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The aim of this paper is to tackle part of the program set by Diekmann et al.in their seminal paper Diekmann et al. (2001). We quote "It remains toinvestigate whether, and in what sense, the nonlinear determin-istic modelformulation is the limit of a stochastic model for initial population sizetending to infinity" We set a precise and general framework for a stochasticindividual based model : it is a piecewise deterministic Markov process definedon the set of finite measures. We then establish a law of large numbers underconditions easy to verify. Finally we show how this applies to old and newexamples.

Some deterministic structured population models which are limit of stochastic individual based models