Running Head: GROWTH IN RANDOM ENVIRONMENTS GROWTH RATES AND ELASTICITIES 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 investigate the stochastic growth rate (λS), the growth rate of average population(λM), the growth rate for average transition rates (λA), and the growth rate of an aggregate represented by a megamatrix (shown here to equal λM). We distinguish these by the temporal or spatial 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 between growth rates: λS < λM but λA can be < or > than λM . We show that stochastic elasticity, E S ij, and megamatrix elasticity, E ij , describe a complex perturbation of both means and variances of rates by the same proportion. Megamatrix elasticities respond slightly, but stochastic elasticities strongly, to changing hurricane frequency. The elasticity E ij of λA does not predict changes in the other elasticities. Because E is difficult to interpret, we introduce elasticities with a more direct interpretation: E for perturbations of means and E 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.