## Introduction

Computation of population growth rate and its sensitivity to changes in vital rates (e.g. survival, growth, development and reproduction) is an important approach in conservation biology (e.g. Morris & Doak 2002), ecotoxicology (e.g. Kooijman & Metz 1984; Klok *et al.* 1997), pest management (e.g. Shea & Kelly 1998), as well as evolutionary ecology (e.g. van Tienderen 2000). These analyses most often exploit matrix models (Caswell 2001) to project long-term changes in population density under the assumption that environmental conditions and hence vital rates remain unchanged. Matrix models allow for an easy computation of the asymptotic population growth factor *λ* as the dominant eigenvalue of the projection matrix [here and below I will reserve the term *population growth rate* for the quantity *r* = log (*λ*), as technically *λ* itself is not a rate]. *λ* succinctly summarizes the influence of life-history components on population performance. The dominant right and left eigenvector of the matrix represent the stable distribution of the exponentially growing population and the reproductive value of individuals in different life-history stages, respectively (Caswell 2001). The eigenvectors can moreover be used to compute the sensitivity of *λ* to changes in vital rates (Caswell 1978). Expressions for these measures of population performance (growth factor *λ* and its sensitivity, stable distribution and reproductive value) were first derived for constant environments and later extended to environments in which vital rates vary periodically (Caswell & Trevisan 1994) or stochastically (Tuljapurkar 1990) or are influenced by population density (Takada & Nakajima 1992).

Matrix models are based on two types of discretization: (i) the population is subdivided into distinct classes or stages, most often age, size or stage of development and (ii) population dynamics are described on a discrete-time basis. Subdivision into population stages inevitably introduces discretization errors if individuals are in reality classified by a continuously varying trait such as body size. Methods have been developed to objectively select a stage classification that most faithfully represents such a continuous population distribution (Vandermeer 1978), but these approaches at best minimize the problems that these errors introduce into the population projection and do not eliminate them (Easterling *et al.* 2000; Pfister & Stevens 2003). Integral projection models (Easterling *et al.* 2000; Ellner & Rees 2006) are akin to matrix models, but represent the population by a continuous distribution and thus solve the problems associated with discrete stages. Ellner & Rees (2006) summarize basic theory and computational details for integral projection models, analogous to the theory for matrix population models (Caswell 2001), in which individuals are classified by an arbitrary number of variables.

Both matrix and integral projection models describe dynamics on a discrete-time basis, projecting the population state from time *t* to some future time *t* + 1. This time discretization has at least two advantages. Observational data on survival, reproductive output and stage transitions of individuals are usually collected on a discrete-time basis as well and can hence be used directly to parameterize the models. In addition, by parameterizing the population model on the basis of such observational data it also accounts phenomenologically for any unexplained and possibly stochastic variability in vital rates among individuals in the same stage. Matrix and integral projection models are therefore the ideal tool to extrapolate observations on vital rates to future population performance. However, demographic analyses may also start out by formulating a mathematical model of the individual life history, such as a dynamic energy budget (DEB) model (Kooijman & Metz 1984; Kooijman 2000), and address the question how particular assumptions about life history affect population performance (e.g. Kooijman & Metz 1984; Klok & De Roos 1996; Hulsmann *et al.* 2005; Rinke & Vijverberg 2005). Formulating a matrix population model may then be complicated, if the individual-level dynamics are described on a continuous-time basis. For example, Klanjscek *et al.* (2006) construct a projection matrix on the basis of a DEB model. To compute the matrix entries the stable age distribution within a particular life-history stage has to be evaluated, which itself depends on the population growth factor *λ* (see also Caswell 2001, p. 164). As a consequence, the matrix construction and computation of *λ* requires a rather complex, multistep iterative process (see also Klok & De Roos 1996; Klok *et al.* 1997, for an earlier example).

If the model of individual life history treats time as a continuous variable, physiologically structured population models (Metz & Diekmann 1986; De Roos 1997) are a more natural choice to describe demography. Physiologically structured population models describe the changes of a continuous population distribution on a continuous-time basis using either partial differential equations (Metz & Diekmann 1986; De Roos 1997) or integral equations (Diekmann *et al.* 1994). However, to compute the population growth rate for such a model requires solving an integral equation, which in its simplest form can be written as:

This is Lotka's integral equation for calculating the population growth rate *r* [= log (*λ*)], in which *b*(*a*) represents the individual fecundity at age *a* and *F*(*a*) the survival probability up to that age. Practical applications of this equation have commonly involved a discretization of the integral occurring in its right-hand side (e.g. Michod & Anderson 1980; Taylor & Carley 1988; Stearns 1992). Caswell (2001, p. 197) discourages this approach and argues that formulating a discrete-time, matrix model should be preferred over any attempt to write discrete versions of Lotka's equation. A condition similar to Lotka's integral equation determines the steady-state of a structured population model in case the population would not grow exponentially but would approach equilibrium as a result of some form of density dependence (see Metz & Diekmann 1986; De Roos *et al.* 1990; De Roos 1997, for examples). For such steady-state analysis of physiologically structured populations Diekmann *et al.* (2003, see also Kirkilionis *et al.* 2001) recently derived methods, in which the integral in the steady-state condition is computed without discretizing it (see Claessen & De Roos 2003 for a more intuitive explanation). In this paper, I exploit the similarity between the aforementioned steady-state condition and Lotka's integral equation to adapt these methods and present a simple computational approach to compute the population growth rate *r* and its associated statistics (sensitivities, stable distribution and reproductive value) using Lotka's integral eqn 1 or generalized versions of it. The idea behind the approach is to evaluate the integral in the right-hand side of the equation by means of numerical integration of an ordinary differential equation (ODE). The method is generally applicable to analyse population growth and performance for a wide range of individual life-history models, provided individual development throughout life is deterministic and the population is growing exponentially in a constant or periodically varying environment. I will illustrate the method using published life-history models and show how it can be applied to cases of varying complexity, including situations in which individuals vary in their trait values and situations in which the life history is characterized by discrete reproduction but development is continuous in time.