The Many Growth Rates and Elasticities of Populations in Random Environments

Despite considerable interest in the dynamics of populations subject to temporally varying environments, alternate population growth rates and their sensitivities remain incompletely understood. For a Markovian environment, we compare and contrast the meanings of the stochastic growth rate (λ ₅), the growth rate of average population (λ_M), the growth rate for average transition rates (λA), and the growth rate of an aggregate represented by a megamatrix (shown here to equal λ _M). We distinguish these growth rates by the averages that define them. We illustrate our results using data on an understory shrub in a hurricane-disturbed landscape, employing a range of hurricane frequencies. We demonstrate important differences among growth rates: λ_S < λ_M, but λ_A can be < or > λ _M. We show that stochastic elasticity, E_ij^S, and megamatrix elasticity, E_ij^M, describe a complex perturbation of both means and variances of rates by the same proportion. Megamatrix elasticities respond slightly and stochastic elasticities respond strongly to changing the frequency of disturbance in the habitat (in our example, the frequency of hurricanes). The elasticity E_ij ^A of λ_A does not predict changes in the other elasticities. Because E^S, although commonly utilized, is difficult to interpret, we introduce elasticities with a more direct interpretation: E^Sμ for perturbations of means and E^Sσ for variances. We argue that a fundamental tool for studying selection pressures in varying environments is the response of growth rate to vital rates in all habitat states.