## Introduction

Determining the likely way that a population will respond to a change in one or more of its demographic parameters is a critical problem both in ‘pure’ ecology and for applied questions such as population management and prediction of the consequences of environmental change. Typically, demographic and or behavioural information is collected and used to parameterize a model. The parameters are then perturbed in turn, and their affect on the model's population growth rate or population size predicted.

Constructing density-independent matrix models is straightforward, as is their perturbation analysis, so they are widely used in conservation biology (for reviews see Benton & Grant 1999a; De Kroon, van Groenendael & Ehrlen 2000; Caswell 2001; Caswell 2000; Fieberg & Ellner 2001). However, there is an increasing recognition that population dynamics are a complex interplay between stage or age structure, environmental variability and density dependence (Benton & Beckerman, in press; Bjørnstad *et al*. 1999; Higgins *et al*. 1997; Leirs *et al*. 1997; Ellner *et al*. 1998; Grenfell *et al*. 1998; Grenfell *et al*. 2000; Bjørnstad & Grenfell 2001; Coulson *et al*. 2001; Fromentin *et al*. 2001; Clutton-Brock & Coulson 2002). The corollary of this is that populations are often unpredictable, and that ‘the devil is in the detail’ (Clutton-Brock & Coulson 2002) when it comes to understanding the way that they behave. It has been claimed that a density-independent analysis will often be a very close approximation to the results of a density-dependent and stochastic analysis (Caswell 2001), but this is certainly not invariably true (Grant & Benton 2003).

If the ‘devil is in the detail’, purely model-based approaches to predicting population dynamics in response to perturbations may leave out biologically important factors, and therefore not properly estimate effects. Conversely, it is often difficult to observe the direct effect of perturbations in the field, where replication and confounding factors are difficult to manage. Laboratory model organisms, on the other hand, may provide a suitable system with which to investigate the population response to perturbation. Our first aim was to perturb the vital rates of our study organism in a controlled and replicated way and observe the population response under both constant and stochastic environmental conditions. Our second aim was to attempt to reconstruct the biology of the model organism in a matrix model using time-series analysis of data on stage abundances in these same experiments. For field populations, we often only have a qualitative understanding of the biology of a species and time-series data of this type. From this start point, can we construct a model to capture the details of the population responses to perturbations?

We have used two sorts of models: simple matrix models based on mean values of the parameters and a simulation model, incorporating information on the response to food and density, parameterized from the time series. We conducted perturbation analyses differently on these models. First, the mean projection matrices, estimated from the time series and from the simulation model, were subjected to typical linear perturbation analysis, which estimates the sensitivity of population growth rate, λ, to independent changes in each of the vital rates. Such sensitivity (or proportional sensitivity = elasticity) analysis has repeatedly been used to estimate the vital rates that would benefit most from management, or the vital rates that are under the strongest selection pressure (Benton & Grant 1999a; De Kroon *et al*. 2000; Caswell 2001). This use is based on two assumptions: (i) that when the population is free to grow (e.g. at low density) the elasticity of λ predicts the change in population growth rate for a given manipulation, and (ii) when λ is constrained by density dependence that a change in λ predicts a long-term change in population size.

Our second approach was to simulate the time-series model and conduct two further types of elasticity analysis. Evolutionary ‘fitness’ is the invasibility of a new type into a resident population (Metz, Nisbet & Geritz 1992; Rand, Wilson & McGlade 1994), rather than the population growth rate *per se*. In a linear situation, strategies with the largest λ will invade residents with smaller λ. In a time-varying situation, this is no longer the case, and fitness is better estimated by the invasion exponent, ϕ, than λ (Benton & Grant 2000; Brommer 2000). Elasticities of λ are only an approximation of the elasticities of ϑ, although the approximation may be quite good (Grant & Benton 2000; Caswell 2001). In addition to the elasticities of ϑ, we used the model output to calculate the way the population size changed with changes in the vital rates. The elasticities of λ, ϑ and population size could then be compared with the elasticities estimated empirically from the population experiments.