Stochastic growth models which generalize Galton-Watson branching processes are discussed. The models have an interpretation in population dynamics and economics. The individuals or particles do interact, e.g. if the individuals represent members of a population, allowance is made for sexual reproduction, so that pairs of individuals are needed to produce offspring. The typical form of the results is that with probability one, either the population remains bounded in size, or it grows at an exponential rate and its composition converges to a fixed point of a suitable transformation.
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Harry Kesten (1971) studied this question.
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