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## Integral projection models for species with complex demography. (2006)

Venue: | American Naturalist |

Citations: | 42 - 6 self |

### BibTeX

@ARTICLE{Ellner06integralprojection,

author = {Stephen P Ellner and Mark Rees},

title = {Integral projection models for species with complex demography.},

journal = {American Naturalist},

year = {2006},

pages = {167--410}

}

### OpenURL

### Abstract

abstract: Matrix projection models occupy a central role in population and conservation biology. Matrix models divide a population into discrete classes, even if the structuring trait exhibits continuous variation (e.g., body size). The integral projection model (IPM) avoids discrete classes and potential artifacts from arbitrary class divisions, facilitates parsimonious modeling based on smooth relationships between individual state and demographic performance, and can be implemented with standard matrix software. Here, we extend the IPM to species with complex demographic attributes, including dormant and active life stages, cross-classification by several attributes (e.g., size, age, and condition), and changes between discrete and continuous structure over the life cycle. We present a general model encompassing these cases, numerical methods, and theoretical results, including stable population growth and sensitivity/ elasticity analysis for density-independent models, local stability analysis in density-dependent models, and optimal/evolutionarily stable strategy life-history analysis. Our presentation centers on an IPM for the thistle Onopordum illyricum based on a 6-year field study. Flowering and death probabilities are size and age dependent, and individuals also vary in a latent attribute affecting survival, but a predictively accurate IPM is completely parameterized by fitting a few regression equations. The online edition of the American Naturalist includes 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- Matrix projection models are probably the most commonly used approach for modeling structured biological populations 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 stages to minimize this problem leads to higher sampling error because fewer data are available on each stage. "Optimal" stage boundaries Integral Models for Complex Demography 411 and elasticities are very sensitive to stage duration ͵ L where is the range of possible states. This is the [L, U] continuous analogue of the matrix model n (t ϩ 1) p i , where a ij is the th entry in the projection a n (t) ( i, j) ij j j 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 Several empirical studies have illustrated how a kernel can be estimated from the same data as a matrix model The currently available general theory for IPMs When several variables are needed to predict demographic performance, estimating a matrix model becomes difficult because many between-class transitions have to be estimated 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, including complex life cycles and multiple attributes affecting individual performance. We begin by developing an IPM for the thistle O. illyricum, which has size-and age-dependent demography as well as substantial latent between-individual variation in survival unrelated to size or age. We then present a general IPM, explain how it can be implemented on a computer, and outline stable population theory for the density-independent model (l, sensitivity/elasticity analysis, etc.) under assumptions very similar to those for matrix models. For density-dependent models, we give a criterion for local stability of a steady state population. The general theory and numerical methods are illustrated using the Onopordum IPM. But for the most part, this article's "Results" are "Methods," with detailed applications appearing elsewhere (e.g., 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 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 Field Study Onopordum illyricum is a monocarpic perennial (reproduction is fatal) across its entire current range Data Analysis Results of the data analysis are summarized in table 1. Seedling size was well described by a normal distribution truncated at 0 ( ). Survival probability increases with P ! .02 plant size and decreases with age ( Integral Models for Complex Demography 413 regression because there was no evidence of betweenindividual variation ( ). Flowering probability in-P 1 .1 creased with plant size ( ) and age ( ), but P ! .0001 P ! .008 site effects were not significant ( ). Seed production P 1 .1 is strongly size dependent ( ) and highly variable P ! .0002 ( Because of the seed bank, the probability of seedling establishment cannot be estimated by the ratio between recruitment and seed production. We therefore set the probability of seedling establishment in the densityindependent IPM to match the observed rate of population increase ( ). We chose not to 1/4 l p (155/140) p 1.026 model the seed bank because estimates of seed germination and death are not available; however, if estimates were available, it would be straightforward to add an additional discrete-state variable representing the number of seeds in the seed bank. Intraspecific competition with neighbors had very little influence on growth and survival. In contrast, despite seed production being highly variable (0-2,750 per quadrat), the number of recruits was remarkably constant and independent of seed production the previous year ( ). P 1 .2 Our density-dependent model therefore assumes that population growth is limited by microsite availability. The American Naturalist Integral Model for Onopordum The fate of Onopordum plants is influenced by their size x, age a, and quality q (measured by survival intercept). Size is continuous whereas age and quality are discrete in our model. We treat quality as discrete for technical reasons (the conditional probability distribution for q(t ϩ given is singular with respect to Lebesgue measure) 1) q(t) that reflect our biological assumption that quality is constant. A dynamic quality variable is treated like size or any other dynamic continuous trait (for an example, see app. A). To deal with multiple traits affecting demography, we generalize the basic model (1) as follows. First, there is a set of functions that gives the distribution of size n (x, t) a, k for individuals of age a ( -7) and quality class k a p 0 ( -Q). Second, there is a set of survival-growth and k p 1 fecundity kernel components that specifies the fate and fecundity of individuals of each possible age # quality combination. Survival takes individuals from population component to . The survival-growth kernel for (a, k) (a ϩ 1, k) ( a, k) individuals is derived from the probability of survival and the size distribution of survivors. In the notation of table 1, That is, to reach size y from size x, the individual must survive, not flower, and make the size transition. x r y Flowering probability is a factor in equation (2) because flowering is fatal. Formulas for the probabilities of survival and flowering are given in table 1, and the growth kernel is given by the conditional size distribution from g (y, x) Births go from each component to all components (a, k) . Seedling size and quality are assigned independently (0, j) (and independent of parent age, size, and quality) according to the distributions given in table 1. The fecundity kernels are therefore , where a j is the fraction of seedlings in quality class j, J 0 is the probability density of seedling size, and is the S(x, a, k) number of seedlings in year per parent of age a, t ϩ 1 size x, and quality class k in year t. Our model breaks down seedling production into survival of the parent, flowering of the parent, seed production, and probability of establishment as a seedling. So, in terms of the demographic models in table 1, we have S (x, a, k) Having specified the survival-growth and fecundity kernels, the model is now complete. Note that the kernel of the integral model is implied directly by the statistical analysis of the data. The model expresses the population-level consequences of individual demography without any additional assumptions or approximations. Why Not Use a Matrix Model? For illustration, we have chosen a species with several features that the basic IPM cannot accommodate, but the model is still defined by a small number of conventional regression models with a total of 17 fitted parameters. In contrast, a conventional matrix model would have an enormous number of parameters (matrix entries) to estimate. At a bare minimum, we might use four size classes and a maximum age of 4. Then, with Q quality classes, there would be 4Q matrices of size for survival and 4 # 4 growth transitions and fecundity parameters for ages 2-4 (total fecundity and the offspring distrisize # quality bution). The total parameter count is . For ac-65Q ϩ 14 curate solution of the integral model, we have used 40 quality classes, which would entail more than 2,500 parameters for a matrix model, but even with three quality classes, the matrix model would have more than 200 parameters. Moreover, if individual quality is dynamic, the number of parameters in a matrix model would be vastly higher (proportional to Q 2 ), while an IPM would typically require only a few extra parameters to describe quality dynamics, as in the model described in apsize # 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., Integral Models for Complex Demography 415 General Integral Model We now describe a general integral model that can accommodate species such as Onopordum with complex demography, including species with complex life cycles and multiple traits (discrete or continuous) affecting demographic performance. We then describe the model's general properties and how it can be implemented on a computer, returning to Onopordum to illustrate practical application of the model. The space of individual states X can include a set of discrete points and a set of continuous Dϩ2 DϩC domain is either a closed interval or a closed rectangle in d-dimensional space. Each set in D or C will be called a component and denoted as Q j , , . j p 1, 2, … N p D ϩ C For example, Q 1 and Q 2 might represent different genders or a pair of discrete morphs in a species with phenotypic plasticity, within which individuals are classified by size or weight. The state of the population is described by a function that gives the distribution of individual n(x, t) ≥ 0 states x at time t; note that this function also consists of the following components: discrete values , n p n(x ) j j and continuous functions , To describe transitions within and among components, there is a set of kernel components , , whenever individuals in Q j contribute to next K ( 0 ij year's population in Q i . In terms of kernel components, the general deterministic model is There are four possible types of kernel components: (e.g., the number of two-leaf seedlings next year per oneleaf seedlings this year); discrete-to-continuous ( , (e.g., the size distribution of re- cruits produced by seeds that germinate); continuousto-discrete ( , ): , the state- ij ij dependent contribution to discrete component i (e.g., the number of resting eggs produced by Daphnia of age j and size x); and continuous-to-continuous ( ): i, j 1 D is a genuinely bivariate function giving the con-K (y, x) ij tribution of state x individuals in component j to state y individuals in component i. Kernel components can be separated into survivalgrowth and fecundity contributions, . We as- sume that all P ij and F ij are continuous functions so that the kernel is continuous. The model as just described is density independent; the density-dependent model is the same except that K can depend on the population state, , or on some measure N of total population In our Onopordum model, the domains are all intervals representing the range of possible sizes, one for each class combination. All kernel components age # quality are therefore of the continuous-to-continuous type. To incorporate the seed bank (as in M. Rees, D. Z. Childs, J. C. Metcalf, K. E. Rose, A. Sheppard, and P. J. Grubb, unpublished manuscript), we would add a discrete component for the number of seeds, a series of continuousto-discrete fecundity kernels for seed production by each combination, a series of discrete-to age # quality -continuous kernels for emergence of seeds into each quality category at age 0, and a discrete-to-discrete kernel (a number) for seeds remaining in dormancy and surviving. Our only restrictive assumption is that continuous domains are bounded. For simplicity, we have assumed rectangular domains, but the theory applies so long as each continuous domain is a bounded and closed subset of Euclidean space. Unbounded domains, however, lead to technical complications (see app. C) that we believe should be avoided. An integral model with bounded domains is a natural generalization of matrix models and shares many of their properties (the domain of a matrix model, a finite set of points, is always bounded in the sense used here, though it can sometimes represent an unbounded set of individual states, e.g., a single matrix stage class or IPM domain for individuals of age A or higher). An integral model with unbounded domains can behave very differently, for example, allowing a population to spread forever in "trait space" without ever reaching a stable distribution (see app. B). There are two ways to make a bounded integral model. First, the modeler can specify a finite range of possible values for all individual attributes. For example, each variable can be truncated at several standard deviations beyond the range of observed values, with the kernel set to 0 outside those limits. Suitable limits can also be set by expanding the range until further increases have no impact on model predictions. Individuals far beyond the range of the data are a fiction that results from using an unbounded statistical distribution to model a finite data set. This is harmless for short-term prediction but can lead to unrealistic long-term behavior. Second, unbounded attributes can be transformed onto a bounded domain using, say, the logistic transformation . Whether x r e /(1 ϩ e ) this produces a model satisfying our assumptions depends on how the kernel is defined outside the range of the data (see app. C). Either way, bounded components result when the model is not allowed to produce individuals very different from those actually observed. The first approach is much simpler and is therefore recommended. The American Naturalist Implementing a General Integral Model Evaluating Integrals Equation where This is a matrix multiplication where K is the matrix whose th entry is and is the vector whose ith entry is . The same idea n(t) n(x , t) i can be used to approximate higher-dimensional integrals for models with multiple state variables by defining mesh points for each variable. The result is again a matrix iteration for the population distribution at the mesh points. The accuracy of the midpoint rule depends on the number of mesh points m. Determining m is a trade-off between accuracy and computational cost, and in practice one should explore a range of mesh sizes to ensure that the population growth rate and other quantities of interest are calculated accurately. In appendix A, we suggest methods for implementing integral models when individuals are cross-classified by age and size or by size and quality. Computing l, w, and v The usual procedure for a matrix model is to compute the complete set of eigenvalues and eigenvectors and then find the dominant pair. For the large matrices representing a complex integral model, it is much more efficient to compute only the dominant pair by iterating the model. Let denote the population state in generation t-either n(t) one vector or the set of vectors for each component of X. Choose any nonzero initial distribution , and n(0) p n 0 let , where is the sum of all entries in u p n / kn k k xk 0 0 0 the vector. The iteration for the population growth rate l t and population structure is then u(t) p n(t)/ kn(t)k k k t where K is the matrix used to iterate the model numerically (e.g., eq. [6]). Iterating equation The dominant left eigenvector v, representing statedependent reproductive value, is the dominant right eigenvalue for the transpose kernel and T K (y, x) { K(x, y) can be obtained by iterating equation Stable Population Growth: Assumptions and Their Meaning General Theory Stable population growth refers to properties centered on the existence of a unique stable population distribution and asymptotic growth rate, to which a density-independent population converges from any initial composition: Here, l and w are the dominant eigenvalue and eigenvector for the kernel, respectively, and C is a constant depending on the initial population. Thus, l is the longterm population growth rate, and w is the stable state distribution, with and . In this section, we l 1 0 w(x) ≥ 0 describe two conditions that imply stable population growth for our general integral model, and in "Stable Population Growth: Results" we state the conclusions. Proofs and additional discussion of our assumptions are in appendixes B and C. For matrix models, equation for all in X, where and K (m) is the m-step- x, y K p K ahead projection kernel defined by the ChapmanKolmogorov formula: Integral Models for Complex Demography 417 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 Technically, suppose that for a parent with state x, the fecundity kernel satisfies , where and J 0 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 there J (y) 0 is mixing at birth and L is power positive, then some iterate of the kernel has a property called u-boundedness What about My Model? An IPM with stable population growth behaves very much like a power-positive matrix model. If it behaves differently, the cause is probably biological rather than mathematical, such as inclusion of postreproductive individuals or semelparous reproduction at a fixed age. The model's behavior will then be totally different from equation A p A # A A p A # A is implemented via separate matrices for each kernel component (as we suggest in app. A for populations with age structure), the same check can be done indirectly by computing the population state after many generations from many different initial conditions. Stable Population Growth: Results Under the assumptions described in "Stable Population Growth: Assumptions and Their Meaning," the long-term behavior of the integral model is identical to that of a power-positive matrix model. These results are derived in appendix C, which is based on work by with the value of G depending on the initial population distribution (see app. B). As in matrix models, R 0 and l are related: and have the same sign, and is also important for analysis of evolutionarily stable strategies (ESS) in density-dependent models. We use it below to identify ESS flowering strategies in the Onopordum model. Sensitivity analysis is also nearly identical to the matrix case, under the appropriate definitions. The question is, How much does l change when we perturb for K(y , x ) This satisfies the familiar sensitivity formula where is the inner product . Av, wS The elasticity function is ( Stability Analysis for Density-Dependent Models In this section, we state a criterion for local stability of equilibria in models with a density-dependent kernel, which is derived in appendix B. For simplicity (following Caswell 2001, chap. 16), we assume that the densitydependent kernel has the form , where N is a K(y, x, N) weighted total population size, N(t) p W(x)n(x, t)dx, ( Here is given by equation where J is analogous to the Jacobian matrix for densitydependent matrix models (Caswell 2001, sec. 16.4). So long as J is continuous, the conclusion from equation J(y, x, N) The stability criterion would generally be applied numerically, by computing J and its dominant eigenvalue. However, in appendix B, we prove that a class of models with density-dependent fecundity, which includes our density-dependent Onopordum model, has at most one positive equilibrium, which is locally stable whenever it exists. Onopordum Model Results We now apply the general theory to analyze the IPM for Onopordum. Some additional results on evolutionary analysis-characterization of optimal and ESS life historiesare also presented and applied. The model has a finite maximum age, and the distribution of offspring states is independent of parent state, so the density-independent model satisfies the mixing at birth assumption and therefore has a unique dominant eigenvalue and associated eigenvectors. The model's stable distribution provides an accurate description of the bimodal distribution of sizes observed in the population ( Integral Models for Complex Demography 419 Elasticity Analysis Using equation The elasticities of the survival-growth component of the kernel can be partitioned into contributions from plants of different ages, sizes, and survival intercepts ( For the reproduction component of the kernel, age 3 plants make the greatest contribution to l ( To model density-dependent recruitment, we assumed that in each year, the probability of establishment was equal to the observed average number of recruits (2.25 m Ϫ2 ) divided by total seed production (m Ϫ2 ). Iteration of the resulting model shows smooth convergence to a stable equilibrium density of 4.8 plants m Ϫ2 , in good agreement with the average density recorded in the field, 4.4 plants m Ϫ2 (Rees et al. 1999). The density-dependent kernel has the form studied analytically in appendix B (eq. [B12]), so our analytical results confirm the numerical observation of a unique stable equilibrium. The Onopordum model illustrates how latent variability between individuals can be parsimoniously modeled using an IPM. To explore this further, we looked at the effects of ignoring individual variability when fitting the survival model and of varying the distribution of survival intercepts while leaving the other parameters of the mixed model fixed. Varying j s , the standard deviation of survival intercepts, results in a rapid increase in l at low levels of variability, with the relationship reaching an asymptote at ( Integral Models for Complex Demography 421 Evolutionary Analysis The field data on Onopordum were originally used to construct an individual-based model in order to study the evolution of the flowering strategy as a function of age and size With an IPM, we can instead use familiar analytic and numerical methods for structured population models. In a density-independent model, the optimal life history maximizes l and so can be found by numerically optimizing l as a function of the model parameters. In our densitydependent model, density dependence acts only on seedling establishment. ESS life histories are therefore characterized by maximization of the net reproductive rate R 0 when offspring are counted before the impact of density dependence (app. B). Because of the mixing at birth in our model, R 0 equals the average per capita lifetime seed production of a cohort of newborns (app. B). We compute this by starting the population as a cohort of newborns and summing up their seed production until all in the founding cohort have died. ESS life histories in the Onopordum model can then be computed by numerical optimization of R 0 , which is far quicker than individual-based simulation of the evolutionary process. This is not just a convenience-it allows far more intensive study of the model. As an example, we revisit the comparison by The characterization of ESSs in terms of R 0 makes it straightforward to quantify the uncertainty in the ESS. We first bootstrapped the original data to compute 1,000 bootstrap estimates of all model parameters (for the survival model, we bootstrapped at the level of individual; for other model components, there is no evidence of heterogeneity between individuals, so we bootstrapped at the level of observations). Then, to find the corresponding ESSs, we computed R 0 and used the Nelder-Mead simplex algorithm to find the ESS flowering parameters for each of the bootstrap parameter sets. The results ( To test whether these differences are statistically signif- Integral Models for Complex Demography 423 icant, we used as test statistics the mean age at flowering and the difference between estimated and ESS flowering probability for an individual with the observed mean age and size at flowering 1.039 These results support the interpretation by Discussion In this article, we have shown how IPMs can be applied to species where individual demography is affected by multiple attributes that vary over the life cycle and can overcome some of the practical limitations of matrix models in such cases. Modern computing power means that the IPM is now practical for empirical applications. As recently as a decade ago, desktop computers would have been inadequate for our Onopordum model-even using just 50 mesh points for size and 20 for quality, there are 1 million fecundity elasticities for each age at which reproduction is possible. The existence of a unique stable distribution and asymptotic growth rate for the integral model rest on two kernel properties: either power positivity or u-boundedness. Power positivity is the analogue of assuming that a projection matrix is primitive and can be tested computationally. The u-boundedness condition is more abstract, but it will generally be satisfied by models with mixing at birth. The mixing at birth condition is likely to hold so long as the range of possible offspring states is the same for all parents, and it can usually be checked in specific models because it involves only properties of newborns. Our application to Onopordum illustrates 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 Elasticities are widely used in comparative studies to partition the contributions of different demographic processes to l at both the species and population levels The use of matrix models in applied situations is often hampered by limited data, and this is often cited as a rationale for constructing a low-dimensional matrix model. This is why Caswell (2001, sec. 3.3) and Morris and Doak (2002, p. 192) discuss ways of choosing among different state variables rather than incorporating multiple state variables and why most of the models reviewed by It should also be noted that if one fits an entire matrix model by regression procedures like those advocated by