Closure schemes for joint density functions in diffusion-limited reactions

Genetic demixing and evolution in linear stepping stone models

Results for mutation, selection, genetic drift, and migration in a one-dimensional continuous population are reviewed and extended. The population is described by a continuous limit of the stepping stone model, which leads to the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation with additional terms describing mutations. Although the stepping stone model was first proposed for population genetics, it is closely related to "voter models" of interest in nonequilibrium statistical mechanics. The stepping stone model can also be regarded as an approximation to the dynamics of a thin layer of actively growing pioneers at the frontier of a colony of micro-organisms undergoing a range expansion on a Petri dish. The population tends to segregate into monoallelic domains. This segregation slows down genetic drift and selection because these two evolutionary forces can only act at the boundaries between the domains; the effects of mutation, however, are not significantly affected by the segregation. Although fixation in the neutral well-mixed (or "zero-dimensional") model occurs exponentially in time, it occurs only algebraically fast in the one-dimensional model. An unusual sublinear increase is also found in the variance of the spatially averaged allele frequency with time. If selection is weak, selective sweeps occur exponentially fast in both well-mixed and one-dimensional populations, but the time constants are different. The relatively unexplored problem of evolutionary dynamics at the edge of an expanding circular colony is studied as well. Also reviewed are how the observed patterns of genetic diversity can be used for statistical inference and the differences are highlighted between the well-mixed and one-dimensional models. Although the focus is on two alleles or variants, q-allele Potts-like models of gene segregation are considered as well. Most of the analytical results are checked with simulations and could be tested against recent spatial experiments on range expansions of inoculations of Escherichia coli and Saccharomyces cerevisiae.

Coagulation reactions in low dimensions: Revisiting subdiffusive A+A reactions in one dimension

We present a theory for the coagulation reaction A+A-->A for particles moving subdiffusively in one dimension. Our theory is tested against numerical simulations of the concentration of A particles as a function of time ("anomalous kinetics") and of the interparticle distribution function as a function of interparticle distance and time. We find that the theory captures the correct behavior asymptotically and also at early times, and that it does so whether the particles are nearly diffusive or very subdiffusive. We find that, as in the normal diffusion problem, an interparticle gap responsible for the anomalous kinetics develops and grows with time. This corrects an earlier claim to the contrary on our part.

Reaction front in an A+B-->C reaction-subdiffusion process

We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.

Majority versus minority dynamics: phase transition in an interacting two-state spin system

We introduce a simple model of opinion dynamics in which binary-state agents evolve due to the influence of agents in a local neighborhood. In a single update step, a fixed-size group is defined and all agents in the group adopt the state of the local majority with probability p or that of the local minority with probability 1-p. For group size G=3, there is a phase transition at p(c)=2/3 in all spatial dimensions. For p>p(c), the global majority quickly predominates, while for p<p(c), the system is driven to a mixed state in which the densities of agents in each state are equal. For p=p(c), the average magnetization (the difference in the density of agents in the two states) is conserved and the system obeys classical voter model dynamics. In one dimension and within a Kirkwood decoupling scheme, the final magnetization in a finite-length system has a nontrivial dependence on the initial magnetization for all p not equal p(c), in agreement with numerical results. At p(c), the exact two-spin correlation functions decay algebraically toward the value 1 and the system coarsens as in the classical voter model.

Finite-size effects of two-particle diffusion-limited reactions

We have studied the finite-size effects of two-particle diffusion-limited reaction A+B-->0, on a one-dimensional lattice using the Monte Carlo method. The density at a finite lattice follows a power law, C(t) approximately t(-x) below the crossover time, and shows an exponential decay above the crossover time. The crossover time depends on the lattice size and the bias field. We found second-order correction terms of the density decay as C(t) approximately t(-1/4)[1+O(t(-1/8))] for the isotropic diffusion of particles and C(t) approximately t(-1/3)[1+O(t(-1/24))] for the maximum drift. We proposed the scaling function of the density given as C(t) approximately L(-1/2)f(o)(t/L(2))+L(-3/4)f(1)(t/L(2)) for the isotropic diffusion and C(t) approximately L(-1/2)f(o)(t/L(3/2))+L(-9/16)f(1)(t/L(3/2)) for the maximum drift where f(o) and f(1) are scaling functions.