Population dynamics in variable environments I. Long-run growth rates and extinction

A model of population growth is studied in which the Leslie matrix for each time interval is chosen according to a Markov process. It is shown analytically that the distribution of total population number is lognormal at long times. Measures of population growth are compared and it is shown that a mean logarithmic growth rate and a logarithmic variance effectively describe growth and extinction at long times. Numerical simulations are used to explore the convergence to lognormality and the effects of environmental variance and autocorrelation. The results given apply to other geometric growth models which involve nonnegative growth matrices.