Journal articleStochastic Processes and their Applications · March 1, 2023
In the system we study, 1’s and 0’s represent occupied and vacant sites in the contact process with births at rate λ and deaths at rate 1. −1’s are sterile individuals that do not reproduce but appear spontaneously on vacant sites at rate α and die at rate ...
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Journal articleElectronic Communications in Probability · January 1, 2023
In [1] we considered periodic trees in which the number of children in successive generations is (n, a1, …, ak ) with maxi ai ≤ Cn1−δ and (log ai)/ log n → bi as n → ∞. Our proof ...
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Journal articleTheoretical population biology · December 2022
Chan, Durrett, and Lanchier introduced a multitype contact process with temporal heterogeneity involving two species competing for space on the d-dimensional integer lattice. Time is divided into two seasons. They proved that there is an open set of the pa ...
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Journal articleProceedings of the National Academy of Sciences of the United States of America · November 2022
The main mathematical result in this paper is that change of variables in the ordinary differential equation (ODE) for the competition of two infections in a Susceptible-Infected-Removed (SIR) model shows that the fraction of cases due to the new variant s ...
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Journal articleElectronic Journal of Probability · January 1, 2022
The evoSIR model is a modification of the usual SIR process on a graph G in which S −I connections are broken at rate ρ and the S connects to a randomly chosen vertex. The evoSI model is the same as evoSIR but recovery is impossible. In [14] the critical v ...
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Journal articleProbability Theory and Related Fields · November 1, 2021
There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term, see e.g., Cox et al. (Astérisque 349:1–127, 2013), Durrett (Ann Appl Prob 19:4 ...
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Journal articlePLoS computational biology · February 2021
Recent work of Sottoriva, Graham, and collaborators have led to the controversial claim that exponentially growing tumors have a site frequency spectrum that follows the 1/f law consistent with neutral evolution. This conclusion has been criticized based o ...
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Journal articleElectronic Journal of Probability · January 1, 2021
In the q-voter model, the voter at x changes its opinion at rate fqx, where fx is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if q < 1 and clus ...
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Journal articleMathematical biosciences and engineering : MBE · December 2020
This research was done during the DOMath program at Duke University from May 18 to July 10, 2020. At the time, Duke and other universities across the country were wrestling with the question of how to safely welcome students back to campus in the Fall. Bec ...
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Journal articleStochastic Processes and their Applications · February 1, 2020
In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the component conta ...
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Journal articleAlea Rio De Janeiro · January 1, 2020
The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution ( ...
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Journal articleElectronic Journal of Probability · January 1, 2020
We consider a contact process on Zd with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species A and/or B. Multiple occupancy by the same species at a single site is prohibited. The ...
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Journal articleElectronic Communications in Probability · January 1, 2020
We study a competitive stochastic growth model called chase-escape in which red particles spread to adjacent uncolored sites and blue particles only to adjacent red sites. Red particles are killed when blue occupies the same site. If blue has rate-1 passag ...
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Journal articleElectronic Communications in Probability · January 1, 2020
A little over 25 years ago Pemantle [6] pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values ⅄1 and ⅄2 for global and local survival were different. He also considered trees wit ...
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Journal articleJournal of mathematical biology · January 2019
This work is inspired by a 2013 paper from Arne Traulsen's lab at the Max Plank Institute for Evolutionary Biology (Wu et al. in PLoS Comput Biol 9:e1003381, 2013). They studied evolutionary games when the mutation rate is so small that each mutation goes ...
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Journal articleAnnals of Applied Probability · January 1, 2019
Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0's are ordinary voters and 1's are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontr ...
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Journal articleAlea · January 1, 2019
Suppose that on the square lattice the edge with midpoint x becomes open at rate ∥x∥σ-1 . Let ρ(x, t) be the probability that the corresponding edge is open at time t and let n(p; t) be the distance at which edges are open with probab ...
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Journal articleAnnals of Applied Probability · October 1, 2018
We study an evolutionary game in which a producer at x gives birth at rate 1 to an offspring sent to a randomly chosen point in x + Nc, while a cheater at x gives birth at rate λ > 1 times the fraction of producers in x + Nd and sends ...
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Journal articleBulletin of mathematical biology · May 2018
Modified T cells that have been engineered to recognize the CD19 surface marker have recently been shown to be very successful at treating acute lymphocytic leukemias. Here, we explore four previous approaches that have used ordinary differential equations ...
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Journal articleStochastic Processes and their Applications · May 1, 2018
In the latent voter model, individuals who have just changed their choice have a latent period, which is exponential with rate λ, during which they will not change their opinion. We study this model on random graphs generated by a configuration model with ...
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Journal articleElectronic Journal of Probability · January 1, 2018
We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. Fix r>0 and place a particle at each point x of a unit intensity Poisson point process P⊆ℝd−B(0,r). Around each point in P, p ...
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Journal articleElectronic Journal of Probability · January 1, 2018
population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. Foxall, Edwards, and van den Driessche [7] found the critical value and studied the subcritical and supercritical regimes. Re ...
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Journal articleNature communications · November 2017
In the absence of antibiotic-mediated selection, sensitive bacteria are expected to displace their resistant counterparts if resistance genes are costly. However, many resistance genes persist for long periods in the absence of antibiotics. Horizontal gene ...
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Journal articleNature communications · October 2017
An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the ...
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Journal articleAdvances in Applied Mathematics · June 1, 2017
We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results that give bounds on the critical values λf for the process to survive from a finite set and λe for the existence of a nontrivial ...
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Journal articleProceedings of the National Academy of Sciences of the United States of America · June 2017
Recently, a rigorous mathematical theory has been developed for spatial games with weak selection, i.e., when the payoff differences between strategies are small. The key to the analysis is that when space and time are suitably rescaled, the spatial model ...
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Journal articleAnnals of Applied Probability · December 1, 2016
The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [Theor. Pop. Biol. 73 (2008) 158-170]. The first step, which is easily accomplished ...
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Journal articleJ Natl Cancer Inst · May 2016
BACKGROUND: Ductal carcinoma in situ (DCIS) is a noninvasive breast lesion with uncertain risk for invasive progression. Usual care (UC) for DCIS consists of treatment upon diagnosis, thus potentially overtreating patients with low propensity for progressi ...
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Journal articleJournal of mathematical biology · April 2016
We study the accumulation and spread of advantageous mutations in a spatial stochastic model of cancer initiation on a lattice. The parameters of this general model can be tuned to study a variety of cancer types and genetic progression pathways. This inve ...
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Journal articleStochastic Processes and their Applications · December 7, 2015
We study evolutionary games on the torus with N points in dimensions d≥3. The matrices have the form Ḡ=1+wG, where 1 is a matrix that consists of all 1’s, and w is small. As in Cox Durrett and Perkins (2011) we rescale time and space and take a limit as N→ ...
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Journal articleAnnals of Applied Probability · December 1, 2015
In this paper, we consider two attractive stochastic spatial models in which each site can be in state 0, 1 or 2: Krone's model in which 0 = vacant, 1 = juvenile and 2 = a mature individual capable of giving birth, and the Staver-Levin forest model in whic ...
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Journal articleBulletin of mathematical biology · October 2015
In this paper, we develop methods for inferring tumor growth rates from the observation of tumor volumes at two time points. We fit power law, exponential, Gompertz, and Spratt’s generalized logistic model to five data sets. Though the data sets are small ...
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Journal articleJournal of Complex Networks · September 1, 2015
Motivated by results of Henry et al. (2011, PNAS, 108, 8605-8610), we propose a general scheme for evolving spatial networks to reduce their total edge lengths. We study the properties of the equilibria of two networks from this class, which interpolate be ...
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Journal articlePLoS Comput Biol · March 2015
Clearance of anogenital and oropharyngeal HPV infections is attributed primarily to a successful adaptive immune response. To date, little attention has been paid to the potential role of stochastic cell dynamics in the time it takes to clear an HPV infect ...
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Journal articleAnnals of Applied Probability · February 1, 2015
We consider a multistage cancer model in which cells are arranged in a d-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of the first time when two neutral mutations have accumulated in some cell in d ...
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Journal articleAlea · January 1, 2015
In our first model, individuals have opinions in [0, 1]d. Connections are broken at rate proportional to their length ℓ, an end point is chosen at random, a new connection to a random individual is proposed. In version (i) the new edge is always ...
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Journal articleElectronic Journal of Probability · December 28, 2014
Here we will use results of Cox, Durrett, and Perkins [56] for voter model perturbations to study spatial evolutionary games on ℤd, d ≥ 3 when the interaction kernel is finite range, symmetric, and has covariance matrix σ2I. The games ...
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Journal articleExperimental Mathematics · October 2, 2014
Motivated by the widespread use of hybrid-discrete cellular automata in modeling cancer, we study two simple growth models on the two-dimensional lattice that incorporate a nutrient, assumed to be oxygen. In the first model, the oxygen concentration u(x, t ...
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Journal articleProceedings of the National Academy of Sciences of the United States of America · September 2014
In 1971, Schelling introduced a model in which families move if they have too many neighbors of the opposite type. In this paper, we will consider a metapopulation version of the model in which a city is divided into N neighborhoods, each of which has L ho ...
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Journal articleTheoretical population biology · May 2014
Inspired by the use of hybrid cellular automata in modeling cancer, we introduce a generalization of evolutionary games in which cells produce and absorb chemicals, and the chemical concentrations dictate the death rates of cells and their fitnesses. Our l ...
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Journal articleElectronic Journal of Probability · March 3, 2014
Consider a combination of the contact process and the voter model in which deaths occur at rate 1 per site, and across each edge between nearest neighbors births occur at rate λ and voting events occur at rate θ. We are interested in the asymptotics as θ→∞ ...
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Journal articlePhysical review. E, Statistical, nonlinear, and soft matter physics · December 2013
We consider an idealized model in which individuals' changing opinions and their social network coevolve, with disagreements between neighbors in the network resolved either through one imitating the opinion of the other or by reassignment of the discordan ...
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Journal articlePhysical Review E Statistical Nonlinear and Soft Matter Physics · June 27, 2013
The quadratic contact process (QCP) is a natural extension of the well-studied linear contact process where infected (1) individuals infect susceptible (0) neighbors at rate λ and infected individuals recover (1ï·0) at rate 1. In the QCP, a combination of ...
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Journal articleJ Theor Biol · December 7, 2012
Ovarian cancer is usually diagnosed at an advanced stage, rendering the possibility of cure unlikely. To date, no cost-effective screening test has proven effective for reducing mortality. To estimate the window of opportunity for ovarian cancer screening, ...
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Journal articleJournal of mathematical biology · March 2012
The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there i ...
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Journal articleProceedings of the National Academy of Sciences of the United States of America · March 2012
We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions ...
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Journal articleAnnals of Applied Probability · December 1, 2011
Aldous [(2007) Preprint] defined a gossip process in which space is a discrete N × N torus, and the state of the process at time t is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each ...
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Journal articleRandom Structures and Algorithms · September 1, 2011
We consider a model for gene regulatory networks that is a modification of Kauffmann's J Theor Biol 22 (1969), 437-467 random Boolean networks. There are three parameters: $n = {\rm the}$ number of nodes, $r = {\rm the}$ number of inputs to each node, and ...
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Journal articleGenetics · June 2011
With rare exceptions, human tumors arise from single cells that have accumulated the necessary number and types of heritable alterations. Each such cell leads to dysregulated growth and eventually the formation of a tumor. Despite their monoclonal origin, ...
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Journal articleAnnals of Applied Probability · April 1, 2011
The goal of cancer genome sequencing projects is to determine the genetic alterations that cause common cancers. Many malignancies arise during the clonal expansion of a benign tumor which motivates the study of recurrent selective sweeps in an exponential ...
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Journal articleAnnals of Probability · January 1, 2011
We consider a branching-selection system in ℝ with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N →∞, the empirical measure proce ...
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Journal articleTheoretical population biology · August 2010
Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis. Other mutations, ...
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Journal articleStochastic Processes and their Applications · July 1, 2010
We study the adaptive dynamics of predatorprey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends t ...
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Journal articleProceedings of the National Academy of Sciences of the United States of America · March 2010
Random graphs are useful models of social and technological networks. To date, most of the research in this area has concerned geometric properties of the graphs. Here we focus on processes taking place on the network. In particular we are interested in ho ...
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Journal articleTheoretical population biology · February 2010
Inspired by previous work of Iwasa et al. (2006) and Haeno et al. (2007), we consider an exponentially growing population of cancerous cells that will evolve resistance to treatment after one mutation or display a disease phenotype after two or more mutati ...
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Journal articleAnnals of Probability · November 1, 2009
If we consider the contact process with infection rate λ on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value λc of the infection rate is positive if the power α>3. Physicis ...
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Journal articleAnnals of Applied Probability · October 1, 2009
We introduce a multitype contact process with temporal heterogeneity involving two species competing for space on the d-dimensional integer lattice. Time is divided into seasons called alternately season 1 and season 2. We prove that there is an open set o ...
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Journal articleAnnals of Applied Probability · August 1, 2009
We investigate an interacting particle system inspired by the gypsy moth, whose populations grow until they become sufficiently dense so that an epidemic reduces them to a low level. We consider this process on a random 3-regular graph and on the d-dimensi ...
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Journal articleTAG. Theoretical and applied genetics. Theoretische und angewandte Genetik · May 2009
We report herein the development of a pepper genetic linkage map which comprises 299 orthologous markers between the pepper and tomato genomes (including 263 conserved ortholog set II or COSII markers). The expected position of additional 288 COSII markers ...
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Journal articleAnnals of Applied Probability · April 1, 2009
We consider the population genetics problem: how long does it take before some member of the population has m specified mutations? The case m = 2 is relevant to onset of cancer due to the inactivation of both copies of a tumor suppressor gene. Models for l ...
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Journal articleAnnals of Applied Probability · April 1, 2009
In this paper I will review twenty years of work on the question: When is there coexistence in stochastic spatial models? The answer, announced in Durrett and Levin [Theor. Pop. Biol. 46 (1994) 363-394], and that we explain in this paper is that this can b ...
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Journal articleAnnals of Applied Probability · February 1, 2009
We consider two processes that have been used to study gene duplication, Watterson's [Genetics 105 (1983) 745-766] double recessive null model and Lynch and Force's [Genetics 154 (2000) 459-473] subfunctionalization model. Though the state spaces of these ...
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Journal articleGenetics · November 2008
Results of Nowak and collaborators concerning the onset of cancer due to the inactivation of tumor suppressor genes give the distribution of the time until some individual in a population has experienced two prespecified mutations and the time until this m ...
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Journal articleStochastic Processes and their Applications · June 1, 2008
Lanchier and Neuhauser have initiated the study of host-symbiont systems but have concentrated on the case in which the birth rates for unassociated hosts are equal. Here we allow the birth rates to be different and identify cases in which a host with a sp ...
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Journal articleAnnals of Applied Probability · February 1, 2008
Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability v, or a voter model on ℤ in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that ...
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Journal articleElectronic Journal of Probability · January 1, 2008
In this paper we study some aspects of the behavior of random walks on large but finite graphs before they have reached their equilibrium distribution. This investigation is motivated by a result we proved recently for the random transposition random walk: ...
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Journal articleGenetics · January 2008
Current methods for detecting fluctuating selection require time series data on genotype frequencies. Here, we propose an alternative approach that makes use of DNA polymorphism data from a sample of individuals collected at a single point in time. Our met ...
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Journal articleStochastic Processes and their Applications · December 1, 2007
In our version of Watts and Strogatz's small world model, space is a d-dimensional torus in which each individual has in addition exactly one long-range neighbor chosen at random from the grid. This modification is natural if one thinks of a town where an ...
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Journal articleStochastic Processes and their Applications · December 1, 2007
Hybrid zones occur when two species are found in close proximity and interbreeding occurs, but the species' characteristics remain distinct. These systems have been treated in the biology literature using partial differential equations models. Here we inve ...
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Journal articleBMC genomics · June 2007
BackgroundDescriptive hierarchical Poisson models and population-genetic coalescent mixture models are used to describe the observed variation in single-nucleotide polymorphism (SNP) density from samples of size two across the human genome.Res ...
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Journal articleGenetics · June 2007
The symmetric island model with D demes and equal migration rates is often chosen for the investigation of the consequences of population subdivision. Here we show that a stepping-stone model has a more pronounced effect on the genealogy of a sample. For s ...
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Journal articleBMC bioinformatics · April 2007
BackgroundWe develop a Bayesian method based on MCMC for estimating the relative rates of pericentric and paracentric inversions from marker data from two species. The method also allows estimation of the distribution of inversion tract lengths.
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Journal articleTheoretical population biology · March 2007
Wagner (1996, Does evolutionary plasticity evolve? Evolution 50, 1008-1023.) and Siegal and Bergman, 2002 and Azevedo et al., 2006 have studied a simple model of the evolution of a network of N genes, in order to explain the observed phenomenon that system ...
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Journal articleAnnals of Applied Probability · February 1, 2007
One possible explanation for the substantial organismal differences between humans and chimpanzees is that there have been changes in gene regulation. Given what is known about transcription factor binding sites, this motivates the following probability qu ...
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Journal articleAnnals of Applied Probability · August 1, 2006
Neuhauser [Probab. Theory Related Fields 91 (1992) 467-506] considered the two-type contact process and showed that on ℤ 2 coexistence is not possible if the death rates are equal and the particles use the same dispersal neighborhood. Here, we s ...
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Journal articleRandom Graph Dynamics · January 1, 2006
The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know ...
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Journal articleProbability Theory and Related Fields · January 1, 2006
Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n ele ...
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Journal articleAnnals of Probability · November 1, 2005
Qian, Luscombe and Gerstein [J. Molecular Biol. 313 (2001) 673-681] introduced a model of the diversification of protein folds in a genome that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are n ...
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Journal articleStochastic Processes and their Applications · October 1, 2005
When an advantageous mutation occurs in a population, the favorable allele may spread to the entire population in a short time, an event known as a selective sweep. As a result, when we sample n individuals from a population and trace their ancestral lines ...
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Journal articleAnnals of Applied Probability · August 1, 2005
When a beneficial mutation occurs in a population, the new, favored allele may spread to the entire population. This process is known as a selective sweep. Suppose we sample n individuals at the end of a selective sweep. If we focus on a site on the chromo ...
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Journal articleStochastic Processes and their Applications · August 1, 2005
Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G1(m, n, t), the set of bipartite graphs with m left vertices, n right vertices, t edges, and each vertex of degree at least one. We give asym ...
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Journal articleJournal of Economic Behavior and Organization · July 1, 2005
A central problem in the biological and social sciences concerns the conditions required for emergence and maintenance of cooperation among unrelated individuals. Most models and experiments have been pursued in a game-theoretic context and involve reward ...
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Journal articleGenetical research · April 2005
There has recently been increased interest in the use of Markov Chain Monte Carlo (MCMC)-based Bayesian methods for estimating genetic maps. The advantage of these methods is that they can deal accurately with missing data and genotyping errors. Here we pr ...
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Journal articleAnnals of Applied Probability · February 1, 2005
This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the two-dimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman's coalesc ...
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Journal articleElectronic Journal of Probability · January 1, 2005
We study two-type branching random walks in which the birth or death rate of each type can depend on the number of neighbors of the opposite type. This competing species model contains variants of Durrett’s predator-prey model and Durrett and Levin’s colic ...
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Journal articleMolecular biology and evolution · December 2004
The human genome is estimated to contain 700 zinc-finger genes, which perform many key functions, including regulating transcription. The dramatic increase in the number of these genes as we move from yeast to C. elegans to Drosophila and to humans, as wel ...
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Journal articleGenetics · September 2004
Using genomic data from homologous microsatellite loci of pure AC repeats in humans and chimpanzees, several models of microsatellite evolution are tested and compared using likelihood-ratio tests and the Akaike information criterion. A proportional-rate, ...
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Journal articleTheoretical population biology · September 2004
The mechanisms responsible for the preservation of duplicate genes have been debated for more than 70 years. Recently, Lynch and Force have proposed a new explanation: subfunctionalization--after duplication the two gene copies specialize to perform comple ...
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Journal articleTheoretical population biology · September 2004
The fixation of advantageous mutations in a population has the effect of reducing variation in the DNA sequence near that mutation. Kaplan et al. (1989) used a three-phase simulation model to study the effect of selective sweeps on genealogies. However, mo ...
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Journal articleGenetics · January 2004
We present a Bayesian approach to the problem of inferring the number of inversions and translocations separating two species. The main reason for developing this method is that it will allow us to test hypotheses about the underlying mechanisms, such as t ...
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Journal articleAnnals of Probability · October 1, 2003
Motivated by the problem of the evolution of DNA sequences, Kauffman and Levin introduced a model in which fitnesses were assigned to strings of 0's and 1's of length N based on the values observed in a sliding window of length K + 1. When K ≥ 1, the lands ...
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Journal articleJournal of Theoretical Probability · July 1, 2003
The gene order of chromosomes can be rearranged by chromosomal inversions that reverse the order of segments. Motivated by a comparative study of two Drosophila species, we investigate the number of reversals that are needed to scramble the gene order when ...
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Journal articleMolecular biology and evolution · May 2003
We use methods of maximum likelihood estimation to fit several microsatellite mutation models to the observed length distribution of dinucletoide repeats in the Drosophila and human genomes. All simple models are rejected by this procedure. Two new models, ...
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Journal articleStochastic Processes and their Applications · December 1, 2002
Motivated by the work of Tilman (Ecology 75 (1994) 2) and May and Nowak (J. Theoret. Biol. 170 (1994) 95) we consider a process in which points are inserted randomly into the unit interval and a new point kills each point to its left independently and with ...
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Journal articleAnnals of Applied Probability · November 1, 2002
We study the stepping stone model on the two-dimensional torus. We prove several new hitting time results for random walks from which we derive some simple approximation formulas for the homozygosity in the stepping stone model as a function of the separat ...
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Journal articleTheoretical population biology · May 2002
In models of competition in which space is treated as a continuum, and population size as continuous, there are no limits to the number of species that can coexist. For a finite number of sites, N, the results are different. The answer will, of course, dep ...
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Journal articleProbability Theory and Related Fields · April 1, 2002
We consider a nearest neighbor walk on a regular tree, with transition probabilities proportional to weights or conductances of the edges. Initially all edges have weight 1, and the weight of an edge is increased to c > 1 when the edge is traversed for the ...
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Journal articleMemoirs of the American Mathematical Society · January 1, 2002
In (1994) Durrett and Levin proposed that the equilibrium behavior of stochastic spatial models could be determined from properties of the solution of the mean field ordinary differential equation (ODE) that is obtained by pretending that all sites are alw ...
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Journal articleJournal of computational biology : a journal of computational molecular cell biology · January 2002
We present a Bayesian approach to the problem of inferring the history of inversions separating homologous chromosomes from two different species. The method is based on Markov Chain Monte Carlo (MCMC) and takes full advantage of all the information from m ...
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Journal articleMemoirs of the American Mathematical Society · January 1, 2002
In (1994) Durrett and Levin proposed that the equilibrium behavior of stochastic spatial models could be determined from properties of the solution of the mean field ordinary differential equation (ODE) that is obtained by pretending that all sites are alw ...
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Journal articleGenetics · January 2002
We derive a formula for the distribution of the length T of the recombination interval containing a target gene and using N gametes in a region where R kilobases correspond to 1 cM. The formula can be used to calculate the number of meiotic events required ...
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Journal articleJournal of Theoretical Probability · December 1, 2001
We investigate how the stationary distribution of a Markov chain changes when transitions from a single state are modified. In particular, adding a single directed edge to nearest neighbor random walk on a finite discrete torus in dimensions one, two, or t ...
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Journal articleGenetics · October 2001
Recently Kruglyak, Durrett, Schug, and Aquadro showed that microsatellite equilibrium distributions can result from a balance between polymerase slippage and point mutations. Here, we introduce an elaboration of their model that keeps track of all parts of ...
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Journal articleTheoretical population biology · September 2001
Statistical tests are needed to determine whether spatial structure has had a significant effect on the genetic differentiation of subpopulations. Here we introduce a new family of statistics based on a sum of an exponential function of the distances betwe ...
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Journal articleStochastic Processes and their Applications · May 1, 2001
Single nucleotide polymorphisms (SNPs) are useful markers for locating genes since they occur throughout the human genome and thousands can be scored at once using DNA microarrays. Here, we use branching processes and coalescent theory to show that if one ...
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Journal articleJournal of Applied Probability · January 1, 2001
Sankoff and Ferretti (1996) introduced several models of the evolution of chromosome size by reciprocal translocations, where for simplicity they ignored the existence of centromeres. However, when they compared the models to data on six organisms they fou ...
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Journal articleSystematic biology · December 2000
Nucleotide transitions are frequently down-weighted relative to transversions in phylogenetic analysis. This is based on the assumption that transitions, by virtue of their greater evolutionary rate, exhibit relatively more homoplasy and are therefore less ...
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Journal articleMolecular biology and evolution · August 2000
We fit a Markov chain model of microsatellite evolution introduced by Kruglyak et al. to data on all di-, tri-, and tetranucleotide repeats in the yeast genome. Our results suggest that many features of the distribution of abundance and length of microsate ...
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Journal articleJournal of theoretical biology · July 2000
It is well known that if reacting species experience unequal diffusion rates, then dynamics that lead to a constant steady state in a "well-mixed" environment can in a spatial setting lead to interesting patterns. In this paper, we focus on complementary p ...
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Journal articleGenetics · May 2000
Historically, linkage mapping populations have consisted of large, randomly selected samples of progeny from a given pedigree or cell lines from a panel of radiation hybrids. We demonstrate that, to construct a map with high genome-wide marker density, it ...
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Journal articleEcological Modelling · March 30, 2000
A spatially explicit, interacting particle system model was developed to simulate the heterogeneous transmission of viruses in fungal populations. This model is based primarily on hypoviruses in the chestnut blight fungus, Cryphonectria parasitica, which d ...
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Journal articleJournal of Theoretical Probability · January 1, 2000
We introduce a one dimensional contact process for which births to the right of the rightmost particle and to the left of the leftmost particle occur at rate λe (where e is for external). Other births occur at rate λi (where i is for ...
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Journal articleAnnals of Probability · January 1, 2000
We show that a sequence of voter models, suitably rescaled in space and time, converges weakly to super-Brownian motion. The result includes both nearest neighbor and longer range voter models and complements a limit theorem of Mueller and Tribe in one dim ...
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Journal articleHeredity · January 2000
We introduce a spatially explicit model of natural hybrid zones that allows us to consider how patterns of allele frequencies and linkage disequilibria change over time. We examine the influence of hybrid zone origins on patterns of variation at two loci, ...
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Journal articleTheoretical population biology · June 1999
In sessile organisms such as plants, interactions occur locally so that important ecological aspects like frequency dependence are manifest within local neighborhoods. Using probabilistic cellular automata models, we investigated how local frequency-depend ...
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Journal articleJournal of Applied Probability · January 1, 1999
We introduce a continuous-time Markov chain model for the evolution of microsatellites, simple sequence repeats in DNA. We prove the existence of a unique stationary distribution for our model, and fit the model to data from approximately 106 ba ...
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Journal articleJournal of Applied Probability · January 1, 1999
We introduce a continuous-time Markov chain model for the evolution of microsatellites, simple sequence repeats in DNA. We prove the existence of a unique stationary distribution for our model, and fit the model to data from approximately 106 ba ...
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Journal articleJournal of Theoretical Probability · January 1, 1999
The coagulation-fragmentation process models the stochastic evolution of a population of N particles distributed into groups of different sizes that coagulate and fragment at given rates. The process arises in a variety of contexts and has been intensively ...
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Journal articleProbability Theory and Related Fields · January 1, 1999
We show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swind ...
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Journal articleSIAM Review · January 1, 1999
In the models we will consider, space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites. Our main aim here is to explain an idea of Durrett and L ...
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Journal articleProceedings of the National Academy of Sciences of the United States of America · September 1998
We describe and test a Markov chain model of microsatellite evolution that can explain the different distributions of microsatellite lengths across different organisms and repeat motifs. Two key features of this model are the dependence of mutation rates o ...
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Journal articleTheoretical population biology · February 1998
Using several variants of a stochastic spatial model introduced by Silvertown et al., we investigate the effect of spatial distribution of individuals on the outcome of competition. First, we prove rigorously that if one species has a competitive advantage ...
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Journal articleAnnals of Probability · January 1, 1998
The voter model, with mutations occurring at a positive rate a, has a unique equilibrium distribution. We investigate the logarithms of the relative abundance of species for these distributions in d ≥ 2. We show that, as α → 0, the limiting distribution is ...
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Journal articleJournal of Theoretical Biology · March 21, 1997
In a homogeneously mixing population of E. coli, colicin-producing and colicin-sensitive strategies both may be evolutionarily stable for certain parameter ranges, with the outcome of competition determined by initial conditions. In contrast, in a spatiall ...
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Journal articleAnnals of Probability · January 1, 1997
We study a family of differential operators {Lα, α ≥ 0} in the unit ball D of Cn with n ≥ 2 that generalize the classical Laplacian, α = 0, and the conformal Laplacian, α = 1/2 (that is, the Laplace-Beltrami operator for Bergman metri ...
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Journal articleAnnals of Applied Probability · January 1, 1997
Barley yellow dwarf is a widespread disease that affects small grains and many grass species, as well as wheat, barley and oat. The disease is caused by an aphid transmitted virus. Rochow conducted a study near Ithaca, New York, which showed that a shift i ...
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Journal articleJournal of Theoretical Probability · January 1, 1997
We use supercritical branching processes with random walk steps of geometrically decreasing size to construct random measures. Special cases of our construction give close relatives of the super-(spherically symmetric stable) processes. However, other case ...
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Journal articleJournal of Theoretical Biology · March 14, 1996
Inspired by earlier work of Hubbell, we introduce a simple spatial model to explain observed species-area curves. As in the theory of MacArthur and Wilson, our curves result from a balance between migration and extinction. Our model predicts that the wide ...
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Journal articleAnnals of Probability · January 1, 1996
The relationship between species number and area is an old problem in biology. We propose here an interacting particle system - the multitype voter model with mutation - as a mathematical model to study this problem. We analyze the species area curves of t ...
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Journal articleBernoulli · January 1, 1995
We study the dynamics of hybrid zones in the absence of selection. In dimensions d > 1 the width of the hybrid zone grows as √t but in one dimension the width converges to a non-degenerate limit. We believe that tight interfaces are common in one-dimension ...
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Journal articleProbability Theory and Related Fields · December 1, 1994
In this paper we consider a modification of Ziff, Gulari and Barshad's (1986) model of oxidation of carbon monoxide on a catalyst surface in which the reactants are mobile on the catalyst surface. We find regions in the parameter space in which poisoning o ...
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Journal articleTheoretical Population Biology · January 1, 1994
We consider and compare four approaches to modeling the dynamics of spatially distributed systems: mean field approaches (described by ordinary differential equations) in which every individual is considered to have equal probability of interacting with ev ...
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Journal articleExperimental Mathematics · January 1, 1993
We study two families of excitable cellular automata known as the Greenberg-Hastings model and the cyclic cellular automaton. Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by th ...
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Journal articleProbability Theory and Related Fields · September 1, 1992
We consider a system that models the shape of a growing polymer. Our basic problem concerns the asymptotic behavior of Xt, the location of the end of the polymer at time t. We obtain bounds on Xt in the (physically uninteresting) case ...
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Journal articleJournal of Theoretical Probability · January 1, 1992
In this paper we consider the Greenberg-Hastings and cyclic color models. These models exhibit (at least) three different types of behavior. Depending on the number of colors and the size of two parameters called the threshold and range, the Greenberg-Hast ...
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Journal articleJournal of Theoretical Probability · October 1, 1991
In this paper, we obtain some rigorous results for a cellular automaton known as the Greenberg-Hastings Model. The state space is {0, 1, 2}Zd. The dynamics are deterministic and discrete time. A site which is 1 changes to 2, a site which is 2 ch ...
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Journal articleProbability Theory and Related Fields · September 1, 1991
Durrett (1984) proved the existence of an invariant measure for the critical and supercritical contact process seen from the right edge. Galves and Presutti (1987) proved, in the supercritical case, that the invariant measure was unique, and convergence to ...
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Journal articleProbability Theory and Related Fields · March 1, 1991
In this paper we consider a hierarchical competition model. Durrett and Swindle have given sufficient conditions for the existence of a nontrivial stationary distribution. Here we show that under a slightly stronger condition, the complete convergence theo ...
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Journal articleStochastic Processes and their Applications · January 1, 1991
We consider Markov processes ηt ⊂ Zd in which (i) particles die at rate δ ≥ 0, (ii) births from x to a neighboring y occur at rate 1, and (iii) when a new particle lands on an occupied site the particles annihilate each other and a va ...
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Journal articleJournal of Theoretical Probability · January 1, 1991
Consider the family tree T of a branching process starting from a single progenitor and conditioned to have v=v(T) edges (total progeny). To each edge we associate a weight W(e). The weights are i.i.d. random variables and independent of T. The weighte ...
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Journal articleStochastic Processes and their Applications · January 1, 1991
In this paper we consider a process in which each site x ε{lunate} Zd can be occupied by grass, bushes or trees and ask the question: Are there equilibria in which bushes and trees are both present? The answer is sometimes yes and sometimes no. ...
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Journal articleActa Mathematica Sinica · December 1, 1990
The complete convergence theorem implies that starting from any initial distribution the one dimensional contact process converges to a limit as t→∞. In this paper we give a necessary and sufficient condition on the initial distribution for the convergence ...
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Journal articleProbability Theory and Related Fields · March 1, 1990
We consider a system of independent random walks on ℤ. Let ξn(x) be the number of particles at x at time n, and let Ln(x)=ξ0(x)+ ... +ξn(x) be the total occupation time of x by time n. In this paper we study the ...
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Journal articleProbability Theory and Related Fields · March 1, 1990
Reaction-diffusion processes were introduced by Nicolis and Prigogine, and Haken. Existence theorems have been established for most models, but not much is known about ergodic properties. In this paper we study a class of models which have a reversible mea ...
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Journal articleJournal of Statistical Physics · June 1, 1989
Oriented percolation has two correlation lengths, one in the "space" and one in the "time" direction. In this paper we define these quantities for the two-dimensional model in terms of the exponential decay of suitably chosen quantities, and study the rela ...
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Journal articleProbability Theory and Related Fields · December 1, 1988
The following results are proved: 1) For the upper invariant measure of the basic one-dimensional supercritical contact process the density of 1's has the usual large deviation behavior: the probability of a large deviation decays exponentially with the nu ...
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Journal articleProbability Theory and Related Fields · December 1, 1988
Gray and Griffeath studied attractive nearest neighbor spin systems on the integers having "all 0's" and "all 1's" as traps. Using the contour method, they established a necessary and sufficient condition for the stability of the "all 1's" equilibrium unde ...
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Journal articleProbability Theory and Related Fields · March 1, 1988
In 1974, Mandelbrot introduced a process in [0, 1]2 which he called "canonical curdling" and later used in this book(s) on fractals to generate self-similar random sets with Hausdorff dimension D∈(0,2). In this paper we will study the connectivi ...
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Journal articleJournal of Physics A Mathematical and General · December 1, 1987
The authors consider inhomogeneous percolation models with density p c+f(x) and examine the forms of f(x) which produce incipient structures. Taking f(x) approximately= mod x mod - lambda and assuming the existence of a correlation le ...
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Journal articleProbability Theory and Related Fields · May 1, 1987
In the processes under consideration, a particle of size L splits with exponential rate Lα, 0<α<∞, and when it splits, it splits into two particles of size LV and L(1-V) where V is independent of the past with d.f. F on (0, 1). Let Zt ...
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Journal articleJournal of Statistical Physics · December 1, 1986
We study the two-dimensional first passage problem in which bonds have zero and unit passage times with probability p and 1-p, respectively. We prove that as the zero-time bonds approach the percolation threshold pc, the first passage time exhib ...
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Journal articleCommunications in Mathematical Physics · March 1, 1986
In this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have the pleasant feature that quite a lot can be said about them. ...
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Journal articleZeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebiete · September 1, 1985
In this paper we will give some results concerning the critical exponents of percolation processes which are valid for "any" model. These results show that in several respects the behavior which occurs for percolation on the binary tree provides bounds on ...
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Journal articleZeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebiete · September 1, 1983
Let W 1,..., W N be N nonnegative random variables and let {Mathematical expression} be the class of all probability measures on [0, ∞). Define a transformation T on {Mathematical expression} by letting Tμ be the distribution of W
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Journal articleZeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebiete · June 1, 1983
In recent years several authors have obtained limit theorems for Ln, the location of the rightmost particle in a supercritical branching random walk but all of these results have been proved under the assumption that the offspring distribution h ...
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Journal articleMathematical Proceedings of the Cambridge Philosophical Society · January 1, 1983
Let pc(d) be the critical probability for oriented percolation in Zdand let (d) be the time constant for the first passage process based on the exponential distribution. In this paper we show that as d →∞, dpc(d)→ 1 and dμ{d)→γ where ...
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Journal articleStochastic Processes and their Applications · January 1, 1981
In 1970, Spitzer wrote a paper called "Interaction of Markov processes" in which he introduced several classes of interacting particle systems. These processes and other related models, collectively referred to as infinite particle systems, have been the o ...
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Journal articleZeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebiete · January 1, 1980
In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if Sn is a random walk with negative mean and finite variance then there is a constant α so that (S[n.]/αn1/2|N>n) converges weakly to a ...
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Journal articleLaryngoscope · 1979
Using a newly developed brain stem evoked response (BSER) parameter in preliminary testing, the authors can individually identify the auditory thresholds of 500 Hz, 1000 Hz and 2000 Hz to within 15-25 db. The authors have developed an accurate method of br ...
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Journal articleStochastic Processes and their Applications · January 1, 1979
In recent years several authors have obtained limit theorems for the location of the right most particle in a supercritical branching random walk. In this paper we will consider analogous problems for an exponentially growing number of independent random w ...
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Journal articleStochastic Processes and their Applications · January 1, 1978
In this paper we will obtain results concerning the distribution of generations and the degree of relationship of the individuals in a critical branching process {Z(t), t≥0} and we will apply these results to obtain a "central limit theorem" for critical b ...
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Journal articleStochastic Processes and their Applications · January 1, 1977
Suppose {Xnn≥-0} are random variables such that for normalizing constants an>0, bn, n≥0 we have Yn(·)=(X[n, ·]-bn/an ⇒ Y(·) in D(0.∞) . Then an and bn must i ...
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Journal articleJ Appl Probab · January 1, 1976
A waiting time without memory, or age-independent residual life-time, is a positive-valued random variable T with the property that for any x, y greater than 0, given that T greater than x, the conditional probability of T greater than x plus y is the same ...
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