Abstract:
We present an explicit solution to a classic model of cell-population growth introduced by Luria and Delbrueck 70 years ago to study the emergence of mutations in bacterial populations. In this model a wild-type population is assumed to grow exponentially in a deterministic fashion. Proportional to the wild-type population size, mutants arrive randomly and initiate new sub-populations of mutants that grows stochastically according to a supercritical birth and death process. We give an exact expression for the generating function of the total number of mutants at a given wild type population size. We present a simple expression for the probability of finding no mutants, and a recursion formula for the probability of finding a given number of mutants. In the "large population-small mutation"-limit we recover recent results of Kessler and Levin for a fully stochastic version of the process.

Abstract:
A coarse grained description of a two-dimensional prey-predator system is given in terms of a 3-state lattice model containing two control parameters: the spreading rates of preys and predators. The properties of the model are investigated by dynamical mean-field approximations and extensive numerical simulations. It is shown that the stationary state phase diagram is divided into two phases: a pure prey phase and a coexistence phase of preys and predators in which temporal and spatial oscillations can be present. The different type of phase transitions occuring at the boundary of the prey absorbing phase, as well as the crossover phenomena occuring between the oscillatory and non-oscillatory domains of the coexistence phase are studied. The importance of finite size effects are discussed and scaling relations between different quantities are established. Finally, physical arguments, based on the spatial structure of the model, are given to explain the underlying mechanism leading to oscillations.

Abstract:
A stochastic evolutionary dynamics of two strategies given by 2 x 2 matrix games is studied in finite populations. We focus on stochastic properties of fixation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with arbitrary hopping rates. The time of fixation is found to be identical for both strategies in any particular game. The asymptotic behavior of the fixation time and fixation probabilities in the large population size limit is also discussed. We show that fixation is fast when there is at least one pure evolutionary stable strategy (ESS) in the infinite population size limit, while fixation is slow when the ESS is the coexistence of the two strategies.

Abstract:
Synthetic bio-molecular spiders with "legs" made of single-stranded segments of DNA can move on a surface which is also covered by single-stranded segments of DNA complementary to the leg DNA. In experimental realizations, when a leg detaches from a segment of the surface for the first time it alters that segment, and legs subsequently bound to these altered segments more weakly. Inspired by these experiments we investigate spiders moving along a one-dimensional substrate, whose legs leave newly visited sites at a slower rate than revisited sites. For a random walk (one-leg spider) the slowdown does not effect the long time behavior. For a bipedal spider, however, the slowdown generates an effective bias towards unvisited sites, and the spider behaves similarly to the excited walk. Surprisingly, the slowing down of the spider at new sites increases the diffusion coefficient and accelerates the growth of the number of visited sites.

Abstract:
We study a two-type branching process which provides excellent description of experimental data on cell dynamics in skin tissue (Clayton et al., 2007). The model involves only a single type of progenitor cell, and does not require support from a self-renewed population of stem cells. The progenitor cells divide and may differentiate into post-mitotic cells. We derive an exact solution of this model in terms of generating functions for the total number of cells, and for the number of cells of different types. We also deduce large time asymptotic behaviors drawing on our exact results, and on an independent diffusion approximation.

Abstract:
Multiple-type branching processes that model the spread of infectious diseases are investigated. In these stochastic processes, the disease goes through multiple stages before it eventually disappears. We mostly focus on the critical multistage Susceptible-Infected-Recovered (SIR) infection process. In the infinite population limit, we compute the outbreak size distributions and show that asymptotic results apply to more general multiple-type critical branching processes. Finally using heuristic arguments and simulations we establish scaling laws for a multistage SIR model in a finite population.

Abstract:
Synthetic bio-molecular spiders with "legs" made of single-stranded segments of DNA can move on a surface covered by single-stranded segments of DNA called substrates when the substrate DNA is complementary to the leg DNA. If the motion of a spider does not affect the substrates, the spider behaves asymptotically as a random walk. We study the diffusion coefficient and the number of visited sites for spiders moving on the square lattice with a substrate in each lattice site. The spider's legs hop to nearest-neighbor sites with the constraint that the distance between any two legs cannot exceed a maximal span. We establish analytic results for bipedal spiders, and investigate multileg spiders numerically. In experimental realizations legs usually convert substrates into products (visited sites). The binding of legs to products is weaker, so the hopping rate from the substrates is smaller. This makes the problem non-Markovian and we investigate it numerically. We demonstrate the emergence of a counter-intuitive behavior - the more spiders are slowed down on unvisited sites, the more motile they become.

Abstract:
An explicit solution for a general two-type birth-death branching process with one way mutation is presented. This continuous time process mimics the evolution of resistance to treatment, or the onset of an extra driver mutation during tumor progression. We obtain the exact generating function of the process at arbitrary times, and derive various large time scaling limits. In the simultaneous small mutation rate and large time scaling limit, the distribution of the mutant cells develops some atypical properties, including a power law tail and diverging average.

Abstract:
The homogeneous ordered state transforms into a polydomain state via a nucleation mechanism in two-dimensional lattice gas if the particle jumps are biased by an external field $E$. A simple phenomenological model is used to describe the time evolution of a circular interface separating the ordered regions. It is shown that the area of a domain increases if its radius exceeds a critical value proportional to $1/E$ which agrees qualitatively with Monte Carlo simulations.

Abstract:
We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual "imports" its state from a randomly-chosen neighbor. Here the average time T_N to reach consensus for a network of N nodes with an uncorrelated degree distribution scales as N mu_1^2/mu_2, where mu_k is the kth moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is "exported" to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree k to overspread the population--the fixation probability--is proportional to k for the voter model and to 1/k for the invasion process.