Dependence and Interaction in Branching Processes
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 e
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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. Keywords Age-structure • General branching processes • Dependence • Carrying capacity
Mathematics Subject Classification (2010): Primary 60J80; Secondary 92D25
P. Jagers () Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden e-mail: [email protected] F.C. Klebaner School of Mathematical Sciences, Monash University, Clayton, 3058 VIC, Australia e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 19, © Springer-Verlag Berlin Heidelberg 2013
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P. Jagers and F.C. Klebaner
1 Introduction A drastic pedagogical example, illustrating the role of independence in branching processes is the “follow-the-generation-leader process”. It also serves to demonstrate that non-linearity on the expectation level is not the same as dependence between individuals. It is defined like a Galton-Watson branching process, but without the requirement that individuals reproduce independently. Instead they all reproduce in the same manner in each generation, follow the leader of their generation as it were. The leaders of different generations have independent and identically distributed offspring numbers, P say k with probability pk ; 0 < p0 < 1. If the process is supercritical, m D kpk > 1, expectations, or for that sake the corresponding deterministically modelled population, will grow geometrically, like mn . The actual population will, however, die out at the first instant the generation leader chooses to have no children. To a probabilist such phenomena will come as no surprise – from the point of view of prevailing deterministic population dynamics, based on differential operators, they may be illuminating. If dependence, on the contrary, is local in the pedigree, so that e.g. only siblings may influence each other, the branching character remains. Indeed, as has been developed by Olofsson, [14] e.g., this situation can b
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