theoretical population biology 50, 254280 (1996) Establishment Probability in Fluctuating Environments: A Branching Process Model Patsy Haccou Section Theoretical Biology, Institute of Evolutionary and Ecological Sciences, Leiden University, P.O. Box 9516, 2300 RA Leiden, The Netherlands and Yoh Iwasa Department of Biology, Faculty of Science, Kyushu University, Fukuoka, 812, Japan Received June 9, 1995 We study the establishment probability of invaders in stochastically fluctuating environments and the related issue of extinction probability of small populations in such environments, by means of an inhomogeneous branching process model. In the model it is assumed that individuals reproduce asexually during discrete reproduction periods. Within each period, individuals have (independent) Poisson distributed numbers of offspring. The expected numbers of offspring per individual are independently identically distributed over the periods. It is shown that the establishment probability of an invader varies over the reproduction periods accord- ing to a stable distribution. We give a method for simulating the establishment probabilities and approximations for the expected establishment probability. Furthermore, we show that, due to the stochasticity of the establishment success over different periods, the expected success of sequential invasions is larger then that of simultaneous invasions and we study the effects of environmental fluctua- tions on the extinction probability of small populations and metapopulations. The results can easily be generalized to other offspring distributions than the Poisson.  1996 Academic Press, Inc. INTRODUCTION Success of invasion to an empty habitat is an important life history characteristic of many organisms. So-called ``r-selected'' species are considered to have life history traits that lead to a higher success of invasion to a novel habitat (MacArthur and Levins, 1967; MacArthur, 1972). Populations of many species are maintained by recurrent events of extinction of local populations and subsequent invasion from other article no. 0031 254 0040-580996 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.