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Elasticity Analysis in Population Biology: Methods and Applications [1].

Ecologists often need to make quantitative predictions about how populations will change, both to understand selective and demographic pressures and to make recommendations for species management. Through the use of matrix projection models and other similar analyses, ecologists are contributing to recommendations on the viability of a variety of species, as well as amassing a valuable database of the comparative demography of our planet's biodiversity. Recent advances in the analytical and simulation tools available to population ecologists have greatly enhanced our ability to compare population growth rates within and among species, even for populations with multiple life stages and complex life histories. Perturbation analysis of matrix models is an exciting new addition to the population ecologist's toolbox. Elasticity analysis in particular allows us to estimate and compare the effects of changes in survival, growth, or reproduction of particular life stages, as well as the proportional contribution of different aspects of the life cycle to population growth rate. Its applications are far-reaching, from comparative demography and life-history evolution to the conservation and management of threatened populations. This Special Feature highlights recent advances in elasticity analysis, focusing on areas where general patterns emerge and where cautionary applications are needed.

Elasticity analysis of matrix projection models examines the effects of proportional changes in demographic transitions (survival, growth, and reproduction parameters) on the asymptotic population growth rate, [lambda]. Because the elasticities of a projection matrix sum to unity, they can be interpreted as the relative contributions of the matrix transitions to [lambda]. The effect that each parameter, or suite of relevant parameters, has on population growth has been used as an index for evaluating the "importance" of certain life stages or demographic rates for management and research. In addition, [lambda] is a measure of fitness, and elasticities of vital rates quantify their contributions to fitness, revealing important links between life-history evolution and demography. Elasticity analysis has gained recent popularity as a population assessment tool for two primary reasons: ease of computation and clear connections with empirical data. Demographic data for many organisms can be translated into matrix form once the life stages or age classes are delineated, and projections allow the comparison of population growth rates and the importance of certain life-cycle parameters. Compared to many models used to determine population trends and viability, matrix models require less data and can be generalized for populations with a wide variety of life-history traits.

This Special Feature covers the methods, applications, and limitations of elasticity analysis, introducing new methodology and comparing the performance of more complex and biologically realistic models with deterministic mean matrix models. The introductory paper by de Kroon, van Groenendael, and Ehrlen reviews the basic mathematical properties of elasticities. Loop analysis is discussed as a logical extension of elasticity analysis that quantifies the relative contribution of alternative life-history pathways to population growth rate. The authors provide a wealth of discussion on the implications and limitations of elasticity analysis when applied to life-history studies and conservation biology, thereby setting the stage for the remaining papers.

Methods that examine the importance of variance and covariance of demographic rates are highlighted in Caswell's review of prospective and retrospective perturbation analyses. Prospective analyses, such as elasticity analysis, explore the functional dependence of population growth on the demographic parameters, while retrospective analyses decompose the variation in population growth caused by observed variations in the demographic transitions. The distinction of the two approaches is important for interpretation and application to conservation issues, and Caswell discusses where and when each analysis is most appropriate. In the following paper, Wisdom, Mills, and Doak present a new method that combines prospective and retrospective analyses in an attempt to account for proportional contributions to population growth as well as vital-rate variance and uncertainty in conservation applications.

Connecting population demography with life-history evolution and genetics is an important step in our efforts to find general rules of thumb for population management. The next two papers examine elasticity patterns from life-table information for a range of bird (Saether and Bakke) and mammal (Heppell, Caswell, and Crowder) populations. The relative values of fertility, juvenile survival, and adult survival elasticities fall predictably along a gradient from "fast" species with large clutches, early maturation, and short life spans to "slow" species with low annual reproduction, later maturation, and long life spans. For poorly known species, it may be possible to use these patterns to make general predictions (cautiously) about how perturbations will affect these populations. More importantly, classifying species according to their predicted elasticity patterns may be extremely useful for choosing "model" species to serve as parameter sources for more complex simulations.

The relationships among elasticities, sensitivities, fitness, and selection gradients are often confused but provide critical links between demographic and evolutionary dynamics. Van Tienderen provides a thorough review of the application of sensitivity and elasticity analyses to studies of fitness and population growth and offers a new integrated method for combining these approaches to assess both direct and indirect effects of changes in correlated vital rates. A reformulation of the concept of elasticity is required at equilibrium density or in cases of multiple stable states in which there are no net changes in population numbers. Grant and Benton present a method by which the elasticity of fitness can be calculated based on the susceptibility of a life-history strategy to invasion by alternative strategies.

Several papers in this feature present novel methods to compare the behavior of elasticities under biologically realistic conditions. When elasticities are compared for different models qualitatively, it appears that mean matrix elasticities are robust to certain covariances (Wisdom et al.), moderate levels of stochasticity, and equilibrium density dependence (Grant and Benton). In contrast, factors that can have a large effect on elasticity values include nonequilibrium density dependence (Grant and Benton), uncertainty in parameter estimates (Wisdom et al.), and stage length (Easterling, Ellner, and Dixon). Because stage length and the divisions used to define a matrix model can alter results, the integral model proposed in the final paper by Easterling et al. is an especially appealing alternative for species where size or stage divisions are problematic because vital rates are a continuous function of size.

All of the papers in this Special Feature make reference to the most common application of elasticity analysis: management and conservation of threatened species. There are two recurring needs in our efforts to conserve declining populations: simple quantitative tools that are general for a wide range of species (the majority of which have unknown demographic parameters and variances) and an understanding of when more complicated features, such as stochasticity and density dependence, need to be incorporated. One important message echoed by all of the papers in this feature is that uncritical interpretation of elasticities, particularly efforts to choose the "best" life stage for management efforts, can lead to poor or incorrect management prescriptions. In natural populations vital rates are rarely constant, and variability can exert strong selective pressures on the evolution of life histories and population growth. However, persistent patterns are found throughout this Special Feature and the published li terature. For example, elasticity patterns generated by similar vital rates may be equivalent among disparate taxa, and variability in vital rates appears to be negatively correlated with contribution to the population growth rate. These patterns suggest that general management recommendations can be made from knowledge of a species' life history. Where variability and density dependence are suspected to be important, the contributions in this feature detail methodologies for investigating their importance. Of course, it is an unrealistic goal to think that we could determine the spatial and temporal patterns of demographic variance and covariance and the strength and timing of density effects for the myriad species whose evolutionary history or ecological fate we are interested in. Rather, we must rely on the type of insight these papers afford: where we might expect general patterns and where the complexities of nature urge a deeper investigation.

(1.) Reprints of this 104-page Special Feature are available for $15.50 each. Prepayment is required. Order reprints from the Ecological Society of America, Attention: Reprint Department, 1707 H Street, N.W., Suite 400, Washington, DC 20006.
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Geographic Code:1USA
Date:Mar 1, 2000

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