arXiv subjects

Richard Durrett

Publications and source records attributed to Richard Durrett.

8 recordsLinked to original sources

arXiv2016

Phase transitions for a planar quadratic contact process

We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results that give bounds on the critical values $λ_f$ for the process to survive from a finite set and $λ_e$ for the existence of a nontrivial stationary distribution. Our first result comes from a standard block construction, while the second involves a comparison with the "generic population model" of Bramson and Gray (1991). An interesting new feature of our work is the suggestion that, as in the one dimensional contact process, edge speeds characterize critical values. We are able to prove the following for our quadratic contact process when the range is large but suspect they are true for two dimensional finite range attractive particle systems that are symmetric with respect to reflection in each axis. There is a speed $c(θ)$ for the expansion of the process in each direction. If $c(θ) > 0$ in all directions, then $λ> λ_f$, while if at least one speed is positive, then $λ> λ_e$. It is a challenging open problem to show that if some speed is negative, then the system dies out from any finite set.

math.PR↗

arXiv2015

Spatial Moran models I. Stochastic tunneling in the neutral case

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 dimensions $d\ge 2$, extending work done by Komarova [Genetics 166 (2004) 1571-1579] for $d=1$.

math.PR↗

arXiv2011

Voter Model Perturbations and Reaction Diffusion Equations

We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins and the second confirms a conjecture of Ohtsuki et al in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.

math.PR↗

arXiv2008

One-dimensional stepping stone models, sardine genetics and Brownian local time

Consider a one-dimensional stepping stone model with colonies of size $M$ and per-generation migration probability $ν$, or a voter model on $\mathbb{Z}$ in which interactions occur over a distance of order $K$. Sample one individual at the origin and one at $L$. We show that if $Mν/L$ and $L/K^2$ converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.

math.PR↗

arXiv2007

Waiting for regulatory sequences to appear

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 question: given a 1000 nucleotide region in our genome, how long does it take for a specified six to nine letter word to appear in that region in some individual? Stone and Wray [Mol. Biol. Evol. 18 (2001) 1764--1770] computed 5,950 years as the answer for six letter words. Here, we will show that for words of length 6, the average waiting time is 100,000 years, while for words of length 8, the waiting time has mean 375,000 years when there is a 7 out of 8 letter match in the population consensus sequence (an event of probability roughly 5/16) and has mean 650 million years when there is not. Fortunately, in biological reality, the match to the target word does not have to be perfect for binding to occur. If we model this by saying that a 7 out of 8 letter match is good enough, the mean reduces to about 60,000 years.

math.PR↗

arXiv2006

A new coexistence result for competing contact processes

Neuhauser [Probab. Theory Related Fields 91 (1992) 467--506] considered the two-type contact process and showed that on $\mathbb{Z}^2$ coexistence is not possible if the death rates are equal and the particles use the same dispersal neighborhood. Here, we show that it is possible for a species with a long-, but finite, range dispersal kernel to coexist with a superior competitor with nearest-neighbor dispersal in a model that includes deaths of blocks due to ``forest fires.''

math.PR↗

arXiv2005

The stepping stone model. II: Genealogies and the infinite sites model

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 coalescent. With DNA sequence data in mind, we investigate mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site. Our results suggest that the spatial structure of the human population contributes to the haplotype structure and a slower than expected decay of genetic correlation with distance revealed by recent studies of the human genome.

math.PR↗

arXiv1993

Asymptotic Behavior of Excitable Cellular Automata

We study two families of excitable cellular automata known as the Greenberg-Hastings Model (GHM) and the Cyclic Cellular Automaton (CCA). Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by the range of interaction, the "shape" of its neighbor set, threshold value for contact updating, and number of possible states per site. GHM and CCA are mathematically tractable prototypes for the spatially distributed periodic wave activity of so-called excitable media observed in diverse disciplines of experimental science. Earlier work by Fisch, Gravner, and Griffeath studied the ergodic behavior of these excitable cellular automata on Z^2, and identified two distinct (but closely-related) elaborate phase portraits as the parameters vary. In particular, they noted the emergence of asymptotic phase diagrams (and Euclidean dynamics) in a well-defined threshold-range scaling limit. In this study we present several rigorous results and some experimental findings concerning various phase transitions in the asymptotic diagrams, focusing on evaluating the limiting threshold cutoff for existence of the spirals that characterize many excitable media. For mathematical expediency our main results are formulated in terms of spo(p), the cutoff for existence of stable periodic objects that arise as spiral cores.

patt-sol↗