Abstract
Independence of reproducing individuals can be viewed as the very defining property of branching processes. It is crucial for the most famous results of the theory, the determination of the extinction probability and the dichotomy between extinction and exponential increase. In general processes, stabilisation of the age-distribution under growth follows, and indeed of the over-all population composition, and so do the many fine results of the area, like conditional stabilisation of the size of non-extinct subcritical processes. The last two decades have witnessed repeated attempts at treating branching processes with various kinds of dependence between individuals, ranging from local dependence between close relatives only to population size dependence. Of particular interest are very recent findings on processes that change from being supercritical to subcriticality at some threshold size, the carrying capacity of the habitat. We overview the development with an emphasis on these recent results.
Mathematics Subject Classification (2010): Primary 60J80; Secondary 92D25
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Acknowledgements
The first author spent spring 1966 studying at the Moscow State University, under the guidance of Yuri V. Prohorov and Boris A. Sevast’yanov, and wrote his first paper. He learnt very much from a thrilling and many-faceted environment.
PJ acknowledges the support of the Swedish Research Council and the Swedish Foundation for Strategic Research under their grants to the Gothenburg Stochastic and Mathematical Modelling Centres. FK was supported by Australian Research Council Grant DP0881011.
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Jagers, P., Klebaner, F.C. (2013). Dependence and Interaction in Branching Processes. In: Shiryaev, A., Varadhan, S., Presman, E. (eds) Prokhorov and Contemporary Probability Theory. Springer Proceedings in Mathematics & Statistics, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33549-5_19
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