## Introduction

Two widely observed empirical patterns in ecology are Taylor's law (TL; Taylor 1961; Eisler et al. 2008) and density-mass allometry (Blackburn and Gaston 1999; Jennings et al. 2007; Belgrano and Reiss 2011). TL asserts that, in an ensemble of populations, the variance of the population density is a power-law function of the mean density of those populations. Density-mass allometry asserts that population density is a power-law function of mean body mass, and can refer to single or mixed species as well as to individuals regardless of species. Composing TL with density-mass allometry predicts that the variance of population density should be a power-law function of mean body size and that the parameters of that power law should be predictable from the parameters of TL and of density-mass allometry (Marquet et al. 2005 and, independently, Cohen et al. 2012). Cohen et al. (2012) confirmed this relationship using detailed forestry data.

Taylor's law has been confirmed for hundreds of species or groups of related species in field observations and laboratory experiments (Reed and Hobbs 2004; Benton and Beckerman 2005; Marquet et al. 2005; Eisler et al. 2008; Ramsayer et al. 2011; Kaltz et al. 2012), and numerous models have been proposed to explain TL under various assumptions (e.g. Anderson et al. 1982; Ballantyne IV 2005; Engen et al. 2008). However, there is no consensus about why TL is so widely observed, how its estimated parameters should be interpreted in terms of underlying population dynamics, and when it might fail to be valid.

Density-mass allometry is seen in at least two different contexts. First, the allometry is widely observed across taxa, when each taxon is described by an average body size and a population density, although the exponent of the power law appears to differ for different groups of organisms (Damuth 1981, 1987; Lawton 1989; Marquet et al. 1990; Silva and Downing 1995; Dunham and Vinyard 1997; Enquist et al. 1998; Hendriks 1999; Schmid et al. 2000; Morand and Poulin 2002; Niklas et al. 2003; Makarieva et al. 2005; Reuman et al. 2008, 2009). This form of density-mass allometry is sometimes referred to as Damuth's law. Second, density-mass allometry is often observed using the densities of individuals grouped by body mass irrespective of taxon. In plant ecology, this form of density-mass allometry is called the self-thinning law (Adler 1996); in marine ecosystems, it is called a size spectrum (Sheldon and Parsons 1967), and this is the form of density-mass allometry used in this article.

Although the allometry of density with taxon average body mass (Damuth's law) has been controversial (Marquet et al. 1995, 2005), the allometry of density with individual body mass in marine ecosystems is clearly supported by empirical data showing that the total biomass in logarithmic bins of body mass is approximately the same across a wide range of body sizes (Sheldon et al. 1972, 1977; Boudreau and Dickie 1992; Kerr and Dickie 2001). Equivalently, the density of organisms of a given mass (per unit volume per unit body mass) is a power-law function of body mass with an exponent close to −2 (Sheldon and Parsons 1967; Platt and Denman 1978; San Martin et al. 2006).

Dynamic models of size spectra in marine ecosystems are based on a size-specific account of predation and growth as organisms eat one another, typically using the McKendrick–von Foerster partial differential equation (Benoît and Rochet 2004; Andersen and Beyer 2006; Blanchard et al. 2009; Law et al. 2009; Hartvig et al. 2011; Zhang et al. 2012). Such models are deterministic and can display at least two modes of behavior, depending on parameter values. In the stable mode, the biomass in any bin of body size converges to a fixed limit in time. In the oscillatory mode, the biomass in any bin of body size converges to a periodic cycle. Other dynamics, such as divergence or chaos, may be possible and have not yet been investigated in detail.

Because the oscillatory mode of dynamic size spectrum models predicts temporally varying population densities, it is natural to ask, for each body-size bin, (a) how the variance of population density is related to the mean of population density (does the model obey TL?), (b) how the mean population density is related to body size (does the model obey density-mass allometry?), and (c) how the variance of population density is related to body size (does the model obey variance-mass allometry?). Question (b) has been investigated in depth (Sheldon et al. 1972, 1977; Boudreau and Dickie 1992; Andersen and Beyer 2006), but neither question (a) nor question (c) has. Positive answers to these questions would provide novel interpretations of TL and variance-mass allometry in terms of the dynamic processes in size spectrum models. Negative answers would challenge the application of TL to dynamic size spectrum models.

The relationships among variability, density, and body size in marine ecosystems are important from a practical perspective because fish stocks appear to show increased variation under pressure of fishing (Hsieh et al. 2006; Andersen and Pedersen 2010; Rochet and Benoît 2012). However, the natural scaling of variability with density and body size in these ecosystems is largely unknown. Moreover, the traditional method of managing fisheries, protecting young, small fish and harvesting old, large ones is coming under increasing scrutiny. One suggestion is that moving away from highly selective fishing, and instead spreading the fishing effort widely over species and sizes, would ensure a sustainable fishery while reducing waste and conserving biodiversity. This approach to fishing is referred to as balanced harvesting (Zhou et al. 2010; Garcia et al. 2012). Theoretical results from a size spectrum model suggest that this approach has the potential to reduce the disruption to the natural size structure of the system (Law et al. 2012), but the effects of harvest patterns on the scaling of variability with body size and density remain unknown.

Here, we examined the extent to which TL and variance-mass allometry apply to size spectrum models. We did this first using a mathematical argument about dynamics close to the boundary between stability and instability. Second, we used numerical methods applied to a model given by Law et al. (2012) over wide ranges of assumptions about predator–prey mass ratios, life histories, and the forms and intensity of fishing. This model is based on an extension of the McKendrick–von Foerster equation to include a diffusion term (Datta et al. 2010) and a model for size-dependent reproduction (Hartvig et al. 2011). Our results showed that the model of Law et al. (2012) robustly predicted that size spectra should conform to TL, density-mass allometry, and variance-mass allometry. Moreover, simulation of a size-at-entry fishery, in which only fish above a minimum body mass can legally be caught, disrupted these relationships more than did balanced harvesting.