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... important when individuals exhibit substantial variability in multiple traits affecting vital rates, which is often the case. Considerable sophistication is now possible in individual-level demographic modeling because of developments during the past decade in statistical theory and software. For example, hierarchical or mixed models (Pinheiro and Bates 2000; Clark 2003) can be fitted where individual-specific parameters are drawn from a distribution. Arbitrary distributions and correlation structures for latent trait variability can be fitted via Markov chain Monte Carlo in either Bayesian (Gelman et al. 2004) or frequentist (de Valpine 2004) paradigms. Generalized least squares allow models to be fitted where the error structure is heteroscedastic or correlated, as in the growth model of Pfister and Stevens (2002, 2003). Our Onopordum data were fitted well by linear models, but nonlinear and nonparametric mixed models can be used when needed (e.g., Wood 2004, 2005). Model-averaging approaches can be represented in the IPM simply by weighted averaging of the kernels implied by each demographic model under consideration. Elasticities are widely used in comparative studies to partition the contributi...

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...riance j 2 of log size next year given current size, the survival probability s,y flowering probability pf, and fecundity fn. populations with continuous trait variation. Subsequent articles will consider stochastic integral models and databased models for species with complex life cycles. To make the article more accessible, most technical details are in the appendixes. Appendix A covers computational methods and should be read before building an IPM from your own data. The mathematical level is the same as the main text, roughly that of Caswell (2001). Script files in R (R Core Development Team 2005) for the methods described in appendix A and for our Onopordum model are provided as a zip archive in the online edition of the American Naturalist. Appendixes B and C in the online edition of the American Naturalist are mathematical derivations, written for theoreticians. Modeling Complex Demography in Onopordum illyricum To motivate the general framework, we develop in this section an IPM for the thistle O. illyricum derived from a 6-year field study. The field study and data analysis summarized below, including model selection, are described in detail by Rees et al. (1999). This example ill...

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...ludes a zip archive of R scripts illustrating our suggested methods. Keywords: structured populations, integral model, matrix model, sensitivity analysis, latent variability, thistle. * Corresponding author; e-mail: spe2@cornell.edu. † E-mail: m.rees@sheffield.ac.uk. Am. Nat. 2006. Vol. 167, pp. 410–428. 2006 by The University of Chicago. 0003-0147/2006/16703-40857$15.00. All rights reserved. Matrix projection models are probably the most commonly used approach for modeling structured biological populations (Caswell 2001) and play a central role in population and conservation biology (e.g., Morris and Doak 2002). The popularity of matrix models is easy to understand. They are conceptually the simplest way to represent population structure, can be parameterized directly from observational data on the fate and reproductive output of individuals, and yield a great deal of useful information. The dominant eigenvalue l of the projection matrix gives the population’s projected long-term growth rate; the dominant right and left eigenvectors are, respectively, the stable stage distribution w and relative reproductive value v ; and the eigenvectors determine the effect on l of changes in individual matrix ent...

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...m growth rate; the dominant right and left eigenvectors are, respectively, the stable stage distribution w and relative reproductive value v ; and the eigenvectors determine the effect on l of changes in individual matrix entries, which are often the key quantities for management applications. These and other metrics can be used as response variables to summarize population responses to changes in environmental conditions (Caswell 2001, chap. 10). Density dependence, stochasticity, and spatial structure can all be incorporated, and there is a growing body of theory for these situations (e.g., Tuljapurkar 1990; Cushing 1998; Caswell 2001; Tuljapurkar et al. 2003; Doak et al. 2005). A matrix model divides the population into a set of classes or “stages,” even when individuals are classified using a continuously varying trait such as body size. Indeed, the majority of empirical case studies reviewed by Caswell (2001) use size-based classifications rather than an actual discrete stage of the life cycle. In such cases, the definition of stages is to some degree arbitrary, giving rise to some potential problems, including the following: first, treating a range of heterogeneous individuals as a discrete ...

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...tly, the appropriate model complexity for the available data can be identified using wellestablished statistical criteria and software rather than the typically ad hoc process of choosing the number of size classes and their boundaries. The currently available general theory for IPMs (Easterling 1998; Easterling et al. 2000) applies only to models where individuals are characterized by one continuous quantity, but for many species, demographic rates are affected by multiple attributes. These might be observed attributes such as age and size (Rees et al. 1999; Rose et al. 2002) or age and sex (Coulson et al. 2001) or unobserved variables that reflect individual quality. For example, the probability of survival in the thistle Onopordum illyricum depends on size, age, and an unobserved measure of individual quality (Rees et al. 1999). In the kittiwake Rissa tridactyla, survival was age dependent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overestimation of population variability and extinction risk (Fo...

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...be fitted where the error structure is heteroscedastic or correlated, as in the growth model of Pfister and Stevens (2002, 2003). Our Onopordum data were fitted well by linear models, but nonlinear and nonparametric mixed models can be used when needed (e.g., Wood 2004, 2005). Model-averaging approaches can be represented in the IPM simply by weighted averaging of the kernels implied by each demographic model under consideration. Elasticities are widely used in comparative studies to partition the contributions of different demographic processes to l at both the species and population levels (Silvertown et al. 1993; Silvertown and Dodd 1996). Because Onopordum individuals are characterized by size, age, and survival intercept, we can partition elasticities according to each of these attributes and so obtain a very detailed understanding of the contributions to l from individuals in different states. These elasticities are in some ways easier to interpret than those of matrix models, whose values depend on the number of stages (Enright et al. 1995). For Onopordum, survival and growth contributed more to l than did reproduction (75% vs. 25% of the total elasticity, respectively), which is remarkably simil...

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...arbitrary, giving rise to some potential problems, including the following: first, treating a range of heterogeneous individuals as a discrete stage inevitably creates some degree of error. Increasing the number of stages to minimize this problem leads to higher sampling error because fewer data are available on each stage. “Optimal” stage boundaries (Vandermeer 1978; Moloney 1986) minimize but cannot eliminate the resulting errors in population projection and can be difficult to implement (Pfister and Stevens 2003). Optimal boundaries for population projection may be poor for other purposes (Easterling et al. 2000). Second, the sensitivities Integral Models for Complex Demography 411 and elasticities are very sensitive to stage duration (Enright et al. 1995), affecting comparisons both within and between species. Easterling et al. (2000) proposed that these issues could be avoided by using a continuous individuallevel state variable x. The population vector is replaced by a distribution function , where is the num-n(x, t) n(x, t)dx ber of individuals with their state variable in the range . The projection matrix A is replaced by a pro-[x, x dx] jection kernel , where P repre-K(y, x) p P(y, x) F(y, x...

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...ates are affected by multiple attributes. These might be observed attributes such as age and size (Rees et al. 1999; Rose et al. 2002) or age and sex (Coulson et al. 2001) or unobserved variables that reflect individual quality. For example, the probability of survival in the thistle Onopordum illyricum depends on size, age, and an unobserved measure of individual quality (Rees et al. 1999). In the kittiwake Rissa tridactyla, survival was age dependent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overestimation of population variability and extinction risk (Fox and Kendall 2002; Kendall and Fox 2002, 2003), underestimation of the uncertainty in population forecasts (Clark 2003), very large biases in estimates of demographic rates (Clark et al. 2003, 2004), and incorrect predictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class tr...

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...easure of individual quality (Rees et al. 1999). In the kittiwake Rissa tridactyla, survival was age dependent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overestimation of population variability and extinction risk (Fox and Kendall 2002; Kendall and Fox 2002, 2003), underestimation of the uncertainty in population forecasts (Clark 2003), very large biases in estimates of demographic rates (Clark et al. 2003, 2004), and incorrect predictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rarely beensize # age used, despite numerous studies documenting size- and ...

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...envectors are, respectively, the stable stage distribution w and relative reproductive value v ; and the eigenvectors determine the effect on l of changes in individual matrix entries, which are often the key quantities for management applications. These and other metrics can be used as response variables to summarize population responses to changes in environmental conditions (Caswell 2001, chap. 10). Density dependence, stochasticity, and spatial structure can all be incorporated, and there is a growing body of theory for these situations (e.g., Tuljapurkar 1990; Cushing 1998; Caswell 2001; Tuljapurkar et al. 2003; Doak et al. 2005). A matrix model divides the population into a set of classes or “stages,” even when individuals are classified using a continuously varying trait such as body size. Indeed, the majority of empirical case studies reviewed by Caswell (2001) use size-based classifications rather than an actual discrete stage of the life cycle. In such cases, the definition of stages is to some degree arbitrary, giving rise to some potential problems, including the following: first, treating a range of heterogeneous individuals as a discrete stage inevitably creates some degree of error. Increa...

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...g 1998). An IPM is implemented on the computer as a matrix iteration, but this is just a technique for computing integrals and not a discretization of the life cycle. Several empirical studies have illustrated how a kernel can be estimated from the same data as a matrix model (Easterling et al. 2000; Rees and Rose 2002; Childs et al. 2003, 2004; Rose et al. 2005). The functions making up the kernel can often be estimated by regression; for example, size-dependent survival and fecundity can be fitted by generalized linear or additive models and growth by parametric or nonparametric regression (Metcalf et al. 2003). Consequently, the appropriate model complexity for the available data can be identified using wellestablished statistical criteria and software rather than the typically ad hoc process of choosing the number of size classes and their boundaries. The currently available general theory for IPMs (Easterling 1998; Easterling et al. 2000) applies only to models where individuals are characterized by one continuous quantity, but for many species, demographic rates are affected by multiple attributes. These might be observed attributes such as age and size (Rees et al. 1999; Rose et al. 2002) or ag...

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...rojection a n (t) (i, j)ij jj matrix A. Under similar assumptions to matrix models, the integral projection model (IPM) predicts a population growth rate l with associated eigenvectors and statedependent sensitivity and elasticity functions (Easterling 1998). An IPM is implemented on the computer as a matrix iteration, but this is just a technique for computing integrals and not a discretization of the life cycle. Several empirical studies have illustrated how a kernel can be estimated from the same data as a matrix model (Easterling et al. 2000; Rees and Rose 2002; Childs et al. 2003, 2004; Rose et al. 2005). The functions making up the kernel can often be estimated by regression; for example, size-dependent survival and fecundity can be fitted by generalized linear or additive models and growth by parametric or nonparametric regression (Metcalf et al. 2003). Consequently, the appropriate model complexity for the available data can be identified using wellestablished statistical criteria and software rather than the typically ad hoc process of choosing the number of size classes and their boundaries. The currently available general theory for IPMs (Easterling 1998; Easterling et al. 2000) applies...

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...s, so property (9) is exactly equivalent to the projection matrix being power positive. A second condition that also guarantees stable population growth in our model is mixing at birth, meaning that the relative frequency of offspring states is similar for all parents. This condition is likely to hold in many species. For example, although many plant species have great plasticity in the number of seeds produced, there is much less plasticity in seed size or quality. In addition, maternal environment effects are often small compared with the effect of the environment in which a seedling grows (Weiner et al. 1997), which means that the distribution of offspring size will be similar for all parents, provided the population is censused sometime after recruitment. Technically, suppose that for a parent with state x, the fecundity kernel satisfies ,A(x)J (y) ≤ F(y, x) ≤ B(x)J (y)0 0 where and J0 is a probability distribution. Then,A, B 1 0 if there is a finite maximum age for reproduction, we can construct a Leslie matrix L from the mean age-specific survival and fecundity of a cohort of newborns with state distribution . In appendix B, we show that if thereJ (y)0 is mixing at birth and L is power positive...

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...y require only a few extra parameters to describe quality dynamics, as in the model described in ap-size # quality pendix A. This enormous difference in parameter count occurs whenever individual performance is affected by multiple state variables. Even with our large data set (1,402 observations on 1,144 individuals), we could not hope to estimate accurately all 192 survival entries in a projection matrix with three quality classes. For the integral model, we have a generalized linear mixed model with four fitted parameters and the option of fitting a nonlinear model if it were needed (e.g., Wood 2004, 2005). The relative ease and parsimony of model parameterization for IPMs is one of the main points of this empirical application. Continuous regression models can be used to parameterize matrix models (Morris and Doak 2002), followed by some way of averaging across a stage class, for example, to estimate the average fecundity of medium-size individuals. But if the actual response function is smooth, then division into stage classes necessarily distorts the functional relationship—indeed, any supposed average fecundity of mediumsize individuals is not well defined because the average depends...

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...inevitably creates some degree of error. Increasing the number of stages to minimize this problem leads to higher sampling error because fewer data are available on each stage. “Optimal” stage boundaries (Vandermeer 1978; Moloney 1986) minimize but cannot eliminate the resulting errors in population projection and can be difficult to implement (Pfister and Stevens 2003). Optimal boundaries for population projection may be poor for other purposes (Easterling et al. 2000). Second, the sensitivities Integral Models for Complex Demography 411 and elasticities are very sensitive to stage duration (Enright et al. 1995), affecting comparisons both within and between species. Easterling et al. (2000) proposed that these issues could be avoided by using a continuous individuallevel state variable x. The population vector is replaced by a distribution function , where is the num-n(x, t) n(x, t)dx ber of individuals with their state variable in the range . The projection matrix A is replaced by a pro-[x, x dx] jection kernel , where P repre-K(y, x) p P(y, x) F(y, x) sents survival and growth from state x to state y and F represents the production of state y offspring by state x parents. The population dynami...

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...ody size. Indeed, the majority of empirical case studies reviewed by Caswell (2001) use size-based classifications rather than an actual discrete stage of the life cycle. In such cases, the definition of stages is to some degree arbitrary, giving rise to some potential problems, including the following: first, treating a range of heterogeneous individuals as a discrete stage inevitably creates some degree of error. Increasing the number of stages to minimize this problem leads to higher sampling error because fewer data are available on each stage. “Optimal” stage boundaries (Vandermeer 1978; Moloney 1986) minimize but cannot eliminate the resulting errors in population projection and can be difficult to implement (Pfister and Stevens 2003). Optimal boundaries for population projection may be poor for other purposes (Easterling et al. 2000). Second, the sensitivities Integral Models for Complex Demography 411 and elasticities are very sensitive to stage duration (Enright et al. 1995), affecting comparisons both within and between species. Easterling et al. (2000) proposed that these issues could be avoided by using a continuous individuallevel state variable x. The population vector is replaced...

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...dictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rarely beensize # age used, despite numerous studies documenting size- and age-dependent demography (Werner 1975; Gross 1981; Klinkhamer et al. 1987; van Groenendael and Slim 1988; McGraw 1989; Lei 1999; Rees et al. 1999; Rose et al. 2002). In contrast, a size- and age-dependent IPM for Carlina vulgaris required only one extra parameter to describe the effect of age on flowering probability (Childs et al. 2003). In addition, many species have complex life cycles where individuals should be classified by different attributes at different points in the life cycle. For example, many plant populations have long-lived seed banks, so an additional discrete-state variable is required to keep track of seed numb...

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...ly, the stable stage distribution w and relative reproductive value v ; and the eigenvectors determine the effect on l of changes in individual matrix entries, which are often the key quantities for management applications. These and other metrics can be used as response variables to summarize population responses to changes in environmental conditions (Caswell 2001, chap. 10). Density dependence, stochasticity, and spatial structure can all be incorporated, and there is a growing body of theory for these situations (e.g., Tuljapurkar 1990; Cushing 1998; Caswell 2001; Tuljapurkar et al. 2003; Doak et al. 2005). A matrix model divides the population into a set of classes or “stages,” even when individuals are classified using a continuously varying trait such as body size. Indeed, the majority of empirical case studies reviewed by Caswell (2001) use size-based classifications rather than an actual discrete stage of the life cycle. In such cases, the definition of stages is to some degree arbitrary, giving rise to some potential problems, including the following: first, treating a range of heterogeneous individuals as a discrete stage inevitably creates some degree of error. Increasing the number of ...

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... (A4)0 a a ap0 where Fa is the matrix whose th entry is .(i, j) hf (x , x )a i j IPM with Size # Quality Classification Attributes representing individual quality will often change slowly over an individual’s lifetime. This leads to positive autocorrelation in growth or reproduction (Pfister and Stevens 2002, 2003): individuals with rapid growth or high fecundity now are likely to be high-quality individuals who will continue to perform well in the future. Negative autocorrelations can occur if current growth or reproduction depletes storage reserves and limits future growth or reproduction (Ehrlen 2000). Pfister and Stevens (2002) documented within-year positive correlation in growth that degraded the forecasting accuracy of a size-classified matrix model but could be accommodated in an individualbased simulation model (Pfister and Stevens 2003). An IPM with size and quality variables provides an alternative model for these situations, whose properties can be determined without recourse to individual-based simulations. Let x and q denote individual size and quality, respectively. These will typically be modeled as continuous variables jointly distributed on some rectangle Q p . An IPM in the...

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...bles that reflect individual quality. For example, the probability of survival in the thistle Onopordum illyricum depends on size, age, and an unobserved measure of individual quality (Rees et al. 1999). In the kittiwake Rissa tridactyla, survival was age dependent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overestimation of population variability and extinction risk (Fox and Kendall 2002; Kendall and Fox 2002, 2003), underestimation of the uncertainty in population forecasts (Clark 2003), very large biases in estimates of demographic rates (Clark et al. 2003, 2004), and incorrect predictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required”...

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...llustrates that when expected growth, survival, and birth rates are a smooth function of continuously varying traits, an integral model is a direct translation of the statistical analysis of individuallevel demographic data. This tight connection is particularly important when individuals exhibit substantial variability in multiple traits affecting vital rates, which is often the case. Considerable sophistication is now possible in individual-level demographic modeling because of developments during the past decade in statistical theory and software. For example, hierarchical or mixed models (Pinheiro and Bates 2000; Clark 2003) can be fitted where individual-specific parameters are drawn from a distribution. Arbitrary distributions and correlation structures for latent trait variability can be fitted via Markov chain Monte Carlo in either Bayesian (Gelman et al. 2004) or frequentist (de Valpine 2004) paradigms. Generalized least squares allow models to be fitted where the error structure is heteroscedastic or correlated, as in the growth model of Pfister and Stevens (2002, 2003). Our Onopordum data were fitted well by linear models, but nonlinear and nonparametric mixed models can be used when needed (e...

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...t 1) pi , where aij is the th entry in the projection a n (t) (i, j)ij jj matrix A. Under similar assumptions to matrix models, the integral projection model (IPM) predicts a population growth rate l with associated eigenvectors and statedependent sensitivity and elasticity functions (Easterling 1998). An IPM is implemented on the computer as a matrix iteration, but this is just a technique for computing integrals and not a discretization of the life cycle. Several empirical studies have illustrated how a kernel can be estimated from the same data as a matrix model (Easterling et al. 2000; Rees and Rose 2002; Childs et al. 2003, 2004; Rose et al. 2005). The functions making up the kernel can often be estimated by regression; for example, size-dependent survival and fecundity can be fitted by generalized linear or additive models and growth by parametric or nonparametric regression (Metcalf et al. 2003). Consequently, the appropriate model complexity for the available data can be identified using wellestablished statistical criteria and software rather than the typically ad hoc process of choosing the number of size classes and their boundaries. The currently available general theory for IPMs (Eas...

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... x) F(y, x) sents survival and growth from state x to state y and F represents the production of state y offspring by state x parents. The population dynamics are then U n(y, t 1) p K(y, x)n(x, t)dx, (1) L where is the range of possible states. This is the[L, U] continuous analogue of the matrix model n (t 1) pi , where aij is the th entry in the projection a n (t) (i, j)ij jj matrix A. Under similar assumptions to matrix models, the integral projection model (IPM) predicts a population growth rate l with associated eigenvectors and statedependent sensitivity and elasticity functions (Easterling 1998). An IPM is implemented on the computer as a matrix iteration, but this is just a technique for computing integrals and not a discretization of the life cycle. Several empirical studies have illustrated how a kernel can be estimated from the same data as a matrix model (Easterling et al. 2000; Rees and Rose 2002; Childs et al. 2003, 2004; Rose et al. 2005). The functions making up the kernel can often be estimated by regression; for example, size-dependent survival and fecundity can be fitted by generalized linear or additive models and growth by parametric or nonparametric regression (Metcalf...

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...tent between-individual differences can lead to systematic overestimation of population variability and extinction risk (Fox and Kendall 2002; Kendall and Fox 2002, 2003), underestimation of the uncertainty in population forecasts (Clark 2003), very large biases in estimates of demographic rates (Clark et al. 2003, 2004), and incorrect predictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rarely beensize # age used, despite numerous studies documenting size- and age-dependent demography (Werner 1975; Gross 1981; Klinkhamer et al. 1987; van Groenendael and Slim 1988; McGraw 1989; Lei 1999; Rees et al. 1999; Rose et al. 2002). In contrast, a size- and age-dependent IPM for Carlina vulgaris required only one extra parameter to describe the effect of age...

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... modeling because of developments during the past decade in statistical theory and software. For example, hierarchical or mixed models (Pinheiro and Bates 2000; Clark 2003) can be fitted where individual-specific parameters are drawn from a distribution. Arbitrary distributions and correlation structures for latent trait variability can be fitted via Markov chain Monte Carlo in either Bayesian (Gelman et al. 2004) or frequentist (de Valpine 2004) paradigms. Generalized least squares allow models to be fitted where the error structure is heteroscedastic or correlated, as in the growth model of Pfister and Stevens (2002, 2003). Our Onopordum data were fitted well by linear models, but nonlinear and nonparametric mixed models can be used when needed (e.g., Wood 2004, 2005). Model-averaging approaches can be represented in the IPM simply by weighted averaging of the kernels implied by each demographic model under consideration. Elasticities are widely used in comparative studies to partition the contributions of different demographic processes to l at both the species and population levels (Silvertown et al. 1993; Silvertown and Dodd 1996). Because Onopordum individuals are characterized by size, age, and surv...

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...rametric regression (Metcalf et al. 2003). Consequently, the appropriate model complexity for the available data can be identified using wellestablished statistical criteria and software rather than the typically ad hoc process of choosing the number of size classes and their boundaries. The currently available general theory for IPMs (Easterling 1998; Easterling et al. 2000) applies only to models where individuals are characterized by one continuous quantity, but for many species, demographic rates are affected by multiple attributes. These might be observed attributes such as age and size (Rees et al. 1999; Rose et al. 2002) or age and sex (Coulson et al. 2001) or unobserved variables that reflect individual quality. For example, the probability of survival in the thistle Onopordum illyricum depends on size, age, and an unobserved measure of individual quality (Rees et al. 1999). In the kittiwake Rissa tridactyla, survival was age dependent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overesti...

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...r structure is heteroscedastic or correlated, as in the growth model of Pfister and Stevens (2002, 2003). Our Onopordum data were fitted well by linear models, but nonlinear and nonparametric mixed models can be used when needed (e.g., Wood 2004, 2005). Model-averaging approaches can be represented in the IPM simply by weighted averaging of the kernels implied by each demographic model under consideration. Elasticities are widely used in comparative studies to partition the contributions of different demographic processes to l at both the species and population levels (Silvertown et al. 1993; Silvertown and Dodd 1996). Because Onopordum individuals are characterized by size, age, and survival intercept, we can partition elasticities according to each of these attributes and so obtain a very detailed understanding of the contributions to l from individuals in different states. These elasticities are in some ways easier to interpret than those of matrix models, whose values depend on the number of stages (Enright et al. 1995). For Onopordum, survival and growth contributed more to l than did reproduction (75% vs. 25% of the total elasticity, respectively), which is remarkably similar to the results obtained ...

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...1) or unobserved variables that reflect individual quality. For example, the probability of survival in the thistle Onopordum illyricum depends on size, age, and an unobserved measure of individual quality (Rees et al. 1999). In the kittiwake Rissa tridactyla, survival was age dependent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overestimation of population variability and extinction risk (Fox and Kendall 2002; Kendall and Fox 2002, 2003), underestimation of the uncertainty in population forecasts (Clark 2003), very large biases in estimates of demographic rates (Clark et al. 2003, 2004), and incorrect predictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers o...

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...opulation responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rarely beensize # age used, despite numerous studies documenting size- and age-dependent demography (Werner 1975; Gross 1981; Klinkhamer et al. 1987; van Groenendael and Slim 1988; McGraw 1989; Lei 1999; Rees et al. 1999; Rose et al. 2002). In contrast, a size- and age-dependent IPM for Carlina vulgaris required only one extra parameter to describe the effect of age on flowering probability (Childs et al. 2003). In addition, many species have complex life cycles where individuals should be classified by different attributes at different points in the life cycle. For example, many plant populations have long-lived seed banks, so an additional discrete-state variable is required to keep track of seed numbers. In this...

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...n (Metcalf et al. 2003). Consequently, the appropriate model complexity for the available data can be identified using wellestablished statistical criteria and software rather than the typically ad hoc process of choosing the number of size classes and their boundaries. The currently available general theory for IPMs (Easterling 1998; Easterling et al. 2000) applies only to models where individuals are characterized by one continuous quantity, but for many species, demographic rates are affected by multiple attributes. These might be observed attributes such as age and size (Rees et al. 1999; Rose et al. 2002) or age and sex (Coulson et al. 2001) or unobserved variables that reflect individual quality. For example, the probability of survival in the thistle Onopordum illyricum depends on size, age, and an unobserved measure of individual quality (Rees et al. 1999). In the kittiwake Rissa tridactyla, survival was age dependent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overestimation of populatio...

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...stle Onopordum illyricum depends on size, age, and an unobserved measure of individual quality (Rees et al. 1999). In the kittiwake Rissa tridactyla, survival was age dependent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overestimation of population variability and extinction risk (Fox and Kendall 2002; Kendall and Fox 2002, 2003), underestimation of the uncertainty in population forecasts (Clark 2003), very large biases in estimates of demographic rates (Clark et al. 2003, 2004), and incorrect predictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rar...

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... and Rose 2002; Cirsium canescens in Rose et al. 2005; Carlina vulgaris in Childs et al. 2003, 2004; Aconitum noveboracense in Easterling et al. 2000) and to Soay sheep (T. Coulson, personal communication); for many other examples, see the article by Metcalf et al. (2003). The model (app. A), or similarsize # quality models classifying individuals by size and energy reserves or by aboveground and belowground sizes, illustrates how temporal correlations in growth, either positive or negative, can be modeled using IPMs. Recent work on model-based measures of individual fitness (Cam et al. 2002; Link et al. 2002) uses modeling approaches to get around the problem that observed values of realized individual fitness are based on samples of size 1 and can be seriously biased (Link et al. 2002). Instead, Link et al. (2002) propose estimating the distribution of individual fitness by combining age-structured projection matrices with mixed models for individual quality variation. The same approach can be used in integral models, which greatly increases the range of life histories to which these approaches can be applied. More speculatively, by using breeding value for heritable traits as one of the state va...

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...2003), underestimation of the uncertainty in population forecasts (Clark 2003), very large biases in estimates of demographic rates (Clark et al. 2003, 2004), and incorrect predictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rarely beensize # age used, despite numerous studies documenting size- and age-dependent demography (Werner 1975; Gross 1981; Klinkhamer et al. 1987; van Groenendael and Slim 1988; McGraw 1989; Lei 1999; Rees et al. 1999; Rose et al. 2002). In contrast, a size- and age-dependent IPM for Carlina vulgaris required only one extra parameter to describe the effect of age on flowering probability (Childs et al. 2003). In addition, many species have complex life cycles where individuals should be classified by different attributes at diff...

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...Rees et al. (1999). This example illustrates how statistical analysis of demographic data translates directly into an integral model that involves far fewer fitted parameters than a conventional matrix model. In later sections, we use the model to demonstrate the mechanics of working with an integral model, with emphases on exploring the effects of latent heterogeneity in ways that would be difficult in a matrix model and on using the model for analyses of life-history evolution. Field Study Onopordum illyricum is a monocarpic perennial (reproduction is fatal) across its entire current range (Pettit et al. 1996). Reproduction occurs only by seed; these form a seed bank (up to 190 seeds m2), with a typical half-life of 2–3 years (Allan and Holst 1996). There were two study sites, but to simplify presentation, we focus on one, Plaine du Crau, an area of sheep-grazed semiarid steppe. Statistical analyses used data from both sites, with site effects fitted when significant. Sampling ran from August 1987 to August 1992, which included the complete lifetime of the 1987 seedling cohort. Each plant’s location and diameters (the longest and its perpendicular) were recorded in August, November, March, and May...

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...dent, and individuals with high survival probability were also more likely to breed, presumably because of between-individual variation in quality (Cam et al. 2002). Failure to account for such latent between-individual differences can lead to systematic overestimation of population variability and extinction risk (Fox and Kendall 2002; Kendall and Fox 2002, 2003), underestimation of the uncertainty in population forecasts (Clark 2003), very large biases in estimates of demographic rates (Clark et al. 2003, 2004), and incorrect predictions of population responses to demographic perturbations (Benton et al. 2004). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rarely beensize # age used, despite numerous studies documenting size- and age-dependent demography (Werner 1975; Gross 1981; Klinkhamer et al. 1987; van Groenendael and Slim 1988; Mc...

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...ral variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rarely beensize # age used, despite numerous studies documenting size- and age-dependent demography (Werner 1975; Gross 1981; Klinkhamer et al. 1987; van Groenendael and Slim 1988; McGraw 1989; Lei 1999; Rees et al. 1999; Rose et al. 2002). In contrast, a size- and age-dependent IPM for Carlina vulgaris required only one extra parameter to describe the effect of age on flowering probability (Childs et al. 2003). In addition, many species have complex life cycles where individuals should be classified by different attributes at different points in the life cycle. For example, many plant populations have long-lived seed banks, so an additional discrete-state variable is required to keep track of seed numbers. In this article, we generalize the IPM to accommodate species with complex demography...

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... dominant right and left eigenvectors are, respectively, the stable stage distribution w and relative reproductive value v ; and the eigenvectors determine the effect on l of changes in individual matrix entries, which are often the key quantities for management applications. These and other metrics can be used as response variables to summarize population responses to changes in environmental conditions (Caswell 2001, chap. 10). Density dependence, stochasticity, and spatial structure can all be incorporated, and there is a growing body of theory for these situations (e.g., Tuljapurkar 1990; Cushing 1998; Caswell 2001; Tuljapurkar et al. 2003; Doak et al. 2005). A matrix model divides the population into a set of classes or “stages,” even when individuals are classified using a continuously varying trait such as body size. Indeed, the majority of empirical case studies reviewed by Caswell (2001) use size-based classifications rather than an actual discrete stage of the life cycle. In such cases, the definition of stages is to some degree arbitrary, giving rise to some potential problems, including the following: first, treating a range of heterogeneous individuals as a discrete stage inevitab...

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...4). When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated (Law 1983; Caswell 2001). For example, “the construction of models using both size and age … may be impractical because of the large numbers of categories required” (Caswell 1988, p. 94). Consequently, matrix models with classification have rarely beensize # age used, despite numerous studies documenting size- and age-dependent demography (Werner 1975; Gross 1981; Klinkhamer et al. 1987; van Groenendael and Slim 1988; McGraw 1989; Lei 1999; Rees et al. 1999; Rose et al. 2002). In contrast, a size- and age-dependent IPM for Carlina vulgaris required only one extra parameter to describe the effect of age on flowering probability (Childs et al. 2003). In addition, many species have complex life cycles where individuals should be classified by different attributes at different points in the life cycle. For example, many plant populations have long-lived seed banks, so an additional discrete-state variable is required to keep track of seed numbers. In this article, we generalize the IPM to accommodate species with complex ...