## Introduction

Stochastic behaviour has become a focus of ecological investigation in recent times (Fox & Gurevitch 2000; Keeling 2000a). Random variations in organisms’ environments may affect their life-histories and therefore their population dynamics. Therefore, quantitatively accurate prediction of population behaviour requires consideration of environmental variation and its impact. This obviously applies to assessing population viability and time to extinction (Boyce 1992; Heino *et al*. 1997; Saether *et al*. 2000), population size (Leirs *et al*. 1997; Grenfell *et al*. 1998; Grant & Benton 2000) and predicting the time course of epidemics (Ellner *et al*. 1998; Keeling & Gilligan 2000) but it also applies to more theoretical studies, such as predicting the outcome of evolution (Benton & Grant 1999). A particular focus in recent times has been attempting to understand the role of environmental stochasticity in producing population synchrony (Haydon & Steen 1997; Grenfell *et al*. 1998, 2000; Lande, Engen & Saether 1999; Kendall *et al*. 2000; Ripa 2000; Greenman & Benton 2001) as synchrony itself affects metapopulation persistence, resilience and susceptibility to control (Heino *et al*. 1997; Myers & Rothman 1995; Bolker & Grenfell 1996).

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.