ of the environment (Dennis & Costantino 1988). When k is large, the distribution becomes symmetrical and differences from a normal distribution may be small, so although a normal distribution may fit (e.g. Nåssel 1998; Block & Allen 2000), it may not be the best fit (Renshaw 1991). Secondly, both distributions are ‘reproductive’, that is to say (under certain conditions at least) the sum of two gamma- or negative binomially distributed variables will itself be gamma- or negative binomially distributed (Johnson & Kotz 1969, 1970).
Under the laboratory conditions used here, we tried to minimize environmental stochasticity, other than that imposed by varying the food supply. For the ‘constant’ tubes, the distribution of population sizes is skewed sufficiently that the normal distribution is a poor fit to the empirical distribution (contraBlock & Allen 2000). This skew increases markedly as the environmental variance increases. We might expect density dependence to prevent extreme values of population sizes and therefore the log-normal distribution not to be a good model of the distribution. This is generally the case (Table 2), with log-normal approximations to the empirical distributions tending to over-estimate the right-hand tail of the distribution (contraKeeling 2000a,b). Instead, the distributions which most closely fit the empirical distributions are the negative binomial and gamma distributions (and differences between the goodnesses-of-fit may be due to differences in the algorithms used to derive estimates and perform the chi-square test). These data therefore add support to the theory which derives the quasi-stationary distribution derived from stochastic logistic equations (Dennis & Costantino 1988; Renshaw 1991). With the reproductive property of the distributions, even if the distributions differ in shape for the different life stages, it is likely that the sum of all stages will also be described by these models.
The relationship of input noise and observed population dynamics will be explored fully elsewhere. However, it is noticeable that the distribution of noise is not congruent with the distribution of population sizes in any case. In addition, descriptors of the shape of the distribution of the food predicted the descriptors of the distribution of population sizes, but only weakly: more variable, more skewed distributions of food led to more variable, more skewed distributions of population sizes. Theoretical work (Ranta et al. 2000; Kaitala & Ranta 2001; Laakso et al. 2001) indicates that in many cases the organism acts as a ‘filter’ changing characteristics of the environmental noise to the extent that a significant statistical association between the forcing and the forced distributions may be lost.
RELATIONSHIP OF MEAN TO VARIANCE
The more variable the environment becomes the more variable the population (Fig. 4, Table 4) and the smaller the mean (Fig. 3, Table 3). Interestingly, the increase in variability is more predictable than the decrease in mean size (the R2s are larger). At the highest stochastic food regimes (21-day variability) female population sizes were 70% of those populations on constant food and with SDs 30% bigger; similarly, male population sizes were 58%, juveniles 67% and eggs 42% of the populations fed constant food, with SDs 15%, 18% and 93% larger, respectively. Although the population sizes decreased markedly in the very variable environments, in no cases (n = 55 tubes in total) was an extinction observed during the course of the experiment. This is due partially to the overlapping generations: even if one life stage disappeared (and different stages reacted differently to changes in variability) there were always individuals in the other life stages.
The relationships between population parameters (CV, mean and variability) and environmental variability are linear in nature (Figs 3 and 4) when the environmental variability is described by the sampling interval (i.e. 7, 14 or 21 days). However, when the variability in the environment is described by the standard deviation of the food supply rather than the sampling period (Table 1), the relationships become increasing or decreasing curves. Small variations have small effects, but as variability increases the effects become ever larger. Food is given on average 2·5 days per week on 7-day variability, but only on 3·8 days per 21 days in 21-day variability. At high variabilities, periods of famine of many days are not uncommon. In good conditions, the mites’ life cycle is about 11 days (egg to egg) (Beckerman et al., unpublished manuscript). Hence, high-variability environments create famine periods which are of similar duration to the generation time: much longer periods (e.g. 4-week variability) would almost certainly result in frequent extinctions as famine periods may extend beyond the average lifespan. The issue of the scale of variability and the impact on organisms has been discussed by Kaitala et al. (1997) and Kaitala & Ranta (2001).
CORRELATION AND VARIANCE
If there is a relationship between forcing noise and population noise then populations undergoing similar sequences of environments should become synchronized: the Moran effect. Environmental synchrony has been discussed very frequently in recent years, partly because of its implications for conservation and management (Kendal et al. 2000). Discussions have focused on the relationship between the synchrony in environmental noise and synchrony in population noise (Grenfell et al. 1998, 2000; Blasius & Stone 2000; Benton et al. 2001; Greenman & Benton 2001) and the relative roles of dispersal and noise in causing environmental synchrony (Haydon & Steen 1997; Lande et al. 1999; Kendall et al. 2000; Ripa 2000). However, there has been little discussion of the role of environmental variance: for a given level of synchrony do environments that vary more cause greater population synchrony? If global warming causes greater variability in climate (at least in temperate zones) (Dai et al. 2001), or if habitat loss concentrates species on the more variable edges of their ranges, will synchrony tend to increase or decrease? Greenman & Benton (2001) suggest that increasing variance is likely to decrease synchrony. Ranta et al. (2000) do not discuss synchrony specifically, but show more variable environments cause greater population variability, so that there is more consistent tracking of environmental noise by population noise. In this situation, it might be reasonable to think increasing variability will cause greater synchrony. Our results (Fig. 4) indicate that there is no consistent relationship between noise variance and population correlation: populations receiving correlated noise maintain similar levels of synchrony, whatever the variance. As with Grenfell et al. (1998, 2000), Benton et al. (2001) and Greenman & Benton (2001), there is a significant decrease in synchrony between environmental noise and population noise, but it is not related to the variance.
Thanks to Alastair Grant, Jon Greenman, Esa Ranta and Ken Wilson for advice and discussion. Funding came from NERC.