## Introduction

Populations of herbivores are well known to fluctuate through time; in some years they flourish, whereas in other years they seem absent from the landscape. Parasitoids are clearly important sources of mortality for many insect herbivores (Southwood 1975; Cornell & Hawkins 1995), and the potentially tight coupling between herbivore hosts and parasitoid reproduction have led host–parasitoid systems to be a model for mathematical theory and the integration of theory and data in population dynamics (Murdoch 1994; Hassell 2000; Murdoch, Briggs & Nisbet 2003). However, despite an abundance of studies showing strong effects of parasitoids on insect hosts, we have little sense of the relative importance of parasitism compared with other ecological factors that also influence host populations. Factors may exert a strong influence on a population and yet may not be regulating if they fail to respond in a density-dependent manner (Murdoch 1994; Walker & Jones 2001). Long-term, field studies of both host and parasite abundances are required that estimate and test population models, such as those of several well-studied forest insects (Turchin *et al.* 2003; Kendall *et al.* 2005). In this study, we use a 21-year host–parasitoid time series to ask how strongly the host–parasitoid interaction, along with impacts of weather and intrinsic density dependence, appears to affect the dynamics of each. To accomplish this, we illustrate and give new tools for the state-space or hierarchical modelling framework for the analysis of population time series, which incorporates variability in both sampling and dynamics. These tools allow classical model selection and hypothesis testing rather than Bayesian results that are often given for hierarchical models. This study is organized so readers can focus on either the ecological or methodological aspect.

### Ecological introduction

Ecologists have long debated the prevalence of consumer-control vs. donor-control, one aspect of top-down vs. bottom-up effects on community organization. Some have argued that herbivore numbers are controlled by their predators and parasites (Hairston, Smith & Slobodkin 1960). Analyses of life tables have supported this view (Southwood 1975; Cornell & Hawkins 1995), but these are limited to causes of mortality such as predation, disease and starvation while deemphasizing factors that affect birth rates such as quality of food and oviposition sites (Price *et al.* 1990), and they have involved problematic statistical analyses (Royama 1996). Time-series analyses combined with other approaches strongly suggest cases of herbivore populations that are controlled by higher trophic levels (Berryman 2002). However, these models are correlational and have not adequately incorporated realistic measures of plant quality and defence (Haukioja 2005).

Other workers have been impressed that herbivores are controlled by the resources available to them and that predator and parasite numbers follow those of their prey (Elton 1927; Lindeman 1942). Hawkins (1992) gives several arguments that hosts may drive parasitoid dynamics (control from below) rather than the other way around. Different workers can reach different conclusions even when examining the best studied systems, such as lynx and hare cycles in boreal Canada, depending upon whether they choose to focus on food quality for the herbivores (Keith 1983) or on predation and food quantity (Boutin *et al.* 2002). Despite many years of work and hundreds of studies, we still lack a consensus about when predators primarily control their insect prey and when availability of herbivorous prey primarily controls populations of their predators and parasitoids. This becomes a difficult empirical problem because the importance of species interactions compared with other abiotic and biotic effects varies from year to year.

Many previous workers have considered the importance of single factors that affect insect populations [e.g. weather (Andrewartha & Birch 1954), interspecific competition (Lawton & Strong 1981)]. More recent studies have considered multiple factors and alternative hypotheses (e.g. see Reynolds *et al.* 2007 for a study that tests several different climatic variables on caterpillar numbers). Studies that have compared long time-series of univoltine host–parasitoid systems to population models representing hypotheses of several different effects, such as abiotic factors (weather), density dependence and species interactions, have yielded important insights (e.g. Turchin 2003; Turchin *et al.* 2003; Bonsall *et al.* 2004; Kendall *et al.* 2005; Munster-Swendsen & Berryman 2005), but there are relatively few such studies. Our study considers the relative importance of several factors in the long-term dynamics of a natural univoltine host–parasitoid interaction; it is the first to use a state-space framework to incorporate both measurement error and stochastic dynamics for univoltine host–parasitoid models (see Gross, Ives & Norpheim 2005, for a multivoltine host–parasitoid state-space model).

In our system, both herbivore density and parasitism levels fluctuate greatly (over three orders of magnitude for the herbivore, with parasitism from nearly 0% to 70%), giving an impression that their dynamics must be connected. Despite the appearance of coupling, other factors could drive dynamics of either or both species. Rather than supporting or rejecting the existence of a role for parasitism, we assess its relative importance. We ask how strong a role does the host–parasitoid interaction play in the dynamics of each species in comparison with other factors, known (weather and density) or unknown.

### Statistical introduction

Time-series analysis can evaluate the viability of hypotheses that predators drive prey populations and/or prey drive predator populations, but a major limitation of most time-series analyses has been the difficulty of incorporating multiple sources of variation. Estimates of predators (parasitoids) and prey (hosts) have uncertainty due to sampling variability, and estimates of population parameters must allow for variability in dynamics, such as from environmental stochasticity. Recently, state-space models have been developed in Bayesian and maximum likelihood frameworks to estimate models of noisy ecological situations (de Valpine & Hastings 2002; Calder *et al.* 2003; de Valpine 2003; Clark *et al.* 2005). The cited papers and a large statistical literature provide a strong case based on simulations and theory that the state-space approach is a solid foundation for incorporating multiple sources of variation, but the approach is still maturing and has not been applied to an univoltine host–parasitoid system.

Development of state-space models has moved contemporaneously with a broadening of views about statistical evidence in ecology. Earlier workers used inferential statistics to reject null hypotheses (Popper 1959; Platt 1964). More recently, ecologists have recognized that many hypotheses about populations and communities cannot be rejected in a meaningful way (Quinn & Dunham 1983; Gotelli & Ellison 2004). Regardless of whether we reject null hypotheses that weather, density dependence and parasitism do not affect herbivore populations, we all agree that all of these factors play some role, whether strong or weak. Ecologists recognize that many factors must be operating simultaneously to affect insect populations (Karban 1989; Hunter & Price 1992; Walker & Jones 2001), so estimates of the magnitude of each factor can be more informative than *P*-values. Only recently have methods become available to fit realistic models to data incorporating variation in both sampling and population dynamics.

To a large extent, such philosophical considerations are tied to practical issues: a Bayesian state-space model implementation yields lots of information but not maximum likelihoods or frequentist *P*-values, whereas the maximum likelihood estimation to approximate likelihood ratio *P*-values or AIC (Akaike Information Criteria) model comparisons requires different computational methods than a Bayesian posterior analysis. Although Bayesian and frequentist approaches have philosophical differences, they can nevertheless be interpreted together (Efron 2005; de Valpine 2009). We present frequentist results using maximum likelihood estimation, model selection using AIC, likelihood ratio hypothesis tests and likelihood profiles for confidence intervals. Maximum likelihood estimation of a hierarchical model can just as well be viewed as an ‘empirical Bayes’ approach. Accomplishing this for a nonlinear and/or non-Gaussian state-space model involves new and recently developed analysis steps for calculating likelihood values and using one-step-ahead predictions to give a simple summary of model fit (see below), which have often been omitted from state-space model results. Our analysis attempts to balance considerations of effect size, statistical significance, practicality and multiple testing (e.g. Ellison 2004; Stephens, Buskirk & del Rio 2007), neither throwing out nor overemphasizing one or another type of statement about models and data.