The Statistical Physics of Fixation and Equilibration in Individual-Based Models
This thesis explores several interdisciplinary topics at the border of theoretical physics and biology, presenting results that demonstrate the power of methods from statistical physics when applied to neighbouring disciplines. From birth-death processes in switching environments to discussions on the meaning of quasi-potential landscapes in high-dimensional spaces, this thesis is a shining example of the efficacy of interdisciplinary research. The fields advanced in this work include game theory, the dynamics of cancer, and invasion of mutants in resident populations, as well as general contributions to the theory of stochastic processes. The background material provides an intuitive introduction to the theory and applications of stochastic population dynamics, and the use of techniques from statistical physics in their analysis. The thesis then builds on these foundations to address problems motivated by biological phenomena.
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1–2 boundary absorbing boundary analysis analytical approximation arrival time distribution basins of attraction behaviour Biol birth–death process boundary conditions cancer Cancer Research UK characterised coexistence game compute concentration simplex coordination game described in Sect deterministic dynamics eigenvalues environment evolution evolutionary dynamics evolutionary game evolutionary game theory exponential distribution extinction finite populations fixation and equilibration fixation probability fixation time distributions Fokker–Planck equation function Gillespie algorithm Hence initial condition integration M.A. Nowak master equation mathematical mean conditional fixation mean fixation metastable Michor most-likely path mutation rates payoff matrix Phys Pn(t population genetics Prisoner’s dilemma probability density quasi-potential quasi-stationary distribution reach fixation region region II selection shown in Fig simulation results small parameter solution stable fixed point stationary distribution stochastic process stochastic tunnelling switching theory thesis time-step timescale trajectory transition rates Traulsen type-2 cells variable wild-type WKB method