Elasticity analysis of density-dependent matrix population models: the invasion exponent and its substitutes

In density-independent models, the population growth rate λ measures population performance, and the perturbation analysis of λ (its sensitivity and elasticity) plays an important role in demography. In density-dependent models, the invasion exponent log λI replaces λ as a measure of population performance. The perturbation analysis of λI reveals the effects of environmental changes and management actions, gives the direction and intensity of density-dependent natural selection on life history traits, and permits calculation of the sampling variance of the invasion exponent. Because density-dependent models require more data than density-independent models, it is tempting to look for substitutes for the invasion exponent, the sensitivity and elasticity of which can be calculated from a density-independent model. Here we examine the accuracy of two such substitutes: the dominant eigenvalue of the projection matrix evaluated at equilibrium ( ) and the dominant eigenvalue of the matrix averaged over the attractor ( ). Using a two-stage model that represents a wide range of life history types, we find that the elasticities of or often agree to within less than 5% error with those of the invasion exponent, even when population dynamics are chaotic. The exceptions are for semelparous life histories, especially when density-dependence affects fertility. This suggests that the elasticity analysis of density-independent models near equilibrium, or averaged over the attractor, provides useful information about the elasticity of the invasion exponent in density-dependent models.