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The long-term success of an introduced population depends on the ecological conditions in its new environment, but is also influenced by stochasticity. This is particularly clear in the first stage of an invasion when the population is still small and either goes extinct quickly or establishes a self-sustaining population. Once established, some populations grow and spread spatially, with potential impacts on native communities and ecosystems. The role of stochasticity during these later invasion stages remains unclear. Furthermore, little is known about the population genetic and evolutionary consequences of stochastic invasion trajectories. With this dissertation, I would like to contribute to a stochastic eco-genetic theory of the entire invasion process—from the first introduction up to potential impacts. The overarching questions in this dissertation are: a) How does a population’s movement through the invasion process depend on ecological factors influencing its average growth rate? b) How does it depend on factors influencing the stochastic variability in the population dynamics? c) How much genetic diversity do introduced populations harbor on average upon reaching a certain point in the invasion process? d) To what extent can the population-genetic consequences of invasion trajectories feed back onto the population dynamics? Together with my advisors and coauthors, I have conducted four studies, each addressing two or more of these questions for specific ecological scenarios. We employ several types of stochastic models: Markov chains, Markov processes, their diffusion approximations, and coalescent-like genealogy simulations. In Chapter 1 (Wittmann et al., 2013a, appeared in Theoretical Population Biology), we focus on a factor influencing the introduced population’s average growth rate: the intensity of competition with an ecologically similar native species. Our results indicate that the expected time until the introduced species drives the native competitor to extinction is smallest for intermediate competition intensity. This phenomenon results from the opposing effects of competition intensity at different points of the invasion process: On the one hand, intense competition renders the establishment of the introduced population more difficult; on the other hand, it facilitates the later exclusion of the native species. In Chapter 1, we also investigate to what extent the native species’ extinction is accelerated if a reduction in population size entails a reduction in genetic diversity and thus a reduced ability to adapt to a changing environment. We find this eco-genetic feedback to be particularly strong at small competition intensities. In Chapter 2 (Wittmann et al., 2013b, in press at Oikos), we compare introduction regimes with the same average number of individuals introduced per time unit, but with a different temporal distribution. Relative to regimes with many small introduction events, regimes with few large introduction events generate more variability in population-size trajectories. We show that this variability helps introduced populations to overcome difficult stages in the invasion process (those with a negative average growth rate), but is disadvantageous during easy stages (those with a positive average growth rate). In the light of our results, we can reinterpret three published data sets on invasion success under different introduction regimes. In Chapters 3 and 4 (Wittmann et al., 2013c,d), we examine levels of genetic diversity in populations that have successfully overcome a strong demographic Allee effect. In this ecological scenario, the average population growth rate is negative below a certain critical population size and positive above, such that the first stage in the invasion process is difficult and the second one easy. In Chapter 3, we assume Poisson-distributed offspring numbers. We show that compared to successful populations without an Allee effect, successful Allee-effect populations are expected to harbor either more or less genetic diversity, depending on the magnitude of typical founder population sizes relative to the critical population size. Part of the explanation is that, counter-intuitively, successful Allee-effect populations escape particularly fast from the range of small population sizes where genetic drift is strongest. In Chapter 3, we also identify conditions under which the critical population size can be estimated from genetic data. In Chapter 4, we consider a range of offspring-number models leading to either more or less variability in population dynamics than the Poisson model. For a fixed founder population size, we observe that the Allee effect has a negative influence on genetic diversity for small amounts of variability, but a positive influence for large amounts of variability. We show that the differences between our various offspring-number models are so substantial that they cannot be resolved by rescaling the parameters of the Poisson model. Taken together, these results offer some general conclusions with respect to the four main questions raised above. a) How fast an introduced population completes the invasion process is mainly determined by the presence and severity of difficult stages. Therefore, an ecological change promotes invasion success if it lessens such difficult stages. b) From the perspective of the introduced population, variability is advantageous during difficult but not during easy stages of the invasion process. c) Because the strength of genetic drift depends on population size, a key to understanding the population genetic consequences of invasion trajectories is to consider how much time the population of interest spends in different population-size ranges. d) Feedbacks between a reduction in population size and a loss of genetic diversity are strongest in ecological scenarios where the population of interest spends considerable time at small population sizes. Some of the most striking results in this dissertation cannot be understood from a deterministic point of view, but only when considering stochasticity. Thus, stochasticity does not just add “noise” to some average outcome, but can qualitatively change the behavior of biological systems.