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Abstract

In this thesis, I study the behaviour of the Stepping-Stone model: a stochastic model from theoretical population genetics. It was introduced by Kimura and Weiss [1][2] as a simple model to investigate the interplay of the evolutionary processes of random genetic drift, mutation, migration and selection. In particular, they were interested in the behaviour of spatially-structured populations when these processes were mediated by local interactions. From the point of view of statistical physics, the Stepping-Stone model can be viewed as a 2-species, non-equilibrium reaction-diffusion model with spatial degrees of freedom and unique fluctuations that arise from irreversible processes at the microscale. My thesis begins with a brief overview of the theory of stochastic processes that includes both the classical treatment using master equations, the Fokker-Planck equation and the Langevin equation, and modern formalisms that map these equations to operators and functional integrals. This is followed by a discussion of the steady-state and critical phenomena in the Wright-Fisher and Moran models, the non-spatial predecessors of the Stepping-Stone model. Critical phenomena in these models is shown to be associated with the breaking of a discrete symmetry that results in a discontinuous order parameter. Next, the Stepping-Stone model is introduced and reformulated using operators and pathintegrals. Kimura and Weiss were able to solve for the steady-state of the Stepping-Stone model, under certain conditions, but the dynamics of the model remained elusive. This model is related however, to the "Voter" model, which has been well-studied and is known to have a critical spatial dimension of 2 in that only for d ≤ 2 does the system asymptotically reach one of the 2 degenerate absorbing states. I extend the results of Kimura and Weiss and obtain exact results for the dynamics of the Stepping-Stone model. Two regimes of the model are analyzed: 1) the neutral regime, where selection has vanished and mutation is viewed as the control parameter. Here a steady state exists, characterized by a correlation length that diverges as the mutation rate, μ, becomes zero; and 2) the broken symmetry or "fixed" state, where selection is small but finite, and mutations so rare that the relevant description concerns the dynamics of "avalanches" or "cascades" of new alleles induced by the initial mutation and perturbing the system from the absorbing state. A unique kind of dynamical critical phenomenon occurs when selective advantage and mutation become negligible. Like the Voter model, it is qualitatively different both above and below a critical dimension d[sub c] = 2. For spatial dimension d ≤ 2 , the critical behaviour is associated with the breaking of a discrete symmetry and corresponds to fixation of one of the two alleles (genotypes). Symmetry breaking in the Stepping-Stone model is shown to be a consequence of the asymptotic return probability of a random walk—identically one for d ≤ 2 and strictly less than one for d > 2. In addition to the correlation length, the steady-state of the neutral regime is further characterized by a measure of variance defined as the amplitude of the two-point correlation function evaluated at vanishing separation. In genetics this measure of variance is known as F[sub ST], the fixation index. Exact results are derived for both the steady-state value of FST, and its asymptotic time-dependence at the critical point, which approaches 1 in both d = 1 and d = 2. At the critical point, 1 — F[sub ST]~ t[sub -1/2] for d = 1. For d = 2, a much slower decay is found, 1 — F[sub ST] ~ 1/ln(t). The d = 3 critical behaviour of F[sub ST] is that it approaches a constant C < 1 as (C — F[sub ST]) ~ t[sub -1/2]. The constant C is non-universal and related to the return probability of a random walk. It approaches 1 for very large values of the dimensionless constant K = [Λ/(2 π² Dn[sub Tg)] where Λ⁻¹ is the spatial scale of the interaction, rg is the generation time, D is the diffusion constant and n is the population density. A related infinite-alleles model was studied by both Sawyer [3] and Nagalaki[4]. These authors found similar asymptotics for the probability that two randomly choses individuals are genetically identical under certain assumptions. Sawyer[3] also proved that fixation in the infinite alleles model occurs iff the random walk followed by the individuals is recurrent. The broken symmetry or "fixed" regime of the Stepping-Stone model has been explored from the point of view of survival of rare mutant alleles, here parameterized by a coefficient of selection s. The exponent u, governing the divergence of the relaxation time as s —> 0 is calculated and found to be v = 2 for d — 1, while for d ≥ 2 it is given by the mean field value of v = 1. Two other exponents for critical spreading processes are determined and scaling arguments are presented and used to find the decay exponents that characterize the time-dependent survival probability and its asymptotic value for very long times. These results also establish the upper critical dimension of the model, d[sub c] = 2. Finally, these results are rederived and supplemented by a dynamical renormalization group (RG) analysis. The critical behaviour in d = 1 is found in both regimes to be controlled by non-trivial fixed points. The critical behaviour of the broken symmetry regime for d ≥ 2 and its RG fixed point is that of a critical branching process. For the neutral regime, however, the renormalization group flow is qualitatively different in rf = 2 and 3, reflecting the existence of broken symmetry for rf = 2. The RG flow for the rf = 3 neutral regime contains a line of fixed points with the effective description at large scales given by a Gaussian version of the time-dependent Landau-Ginsburg model. Finally, the renormalization results are found to be valid to all orders of perturbation theory.