## Introduction

Biological complexity is increasingly recognized as a critical factor underpinning the stability of ecological systems (e.g. Hilborn *et al*. 2003; Ives & Carpenter 2007; Schindler *et al*. 2010). While the diversity–stability relationship for ecosystem properties is generally held to be true, what is not known is the relative increase in benefit from each additional element of biodiversity for stability and persistence (Cardinale *et al*. 2012). For example, Schindler *et al*. (2010) found that sockeye salmon populations in Bristol Bay were twice as stable as a homogeneous population, and management should focus on retaining biological diversity to ensure a tenfold reduction in the frequency of fishery closures. The stabilizing benefit of such population diversity is clearly a critical and undervalued component of ecological systems for resource management to conserve, yet there are few ways to quantify its benefit.

The empirical portfolio effect (PE) is a rapidly popularized metric (e.g. Schindler *et al*. 2010; Carlson & Satterthwaite 2011; IMCC 2011) derived from theory introduced a decade earlier (Doak *et al*. 1998; Tilman, Lehman & Bristow 1998; Tilman 1999) that aims to measure the increase in stability due to subpopulation diversity within a metapopulation (or greater species diversity within a community). For example, we can think of salmon from individual streams as assets (subpopulations) within a portfolio (metapopulation) that comprises the watershed. If subpopulations react differently to environmental variability, then the metapopulation may experience a reduced risk of collapse or decline. Similarly, financial managers choose portfolios of diverse financial assets to reduce their risk of financial losses.

Financial managers estimate the benefit of diversifying a financial portfolio by comparing the variability in returns from investing in a single asset to the variability from investing in a diversified portfolio (Markowitz 1959). In ecology, the empirical PE has been calculated by comparing the temporal coefficient of variation (CV) of metapopulation abundance (the diversified portfolio; Fig. 1a) with the average-CV of subpopulation abundances (the single assets; Fig. 1b) (Secor, Kerr & Cadrin 2009; Schindler *et al*. 2010; Carlson & Satterthwaite 2011). We refer to this approach as the average-CV PE (Fig. 1c). But, ecological and financial systems differ; it is timely to consider whether we can apply the same approach to ecological systems.

One crucial difference between financial and ecological portfolios is how asset variability scales with investment. For a financial asset, the standard deviation of an investor's returns increases linearly with investment because investing in a financial stock does not meaningfully affect the stock's properties. Therefore, as mean financial investment increases, we expect the variance in returns to increase by a power of two. This is not true in ecological systems. As abundance of a subpopulation grows (i.e. as investment in the single asset grows), the standard deviation usually increases nonlinearly according to Taylor's power law: the slope (*z*) of a log–log plot of the variance and mean of subpopulation abundance is typically less than two (Taylor & Woiwod 1980, 1982). This means that larger populations may be less variable than expected if we applied the financial metaphor. The CV is not necessarily a size-independent metric of variability (McArdle, Gaston & Lawton 1990).

The theoretical work of Tilman, Lehman & Bristow (1998) implies an alternative way to measure the empirical PE that accounts for the mean–variance relationship. Rather than assuming we can represent the variability of the theoretical homogeneous metapopulation (the single asset) by the average subpopulation CV, we can estimate the variance of the homogeneous metapopulation by extrapolating the mean–variance relationship to the observed metapopulation size (Fig. 1d). We can then compare this expected homogeneous population variability with the observed metapopulation variability to get what we call the mean–variance PE. This mean–variance PE asks: If the mean–variance relationship continued to scale as we observed for larger and larger subpopulations, how much more variable would we expect the metapopulation to be if it was identically sized but acted with the same dynamics as any one subpopulation? Therefore, although both the mean–variance PE and the average-CV PE get at the benefit of splitting one large population into many subpopulations, only the mean–variance PE accounts for the observed mean–variance scaling relationship – the average-CV PE assumes that *z* = 2. Given this theoretical advantage of the mean–variance PE, what happens when we apply the average-CV PE to empirical data where *z* is typically less than two, as recent literature has done (Secor, Kerr & Cadrin 2009; Schindler *et al*. 2010; Carlson & Satterthwaite 2011)?

Here, we conducted the first large-scale cross-taxa evaluation of the average-CV PE compared with the mean–variance PE for metapopulations, specifically addressing three main questions: (i) How does the average-CV PE differ compared with the mean–variance PE when applied to theoretical systems with varying *z* values? (ii) How prevalent and strong is this difference across 51 metapopulations and 1070 subpopulations of salmon, moths and reef fishes? (iii) Despite its stronger theoretical foundations, is the mean–variance PE a reliable empirical metric of how subpopulation diversity benefits stability? We conclude with a guide to measuring ecological PEs based on question, study system and data type.