Scholars@Duke
Richard Timothy Durrett

Richard Timothy Durrett

James B. Duke Distinguished Professor Emeritus of Mathematics
Mathematics
rtd@math.duke.edu
Dept of Math, Box 90320, Durham, NC 27708-0320
120 Science Drive, Durham, NC 27708-0320

Selected Publications

A stochastic spatial model for the sterile insect control strategy

Journal Article Stochastic 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 ... Full text Cite

Corrigendum to: The contact process on periodic trees (Electronic Communications in Probability)

Journal Article Electronic 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 contained an error. In this note we correct the proof. The theorem has change ... Full text Cite

Competitive exclusion in a model with seasonality: Three species cannot coexist in an ecosystem with two seasons.

Journal Article Theoretical 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 ... Full text Cite

Selective sweeps in SARS-CoV-2 variant competition.

Journal Article Proceedings 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 ... Full text Cite

Susceptible–infected epidemics on evolving graphs

Journal Article Electronic 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 ... Full text Cite

Motion by mean curvature in interacting particle systems

Journal Article Probability 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 ... Full text Cite

Signatures of neutral evolution in exponentially growing tumors: A theoretical perspective.

Journal Article PLoS 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 ... Full text Cite

The q-voter model on the torus

Journal Article Electronic 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 clustering if q > 1. This ... Full text Cite

Controlling the spread of COVID-19 on college campuses.

Journal Article Mathematical 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 ... Full text Cite

Poisson percolation on the oriented square lattice

Journal Article Stochastic 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 ... Full text Cite

The contact process on random graphs and Galton Watson trees

Journal Article Alea (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 ( ... Full text Cite

The symbiotic contact process

Journal Article Electronic 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 name symbi ... Full text Cite

Coexistence in chase-escape

Journal Article Electronic 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 ... Full text Cite

The contact process on periodic trees

Journal Article Electronic 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 with periodic degree sequ ... Full text Cite

Probability: Theory and Examples

Book · April 5, 2019 This lively introduction to measure-theoretic probability theory covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. Concentrating on results that are the most useful for ap ... Full text Cite

Extrapolating weak selection in evolutionary games.

Journal Article Journal 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 ... Full text Cite

The Zealot voter model

Journal Article Annals 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 ... Full text Cite

Poisson percolation on the square lattice

Journal Article Alea · 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 probability p at time t. We ... Full text Cite

A simple evolutionary game arising from the study of the role of igf-II in pancreatic cancer

Journal Article Annals 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 its offspring to a ra ... Full text Cite