### Introduction

- Top of page
- summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References

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.

### Discussion

- Top of page
- summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References

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 (*contra*Block & 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 (*contra*Keeling 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 *R*^{2}s 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.