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Keywords:

  • autocorrelation;
  • fire;
  • Lomatium bradshawii;
  • LTRE analysis;
  • Markovian environments;
  • matrix population models;
  • sensitivity analysis;
  • stochastic growth rate;
  • stochastic models

Summary

1. Life table response experiment (LTRE) analyses decompose treatment effects on a dependent variable (usually, but not necessarily, population growth rate) into contributions from differences in the parameters that determine that variable. 2. Fixed, random and regression LTRE designs have been applied to plant populations in many contexts. These designs all make use of the derivative of the dependent variable with respect to the parameters, and describe differences as sums of linear approximations. 3. Here, I extend LTRE methods to analyse treatment effects on the stochastic growth rate  log λs. The problem is challenging because a stochastic model contains two layers of dynamics: the stochastic dynamics of the environment and the response of the vital rates to the state of the environment. I consider the widely used case where the environment is described by a Markov chain. 4. As the parameters describing the environmental Markov chain do not appear explicitly in the calculation of  log λs, derivatives cannot be calculated. The solution presented here combines derivatives for the vital rates with an alternative (and older) approach, due to Kitagawa and Keyfitz, that calculates contributions in a way analogous to the calculation of main effects in statistical models. 5. The resulting LTRE analysis decomposes  log λs into contributions from differences in: (i) the stationary distribution of environmental states, (ii) the autocorrelation pattern of the environment, and (iii) the stage-specific vital rate responses within each environmental state. 6. As an example, the methods are applied to a stage-classified model of the prairie plant Lomatium bradshawii in a stochastic fire environment. 7.Synthesis. The stochastic growth rate is an important parameter describing the effects of environmental fluctuations on population viability. Like any growth rate, it responds to differences in environmental factors. Without a decomposition analysis there is no way to attribute differences in the stochastic growth rate to particular parts of the life cycle or particular aspects of the stochastic environment. The methods presented here provide such an analysis, extending the LTRE analyses already available for deterministic environments.