Elasticity analysis for density-dependent populations in stochastic environments

Elasticities of lambda indicate the influence of demographic parameters on both individual fitness and population growth for a population with density-independent growth. Here we discuss the extension of elasticity analysis to populations with density-dependent demographic parameters and examine the circumstances in which the standard density-independent analysis gives useful information fur populations that are, in reality, density dependent. In the presence of density dependence the fitness of one strategy depends upon what the rest of the population is doing. Elasticities of fitness must therefore be calculated using invasibility methods, and the resulting elasticities of the invasion exponent l are not necessarily identical with elasticities of mean or equilibrium population size. The former are of interest in evolutionary studies; the latter in population management. If a population has a stable equilibrium and density dependence is a function of the total number of individuals in the population, then elasticities of fitness are equal to elasticities of lambda, and elasticities of equilibrium population size are proportional to these. Even at equilibrium, elasticities of population size are not necessarily proportional to elasticities of lambda when density dependence is a function of the number of individuals in only part of the population, as in populations regulated by competition between juveniles. Elasticities of lambda are also a relatively poor guide to elasticities of population size when population numbers fluctuate. Thus, if density dependence operates at any stage in the life history, ignoring it can result in elasticity analyses that are a misleading guide fur population management. In the presence of stochastic environmental variation, elasticities of lambda calculated for the observed mean projection matrix are often a quite accurate approximation to elasticities of fitness, even when population fluctuations: are large. An elasticity analysis on an observed mean projection matrix will often give useful information regarding selection pressures on individual demographic parameters, although this approximation usually per forms poorly for populations with nonequilibrium dynamics. Whether one is interested in life history evolution or population management, the amount of work involved in carrying out a density-dependent elasticity analysis is small, relative to the amount of work involved in quantifying demographic parameters in the field populations. It will therefore usually be worth carrying out some nonlinear analyses using plausible patterns of density dependence and environmental stochasticity to ensure that an appropriate elasticity analysis is used.