A Path Integral Approach to Age Dependent Branching Processes

TL;DR

This paper develops a stochastic, path integral framework for age-dependent population models using quantum field theory techniques, enabling efficient analysis of complex hereditary interactions and cell division processes.

Contribution

It introduces a novel path integral approach to stochastic age-structured populations, connecting quantum field methods with biological modeling.

Findings

Recapitulates previous stochastic results using Doi-Peliti formalism

Provides an exact perturbative expansion for age-structured moments

Generalizes the approach to binary fission cell division models

Abstract

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 [1,2]. 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 [1,2], 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.