Stochastic Population Dynamics with Delay Reactions

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Abstract

All real-world populations are composed of a finite number of individuals. Due to the intrinsically random nature of interactions between individuals, the dynamics of finite-sized populations are stochastic processes. Additionally, for many types of interaction not all effects occur instantaneously. Instead there are delays before effects are felt. The centrepiece of this thesis is a method of analytically studying stochastic population dynamics with delay reactions.Dynamics with delay reactions are non-Markovian, meaning many of the widely used techniques to study stochastic processes break down. It is not always possible to formulate the master equation, which is a common starting point for analysis of stochastic effects in population dynamics.We follow an alternative method, and derive an exact functional integral approach which is capable of capturing the effects of both stochasticity and delay in the same modelling framework. Our work builds on previous techniques developed in statistical physics, in particular the Martin-Siggia-Rose-Janssen-de Dominicis functional integral. The functional integral approach does not rely on an particular constraints on the population dynamics, for example the choice of delay distribution.Functional integrals can not in general be solved exactly. We show how the functional integral can be used to derive the deterministic, chemical Langevin, and linear-noise approximations for stochastic dynamics with delay.In the later chapters we extend Gillespie’s approximate method of studying stochastic dynamics with delay reactions, which can be used to derive the chemical Langevin equation, by-pass the functional integral. We also derive an extension to the functional integral approach so that it also covers systems with interruptible delay reactions.To demonstrate the applicability of our results we consider various models of population dynamics, arising from ecology, epidemiology, developmental biology, and chemistry. Our analytical calculations are found to provide excellent agreement with exact numerical simulations.

Keyword(s)

delay dynamics; non-Markovian dynamics; non-equilibrium statistical physics; population dynamics; stochastic processes

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