## Introduction

Since the pioneering work of Lotka and Volterra, predator–prey models have helped ecologists understand a great number of ecological phenomena. Predator–prey models show when trophic interactions can lead to population cycles (e.g. May 1972; Maynard Smith & Slatkin 1973; Tanner 1975; Andersson & Erlinge 1977; Hanski *et al*. 2001; King & Schaffer 2001; Turchin & Batzli 2001), and they are the basic building blocks of larger, food-web models that address the links between the stability and diversity of ecosystems (e.g. May 1973; Gross *et al*. 2009; McCann 2011). They have a long history of addressing how changes in primary productivity (enrichment) can affect ecosystems (Rosenzweig 1971; Oksanen *et al*. 1981; Arditi, Ginzberg & Akcakaya 1991; Lundberg & Fryxell 1995; Oksanen & Oksanen 2000; Gross, Ebenhoh & Feudel 2004; Fussmann & Blasius 2005; Murrell 2005), and are even used to study the evolution of behavioural traits, thanks to the development of adaptive dynamics theory (review in Abrams 2000). For all these reasons and more, predator–prey models play a central role in ecological theory, and now span the whole spectrum of ecological models, from theoretical to applied models (e.g. small mammal population dynamics, Hanski *et al*. 2001; King & Schaffer 2001; or marine food webs, Yodzis 1994,1998).

Predator–prey demographic models have developed largely around the concepts of functional and numerical responses of predators (see eqns 1-2 below). Classically, these models are written as systems of ordinary differential equations

The very simple Lotka-Volterra assumes *ϕ*(*V*)=*rV* , *g*(*V*,*P*)=*aV* and *f*(*V*,*P*)=*εg*(*V*,*P*) while the more popular Rosenzweig-MacArthur model assumes logistic growth, *ϕ*(*V*)=*rV*(1−*V*/*K*) where *K* is a carrying capacity, and a saturating functional response, where *h* is a handling time (as described by Holling 1959). Other choices are possible for the functional response (see Appendix S1), notably models that make the functional responses depend on the density of predators (Skalski & Gilliam 2001), the prey-to-predator ratio Abrams & Ginzburg 2000; Arditi & Ginzburg 2012) or densities of other prey species, which is important in food webs (McCann 2011). The numerical response is often used in a more general sense in empirical studies, including emigration/immigration processes of mobile predators (Andersson & Erlinge 1977). In Lotka-Volterra models and their derivatives, the numerical response *f*(*V*,*P*)=*εg*(*V*,*P*) is stunningly simple (linear) and obeys the ‘biomass conversion rule’ (Ginzburg 1998), as it assumes that offspring production is function of resource consumption. Not all models obey that rule (see Appendix S1). Note also that an alternative route to specify precise functional forms is to analyze the behaviour of these models for arbitrary functions (May 1972; Gross *et al*. 2009; Yeakel *et al*. 2011). Both these models and the associated concepts of functional and numerical responses have been quite useful to explain the effect of predators on their prey, for example why and when population oscillations can occur because of predator–prey interactions (e.g. in rodents, Andersson & Erlinge 1977). Saturating functional responses are thought very likely to generate limit cycles when predators possess a clear numerical response (Rosenzweig & MacArthur 1963), although the cycling might occur for other reasons as well, e.g. fixed maturation delays in the numerical response (Maynard Smith 1974; Wang *et al*. 2009).

There is, however, an often unquestioned assumption at the core of the classic predator–prey theory (e.g. Berryman 1992): that populations of prey and predators are well-mixed, i.e. spatial averages of densities are a sufficient description of the system. This owes much to the Lotka-Volterra tradition where the basic encounter rate is expressed in a mass-action fashion, as a product of prey and predator densities. This approach is often called ‘mean-field’ modelling, in connection to the physical origin of these models (Dieckmann, Law & Metz 2000). Many explicitly spatial models are also mean-field models in a sense, but at the subpopulation or patch level – a scale at which mass-action is indeed more likely to happen. More information on mass-action and its limits is given in Appendix S2.

Spatial and/or stochastic extensions to the basic models have fueled the growth of predator–prey theory (e.g. Gurney & Nisbet 1978; de Roos, McCauley & Wilson 1991, 1998; Wilson, McCauley & de Roos 1993; Keitt & Johnson 1995; McCauley, Wilson & de Roos 1996; Cuddington & Yodzis 2000). They show, among other things, why some predator–prey systems might persist at the landscape scale despite local extinction (Gurney & Nisbet 1978; Hassell 2000); produce travelling waves (see Sherratt & Smith 2008, for a review), or when spatial variation above the population level should stabilize populations (review in Briggs & Hoopes 2004). Many authors have used explicitly spatial individual-based models, with clear assumptions on local space-use and movements, to relax the mass-action assumption (e.g. de Roos, McCauley & Wilson 1991; Wilson, McCauley & de Roos 1993; Keitt & Johnson 1995; McCauley, Wilson & de Roos 1996; Cuddington & Yodzis 2000). Despite their important contributions to predator–prey theory, these individual-based models have an arguably limited scope, because they are specified mostly as computer simulations. Their algorithmic nature is appealing as they are relatively easy to implement, and they have few restrictions on the amount of biological detail that can be included; but they are hard to analyse fully and like all theory, care must be taken not to overparameterise the model.

Here we review the predator–prey modelling literature using mathematical tools called spatial moments. Spatial moments, such as spatial covariances, express the degree to which populations of predators and prey are spatially correlated (i.e. clustered/associated, randomly distributed, or segregated). These tools are of intermediate complexity between classical mean-field models and spatially explicit models, and it is often possible to calculate these correlations on real populations to correct predator–prey encounter rates. The most theoretical spatial moment models can be formally derived from individual-level models, and bridge the gap between simple, fully analytical mean-field models and complex individual-based simulations. We believe that starting from the individual and scaling up whilst maintaining a degree of mathematical tractability makes this approach most relevant to ecological theory.

The manuscript is organised around the central concept of *moment closure*. While the term may be new to some, most ecologists have already encountered the basic idea of moment closure, and much ecological data is summarised by some notion of central tendency (first moment) and variance (second moment) whilst ignoring higher order moments (i.e. skew, kurtosis, etc.). In dynamic spatial models, moment closure aims to capture the main features of the spatial structure using lower order spatial moments (the space-averaged, or mean densities, and their covariance across space). The moment closure arises because the dynamics of the mean density depends on the dynamics of spatial covariances, which in turn depends on the dynamics on higher-order moments, in an infinite regress. A mathematical illustration of this linkage is presented in Appendix S3 for interested readers, but the important point to remember is that we need to cut off this infinite chain at some point. Fortunately, it is often possible for the higher-order moments to be well approximated by functions of lower-order moments such as spatial means and spatial covariances, and methods of varying complexity can be used to do so, depending on the modeller's objectives.

In the following, we discuss models according to the closure method they use, starting from the simplest well-mixed models to models including fixed descriptors of spatial structure (first-order closure) to models including dynamic descriptors of spatial structure (second-order closure). First-order closure models, with a rather low degree of complexity, should appeal mostly to empiricists trying to integrate spatial structure into populations models, especially when it comes to the study of cyclic predator–prey systems. Second-order models are currently more theoretical tools, mostly linking deterministic equations to mathematically rigorous individual-based models (IBM). This combination of mathematically well-defined IBMs and new approximation techniques provide new insights into the role of spatial correlations in molding ecological and evolutionary dynamics. The components of some IBMs bear a promising resemblance to current space-use data (home range estimates, dispersal kernels), which suggest they could be parametrized, and potentially simplified thanks to second-order methods in the future.

In their overview of consumer-resource dynamics, Murdoch, Briggs & Nisbet (2003) concluded that these methods were ‘still in their infancy’. However, progress has subsequently been made for both first and second-order closures, and we hope reviewing this body of work will help researchers to get a grip of the new methods, and increase the frequency of their use within the ecological community. Similar endeavours to popularize spatial moment equations have been undertaken before, notably by Dieckmann, Law & Metz (2000), where the focus is mostly on plant population dynamics, or Lion & Baalen (2008), in the context of the evolution of altruism (see also Bolker 2004). Both Bolker (2004) and Lion & Baalen (2008) provide good categorisations of the various kinds of models, especially with respect to the treatment of space.

Whilst it is reasonable to assume in most cases that plant communities cannot obey mass-action mixing, this is less clear for animal communities, because animals are able to move widely. However, although most animals *can* move in ways that would render the community well-mixed, most of them rarely *do* so. Many consumers are anchored to some spatial location (e.g. a nest or refuge), and deplete their resources locally (as in seabirds, e.g. Elliott *et al*. 2009); other animals avoiding conspecifics can produce similar space-use patterns, especially in vertebrates (Brown & Orians 1970; Moorcroft & Lewis 2006). Following Murdoch, Briggs & Nisbet (2003), such social organisation leads to *within-population* spatial structure. A population is here understood to be a demographic unit within which most changes in numbers occur because of birth and death processes, rather than movements (Berryman 2002). *Within-population* spatial structure has been rather well investigated in insect parasitoids (Hassell 2000; Murdoch, Briggs & Nisbet 2003), and we know for instance that host-independent parasitoid aggregation is stabilizing while host-dependent aggregation is often destabilizing (Murdoch, Briggs & Nisbet 2003). We know comparatively less in vertebrate predator–prey systems, on which most of the biological examples in this review will focus. We provide an illustration of how within-population spatial structure can affect predator–prey encounter rates in Fig. 1.

Historically, however, *among-population* spatial structure (at the metapopulation, or regional level) has been investigated first in ecological theory, as unstable well-mixed models predicted the collapse of host–parasitoid systems that persisted in the wild. It has then been suggested that asynchronous fluctuations of locally unstable populations may produce regional persistence (Nicholson 1933), and this has strongly influenced both ecological theory and the modelling habits of theoreticians. In Appendix S4, we review shortly the literature on spatial predator–prey interactions to better connect spatial moment equations to earlier work, and why the neglected within-population spatial structure can matter.

Before heading to the next section and spatial moment methods, we should mention that in comparison to the great number of models parametrised (except in parasitoids and maybe other insects), there is little evidence for or against mass-action in natural systems. A recent exception is Mols *et al*. (2004), where the authors demonstrated that great tits encounter moth caterpillars at a rate less than proportional to prey density; this empirical study inspired several articles on the potential reasons for the absence of linear relationship between predator encounter rate and prey density (Ruxton 2005; Travis & Palmer 2005; Ioannou, Ruxton & Krause 2008). Clearly ecologists have an interest in these matters, but the mass-action assumption is rarely verified and we believe the lack of tests of the mass-action assumption is probably due to the success of the functional response as a concept. Functional responses computed at a large spatial scale (e.g. Redpath & Thirgood 1999) are large-scale phenomenological descriptors, and do not reflect directly individual diet choice and physiological constraints. The encounter rate in absence of handling/satiation can saturate as well in heterogeneous landscapes or with restricted movements (e.g. Keitt & Johnson 1995; Pascual, Roy & Laneri 2011), which means large-scale functional response *g*(*V*) encompasses both kinds of non-linearities, those due to spatial structure and those due to handling/satiation. To obtain a more mechanistic understanding of the functional response, one needs to derive encounter rates, which requires the separation of searching from handling time (as in Mols *et al*. 2004). This is not often done, though it is the only way to test the assumption of mass-action, and understand better the relationship between encounter rate and prey density (see Appendix S5 for more on estimating functional responses from data, and issues related to the spatial scale of measurement).