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A path integral approach to age dependent branching processes

Age dependent population dynamics are frequently modeled with generalizations of the classic McKendrick–von Foerster equation. These are deterministic systems, and a stochastic generalization was recently reported in Greenman and Chou (2016 Phys. Rev. E 93 012112, 2016 J. Stat. Phys. 16449). Here we develop a fully stochastic theory for age-structured populations via quantum field theoretical Doi–Peliti techniques. This results in a path integral formulation where birth and death events correspond to cubic and quadratic interaction terms. This formalism allows us to efficiently recapitulate the results in Greenman and Chou (2016 Phys. Rev. E 93 012112, 2016 J. Stat. Phys. 16449), exemplifying the utility of Doi–Peliti methods. Furthermore, we find that the path integral formulation for age-structured moments has an exact perturbative expansion that explicitly relates to the hereditary structure between correlated individuals. These methods are then generalized with a binary fission model of cell division.