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Probabilistic and piecewise deterministic models in biology. (English. French summary) Zbl 1383.92011

Summary: We present recent results on piecewise deterministic Markov processes (PDMPs), involved in biological modeling. PDMPs, first introduced in the probabilistic literature by M. H. A. Davis [“Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models”, J. R. Stat. Soc., Ser. B 46, No. 3, 353–388 (1984), http://www.jstor.org/stable/2345677], are a very general class of Markov processes and are being increasingly popular in biological applications. They also give new interesting challenges from the theoretical point of view. We give here different examples on the long time behavior of switching Markov models applied to population dynamics, on uniform sampling in general branching models applied to structured population dynamic, on time scale separation in integrate-and-fire models used in neuroscience, and, finally, on moment calculus in stochastic models of gene expression.

MSC:

92B05 General biology and biomathematics
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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