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## Publications of Richard T. Durrett    :chronological  alphabetical  combined listing:

%% Books
@book{fds299418,
Author = {R.T. Durrett},
Title = {Branching process models of cancer},
Publisher = {Springer},
Year = {2015},
Key = {fds299418}
}

@book{fds177897,
Author = {R. Durrett},
Title = {Probability: Theory and Examples},
Series = {4th Edition},
Publisher = {Cambridge U Press},
Year = {2010},
Key = {fds177897}
}

@book{fds177879,
Author = {R. Durrett},
Title = {Elementary Probability for Applications},
Series = {4th Edition},
Pages = {x+243},
Publisher = {Cambridge University Press},
Address = {Cambridge},
Year = {2009},
ISBN = {978-0-521-86756-6},
MRCLASS = {60-01 (60J10)},
MRNUMBER = {MR2537423 (2010i:60003)},
url = {http://www.ams.org/mathscinet-getitem?mr=2537423},
Key = {fds177879}
}

@book{fds177878,
Author = {R. Durrett},
Title = {Probability Models of DNA Sequence Evolution},
Series = {2nd Edition},
Publisher = {Springer},
Year = {2006},
Key = {fds177878}
}

@book{fds177911,
Author = {R. Durrett},
Title = {Essentials of Stochastic Processes},
Publisher = {Springer-Verlag},
Year = {1998},
Key = {fds177911}
}

@book{fds177910,
Author = {R. Durrett},
Title = {Stochastic Calculus: A Practrical Introduction},
Publisher = {CRC Press},
Year = {1996},
Key = {fds177910}
}

@book{fds177912,
Author = {R. Durrett},
Title = {Lecture Notes on Particle Systems and Percolation},
Publisher = {Wadsworth},
Year = {1988},
Key = {fds177912}
}

@book{fds177913,
Author = {R. Durrett},
Title = {Brownian Motion and Martingales in Analysis},
Publisher = {Wadsworth},
Year = {1984},
Key = {fds177913}
}

%% Papers Published
@article{fds341499,
Author = {Cristali, I and Junge, M and Durrett, R},
Title = {Poisson percolation on the oriented square
lattice},
Journal = {Stochastic Processes and Their Applications},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.1016/j.spa.2019.01.005},
Abstract = {© 2019 Elsevier B.V. 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 containing the
origin in the oriented case. We show that the density of
occupied sites at height y in the cluster is close to the
percolation probability in the corresponding homogeneous
percolation process, and we study the fluctuations of the
boundary.},
Doi = {10.1016/j.spa.2019.01.005},
Key = {fds341499}
}

@article{fds335535,
Author = {Wang, Z and Durrett, R},
Title = {Extrapolating weak selection in evolutionary
games.},
Journal = {Journal of Mathematical Biology},
Volume = {78},
Number = {1-2},
Pages = {135-154},
Year = {2019},
Month = {January},
url = {http://dx.doi.org/10.1007/s00285-018-1270-6},
Abstract = {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 to fixation before the next one occurs.
It has been shown that for [Formula: see text] games the
ranking of the strategies does not change as strength of
selection is increased (Wu et al. in Phys Rev 82:046106,
2010). The point of the 2013 paper is that when there are
three or more strategies the ordering can change as
selection is increased. Wu et al. (2013) did numerical
computations for a fixed population size N. Here, we will
instead let the strength of selection [Formula: see text]
where c is fixed and let [Formula: see text] to obtain
formulas for the invadability probabilities [Formula: see
text] that determine the rankings. These formulas, which are
integrals on [0, 1], are intractable calculus problems, but
can be easily evaluated numerically. Here, we use them to
derive simple formulas for the ranking order when c is small
or c is large.},
Doi = {10.1007/s00285-018-1270-6},
Key = {fds335535}
}

@article{fds337720,
Author = {Ma, R and Durrett, R},
Title = {A simple evolutionary game arising from the study of the
role of igf-II in pancreatic cancer},
Journal = {The Annals of Applied Probability},
Volume = {28},
Number = {5},
Pages = {2896-2921},
Publisher = {Institute of Mathematical Statistics},
Year = {2018},
Month = {October},
url = {http://dx.doi.org/10.1214/17-AAP1378},
Abstract = {© Institute of Mathematical Statistics, 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 randomly chosen point in x + Nc. We first study this
game on the d-dimensional torus (Z mod L)d with Nd = (Z mod
L)d and Nc = the 2d nearest neighbors. If we let L → ∞
then t → ∞ the fraction of producers converges to 1/λ.
In d ≥ 3 the limiting finite dimensional distributions
converge as t → ∞ to the voter model equilibrium with
density 1/λ. We next reformulate the system as an
evolutionary game with “birth-death” updating and take
Nc = Nd = N. Using results for voter model perturbations we
show that in d = 3 with N = the six nearest neighbors, the
density of producers converges to (2/λ) − 0.5 for 4/3 <
λ < 4. Producers take over the system when λ < 4/3 and die
out when λ > 4. In d = 2 with N = [−clog N, clog N]2
there are similar phase transitions, with coexistence
occurring when (1 + 2θ)/(1 + θ) < λ < (1 + 2θ)/θ where
θ = (e3/(πc2) − 1)/2.},
Doi = {10.1214/17-AAP1378},
Key = {fds337720}
}

@article{fds339723,
Author = {Talkington, A and Dantoin, C and Durrett, R},
Title = {Ordinary Differential Equation Models for Adoptive
Immunotherapy.},
Journal = {Bulletin of Mathematical Biology},
Volume = {80},
Number = {5},
Pages = {1059-1083},
Year = {2018},
Month = {May},
url = {http://dx.doi.org/10.1007/s11538-017-0263-8},
Abstract = {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 to model this type of therapy,
compare their properties, and modify the models to address
their deficiencies. Although the four models treat the
workings of the immune system in slightly different ways,
they all predict that adoptive immunotherapy can be
successful to move a patient from the large tumor fixed
point to an equilibrium with little or no
tumor.},
Doi = {10.1007/s11538-017-0263-8},
Key = {fds339723}
}

@article{fds330932,
Author = {Huo, R and Durrett, R},
Title = {Latent voter model on locally tree-like random
graphs},
Journal = {Stochastic Processes and Their Applications},
Volume = {128},
Number = {5},
Pages = {1590-1614},
Publisher = {Elsevier BV},
Year = {2018},
Month = {May},
url = {http://dx.doi.org/10.1016/j.spa.2017.08.004},
Abstract = {© 2017 Elsevier B.V. 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 degrees
3≤d(x)≤M. We show that if the number of vertices n→∞
and logn≪λn≪n then there is a quasi-stationary state in
which each opinion has probability ≈1∕2 and persists in
this state for a time that is ≥nm for any
m<∞.},
Doi = {10.1016/j.spa.2017.08.004},
Key = {fds330932}
}

@article{fds339577,
Author = {Beckman, E and Dinan, E and Durrett, R and Huo, R and Junge,
M},
Title = {Asymptotic behavior of the brownian frog
model},
Journal = {Electronic Journal of Probability},
Volume = {23},
Publisher = {Institute of Mathematical Statistics},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1214/18-EJP215},
Abstract = {© 2018, University of Washington. All rights reserved. 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, put a ball of radius r. A particle at the origin
performs Brownian motion. When it hits the ball around x for
some x ∈ P, new particles begin independent Brownian
motions from the centers of the balls in the cluster
containing x. Subsequent visits to the cluster do nothing.
This waking process continues indefinitely. For r smaller
than the critical threshold of continuum percolation, we
show that the set of activated points in P approximates a
linearly expanding ball. Moreover, in any fixed ball the set
of active particles converges to a unit intensity Poisson
point process.},
Doi = {10.1214/18-EJP215},
Key = {fds339577}
}

@article{fds339578,
Author = {Basak, A and Durrett, R and Foxall, E},
Title = {Diffusion limit for the partner model at the critical
value},
Journal = {Electronic Journal of Probability},
Volume = {23},
Publisher = {Institute of Mathematical Statistics},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1214/18-EJP229},
Abstract = {© 2018, University of Washington. All rights reserved.
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. Recently Foxall [4] has shown that
(if there are enough initial infecteds I0) the extinction
time in the critical model is of order √N. Here we improve
that result by proving the convergence of
iN(t)=I(√Nt)/√N to a limiting diffusion. We do this by
showing that within a short time, this four dimensional
process collapses to two dimensions: the number of SI and II
partnerships are constant multiples of the the number of
infected singles. The other variable, the total number of
singles, fluctuates around its equilibrium like an
Ornstein-Uhlenbeck process of magnitude √N on the original
time scale and averages out of the limit theorem for iN(t).
As a by-product of our proof we show that if τN is the
extinction time of iN(t) (on the √N time scale) then τN
has a limit.},
Doi = {10.1214/18-EJP229},
Key = {fds339578}
}

@article{fds339329,
Author = {Cristali, I and Ranjan, V and Steinberg, J and Beckman, E and Durrett,
R and Junge, M and Nolen, J},
Title = {Block size in geometric(P)-biased permutations},
Journal = {Electronic Communications in Probability},
Volume = {23},
Pages = {1-10},
Publisher = {Institute of Mathematical Statistics},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1214/18-ECP182},
Abstract = {© 2018, University of Washington. All rights reserved. Fix
a probability distribution p = (p1, p2, …) on the positive
integers. The first block in a p-biased permutation can be
visualized in terms of raindrops that land at each positive
integer j with probability pj. It is the first point K so
that all sites in [1, K] are wet and all sites in (K, ∞)
are dry. For the geometric distribution pj = p(1 − p)j−1
we show that p log K converges in probability to an explicit
constant as p tends to 0. Additionally, we prove that if p
has a stretch exponential distribution, then K is infinite
with positive probability.},
Doi = {10.1214/18-ECP182},
Key = {fds339329}
}

@article{fds330931,
Author = {Lopatkin, AJ and Meredith, HR and Srimani, JK and Pfeiffer, C and Durrett, R and You, L},
Title = {Persistence and reversal of plasmid-mediated antibiotic
resistance.},
Journal = {Nature Communications},
Volume = {8},
Number = {1},
Pages = {1689},
Year = {2017},
Month = {November},
url = {http://dx.doi.org/10.1038/s41467-017-01532-1},
Abstract = {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 transfer (primarily
conjugation) could explain this persistence, but it has been
suggested that very high conjugation rates would be
required. Here, we show that common conjugal plasmids, even
when costly, are indeed transferred at sufficiently high
rates to be maintained in the absence of antibiotics in
Escherichia coli. The notion is applicable to nine plasmids
from six major incompatibility groups and mixed populations
carrying multiple plasmids. These results suggest that
reducing antibiotic use alone is likely insufficient for
reversing resistance. Therefore, combining conjugation
inhibition and promoting plasmid loss would be an effective
strategy to limit conjugation-assisted persistence of
antibiotic resistance.},
Doi = {10.1038/s41467-017-01532-1},
Key = {fds330931}
}

@article{fds329932,
Author = {Gleeson, JP and Durrett, R},
Title = {Temporal profiles of avalanches on networks.},
Journal = {Nature Communications},
Volume = {8},
Number = {1},
Pages = {1227},
Year = {2017},
Month = {October},
url = {http://dx.doi.org/10.1038/s41467-017-01212-0},
Abstract = {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 temporal profiles of avalanches
of fixed duration. At the critical point of the dynamics,
the rescaled average avalanche shapes for different
durations collapse onto a single universal curve. We apply
Markov branching process theory to derive an equation
governing the average avalanche shape for cascade dynamics
on networks. Analysis of the equation at criticality
demonstrates that nonsymmetric average avalanche shapes (as
observed in some experiments) occur for certain combinations
of dynamics and network topology. We give examples using
numerical simulations of models for information spreading,
neural dynamics, and behavior adoption and we propose simple
experimental tests to quantify whether cascading systems are
in the critical state.},
Doi = {10.1038/s41467-017-01212-0},
Key = {fds329932}
}

@article{fds329933,
Author = {Tomasetti, C and Durrett, R and Kimmel, M and Lambert, A and Parmigiani,
G and Zauber, A and Vogelstein, B},
Title = {Role of stem-cell divisions in cancer risk},
Journal = {Nature},
Volume = {548},
Number = {7666},
Pages = {E13-E14},
Publisher = {Springer Nature},
Year = {2017},
Month = {August},
url = {http://dx.doi.org/10.1038/nature23302},
Doi = {10.1038/nature23302},
Key = {fds329933}
}

@article{fds327001,
Author = {Nanda, M and Durrett, R},
Title = {Spatial evolutionary games with weak selection},
Journal = {Proceedings of the National Academy of Sciences of the
United States of America},
Volume = {114},
Number = {23},
Pages = {6046-6051},
Year = {2017},
Month = {June},
url = {http://dx.doi.org/10.1073/pnas.1620852114},
Abstract = {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 converges to the solution of a partial
differential equation (PDE). This approach can be used to
analyze all 2 × 2 games, but there are a number of 3 × 3
games for which the behavior of the limiting PDE is not
known. In this paper, we give rules for determining the
behavior of a large class of 3 × 3 games and check their
validity using simulation. In words, the effect of space is
equivalent to making changes in the payoff matrix, and once
this is done, the behavior of the spatial game can be
predicted from the behavior of the replicator equation for
the modified game. We say predicted here because in some
cases the behavior of the spatial game is different from
that of the replicator equation for the modified game. For
example, if a rock-paper-scissors game has a replicator
equation that spirals out to the boundary, space stabilizes
the system and produces an equilibrium.},
Doi = {10.1073/pnas.1620852114},
Key = {fds327001}
}

@article{fds323833,
Author = {Bessonov, M and Durrett, R},
Title = {Phase transitions for a planar quadratic contact
process},
Journal = {Advances in Applied Mathematics},
Volume = {87},
Pages = {82-107},
Publisher = {Elsevier BV},
Year = {2017},
Month = {June},
url = {http://dx.doi.org/10.1016/j.aam.2017.01.002},
Doi = {10.1016/j.aam.2017.01.002},
Key = {fds323833}
}

@article{fds323651,
Author = {Durrett, R and Fan, W-TL},
Title = {Genealogies in expanding populations},
Journal = {The Annals of Applied Probability},
Volume = {26},
Number = {6},
Pages = {3456-3490},
Publisher = {Institute of Mathematical Statistics},
Year = {2016},
Month = {December},
url = {http://dx.doi.org/10.1214/16-aap1181},
Doi = {10.1214/16-aap1181},
Key = {fds323651}
}

@article{fds323652,
Author = {Cox, JT and Durrett, R},
Title = {Evolutionary games on the torus with weak
selection},
Journal = {Stochastic Processes and Their Applications},
Volume = {126},
Number = {8},
Pages = {2388-2409},
Publisher = {Elsevier BV},
Year = {2016},
Month = {August},
url = {http://dx.doi.org/10.1016/j.spa.2016.02.004},
Doi = {10.1016/j.spa.2016.02.004},
Key = {fds323652}
}

@article{fds321819,
Author = {Ryser, MD and Worni, M and Turner, EL and Marks, JR and Durrett, R and Hwang, ES},
Title = {Outcomes of Active Surveillance for Ductal Carcinoma in
Situ: A Computational Risk Analysis.},
Journal = {Journal of the National Cancer Institute},
Volume = {108},
Number = {5},
Year = {2016},
Month = {May},
url = {http://dx.doi.org/10.1093/jnci/djv372},
Abstract = {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 progression. One strategy to reduce overtreatment is
active surveillance (AS), whereby DCIS is treated only upon
detection of invasive disease. Our goal was to perform a
quantitative evaluation of outcomes following an AS strategy
for DCIS.Age-stratified, 10-year disease-specific cumulative
mortality (DSCM) for AS was calculated using a computational
risk projection model based upon published estimates for
natural history parameters, and Surveillance, Epidemiology,
and End Results data for outcomes. AS projections were
compared with the DSCM for patients who received UC. To
quantify the propagation of parameter uncertainty, a 95%
projection range (PR) was computed, and sensitivity analyses
were performed.Under the assumption that AS cannot
outperform UC, the projected median differences in 10-year
DSCM between AS and UC when diagnosed at ages 40, 55, and 70
years were 2.6% (PR = 1.4%-5.1%), 1.5% (PR = 0.5%-3.5%), and
0.6% (PR = 0.0%-2.4), respectively. Corresponding median
numbers of patients needed to treat to avert one breast
cancer death were 38.3 (PR = 19.7-69.9), 67.3 (PR =
28.7-211.4), and 157.2 (PR = 41.1-3872.8), respectively.
Sensitivity analyses showed that the parameter with greatest
impact on DSCM was the probability of understaging invasive
cancer at diagnosis.AS could be a viable management strategy
for carefully selected DCIS patients, particularly among
older age groups and those with substantial competing
mortality risks. The effectiveness of AS could be markedly
improved by reducing the rate of understaging.},
Doi = {10.1093/jnci/djv372},
Key = {fds321819}
}

@article{fds243415,
Author = {Durrett, R and Foo, J and Leder, K},
Title = {Spatial Moran models, II: cancer initiation in spatially
structured tissue.},
Journal = {Journal of Mathematical Biology},
Volume = {72},
Number = {5},
Pages = {1369-1400},
Year = {2016},
Month = {April},
ISSN = {0303-6812},
url = {http://dx.doi.org/10.1007/s00285-015-0912-1},
Abstract = {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 investigation contributes to an
understanding of how the selective advantage of cancer cells
together with the rates of mutations driving cancer, impact
the process and timing of carcinogenesis. These results can
be used to give insights into tumor heterogeneity and the
"cancer field effect," the observation that a malignancy is
often surrounded by cells that have undergone premalignant
transformation.},
Doi = {10.1007/s00285-015-0912-1},
Key = {fds243415}
}

@article{fds243417,
Author = {Durrett, R and Zhang, Y},
Title = {Coexistence of grass, saplings and trees in the
Staver–Levin forest model},
Journal = {The Annals of Applied Probability},
Volume = {25},
Number = {6},
Pages = {3434-3464},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
Month = {December},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/14-aap1079},
Doi = {10.1214/14-aap1079},
Key = {fds243417}
}

@article{fds302176,
Author = {Talkington, A and Durrett, R},
Title = {Estimating Tumor Growth Rates In Vivo.},
Journal = {Bulletin of Mathematical Biology},
Volume = {77},
Number = {10},
Pages = {1934-1954},
Year = {2015},
Month = {October},
ISSN = {0092-8240},
url = {http://dx.doi.org/10.1007/s11538-015-0110-8},
Abstract = {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 and there are biases due to
the way the samples were ascertained, there is a clear sign
of exponential growth for the breast and liver cancers, and
a 2/3’s power law (surface growth) for the two
neurological cancers.},
Doi = {10.1007/s11538-015-0110-8},
Key = {fds302176}
}

@article{fds323653,
Author = {Varghese, C and Durrett, R},
Title = {Spatial networks evolving to reduce length},
Journal = {Journal of Complex Networks},
Volume = {3},
Number = {3},
Pages = {411-430},
Publisher = {Oxford University Press (OUP)},
Year = {2015},
Month = {September},
url = {http://dx.doi.org/10.1093/comnet/cnu044},
Doi = {10.1093/comnet/cnu044},
Key = {fds323653}
}

@article{fds243418,
Author = {Ryser, MD and Myers, ER and Durrett, R},
Title = {HPV clearance and the neglected role of stochasticity.},
Journal = {Plos Computational Biology},
Volume = {11},
Number = {3},
Pages = {e1004113},
Year = {2015},
Month = {March},
ISSN = {1553-734X},
url = {http://hdl.handle.net/10161/9545 Duke open
access},
Abstract = {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 infection. In this study, we combine
mechanistic mathematical models at the cellular level with
epidemiological data at the population level to disentangle
the respective roles of immune capacity and cell dynamics in
the clearing mechanism. Our results suggest that chance-in
form of the stochastic dynamics of basal stem cells-plays a
critical role in the elimination of HPV-infected cell
clones. In particular, we find that in immunocompetent
adolescents with cervical HPV infections, the immune
response may contribute less than 20% to virus clearance-the
rest is taken care of by the stochastic proliferation
dynamics in the basal layer. In HIV-negative individuals,
the contribution of the immune response may be
negligible.},
Doi = {10.1371/journal.pcbi.1004113},
Key = {fds243418}
}

@article{fds243416,
Author = {Durrett, R and Moseley, S},
Title = {Spatial Moran models I. Stochastic tunneling in the neutral
case},
Journal = {The Annals of Applied Probability},
Volume = {25},
Number = {1},
Pages = {104-115},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
Month = {February},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/13-aap989},
Doi = {10.1214/13-aap989},
Key = {fds243416}
}

@article{fds323654,
Author = {Magura, SR and Pong, VH and Durrett, R and Sivakoff,
D},
Title = {Two evolving social network models},
Journal = {Alea},
Volume = {12},
Number = {2},
Pages = {699-715},
Year = {2015},
Month = {January},
Abstract = {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 accepted. In version (ii) a new connection of
length ℓ' is accepted with probability minℓ/ℓ', 1. Our
second model is a dynamic version of preferential
attachment. Edges are chosen at random for deletion, then
one endpoint chosen at random connects to vertex z with
probability proportional to f(d(z)), where d(z) is the
degree of z, f(k) = θ(k+1)+(1-θ)(d+1), d is the average
degree. In words, this is a mixture of degree-proportional,
at random rewiring. The common feature of these models is
that they have stationary distributions that satisfy the
detailed balance condition, are given by explicit formulas.
In addition, the equilibrium of the first model is closely
related to long range percolation, of the second to the
configuration model of random graphs. As a result, we obtain
explicit results about the degree distribution,
connectivity, diameter for each model.},
Key = {fds323654}
}

@article{fds243413,
Author = {Durrett, R},
Title = {Spatial evolutionary games with small selection
coefficients},
Journal = {Electronic Journal of Probability},
Volume = {19},
Publisher = {Institute of Mathematical Statistics},
Year = {2014},
Month = {December},
url = {http://dx.doi.org/10.1214/EJP.v19-3621},
Abstract = {© 2015 University of Washington. All rights reserved. Here
we will use results of Cox, Durrett, and Perkins [56] for
voter model perturbations to study spatial evolutionary
games on ℤ<sup>d</sup>, d ≥ 3 when the interaction
kernel is finite range, symmetric, and has covariance matrix
σ<sup>2</sup>I. The games we consider have payoff matrices
of the form 1 + wG where 1 is matrix of all 1’s and w is
small and positive. Since our population size N = ∞, we
call our selection small rather than weak which usually
means w = O(1=N). The key to studying these games is the
fact that when the dynamics are suitably rescaled in space
and time they convergence to solutions of a reaction
diffusion equation (RDE). Inspired by work of Ohtsuki and
Nowak [28] and Tarnita et al [35, 36] we show that the
reaction term is the replicator equation for a modified game
matrix and the modifications of the game matrix depend on
the interaction kernel only through the values of two or
three simple probabilities for an associated coalescing
random walk. Two strategy games lead to an RDE with a cubic
nonlinearity, so we can describe the phase diagram
completely. Three strategy games lead to a pair of coupled
RDE, but using an idea from our earlier work [59], we are
able to show that if there is a repelling function for the
replicator equation for the modified game, then there is
coexistence in the spatial game when selection is small.
This enables us to prove coexistence in the spatial model in
a wide variety of examples including the behavior of four
evolutionary games that have recently been used in cancer
modeling.},
Doi = {10.1214/EJP.v19-3621},
Key = {fds243413}
}

@article{fds243414,
Author = {Aristotelous, AC and Durrett, R},
Title = {Fingering in Stochastic Growth Models},
Journal = {Experimental Mathematics},
Volume = {23},
Number = {4},
Pages = {465-474},
Publisher = {Informa UK Limited},
Year = {2014},
Month = {October},
ISSN = {1058-6458},
url = {http://dx.doi.org/10.1080/10586458.2014.947053},
Doi = {10.1080/10586458.2014.947053},
Key = {fds243414}
}

@article{fds243419,
Author = {Durrett, R and Zhang, Y},
Title = {Exact solution for a metapopulation version of Schelling's
model.},
Journal = {Proceedings of the National Academy of Sciences of the
United States of America},
Volume = {111},
Number = {39},
Pages = {14036-14041},
Year = {2014},
Month = {September},
ISSN = {0027-8424},
url = {http://dx.doi.org/10.1073/pnas.1414915111},
Abstract = {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 houses. There are ρNL red families and ρNL
blue families for some ρ < 1/2. Families are happy if there
are ≤ ρ(c)L families of the opposite type in their
neighborhood and unhappy otherwise. Each family moves to
each vacant house at rates that depend on their happiness at
their current location and that of their destination. Our
main result is that if neighborhoods are large, then there
are critical values ρ(b) < ρ(d) < ρ(c), so that for ρ <
ρ(b), the two types are distributed randomly in
equilibrium. When ρ > ρ(b), a new segregated equilibrium
appears; for ρ(b) < ρ < ρ(d), there is bistability, but
when ρ increases past ρ(d) the random state is no longer
stable. When ρ(c) is small enough, the random state will
again be the stationary distribution when ρ is close to
1/2. If so, this is preceded by a region of
bistability.},
Doi = {10.1073/pnas.1414915111},
Key = {fds243419}
}

@article{fds243421,
Author = {Aristotelous, AC and Durrett, R},
Title = {Chemical evolutionary games.},
Journal = {Theoretical Population Biology},
Volume = {93},
Pages = {1-13},
Year = {2014},
Month = {May},
ISSN = {0040-5809},
url = {http://dx.doi.org/10.1016/j.tpb.2014.02.001},
Abstract = {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 long term aim is to understand how the
details of the interactions in a system with n species and m
chemicals translate into the qualitative behavior of the
system. Here, we study two simple 2×2 games with two
chemicals and revisit the two and three species versions of
the one chemical colicin system studied earlier by Durrett
and Levin (1997). We find that in the 2×2 examples, the
behavior of our new spatial model can be predicted from that
of the mean field differential equation using ideas of
Durrett and Levin (1994). However, in the three species
colicin model, the system with diffusion does not have the
coexistence which occurs in the lattices model in which
sites interact with only their nearest neighbors.},
Doi = {10.1016/j.tpb.2014.02.001},
Key = {fds243421}
}

@article{fds243420,
Author = {Durrett, R and Liggett, T and Zhang, Y},
Title = {The contact process with fast voting},
Journal = {Electronic Journal of Probability},
Volume = {19},
Publisher = {Institute of Mathematical Statistics},
Year = {2014},
Month = {March},
url = {http://dx.doi.org/10.1214/EJP.v19-3021},
Abstract = {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→∞ of the critical value λc(θ) for
the existence of a nontrivial stationary distribution. In
d≥3, λc(θ)→1/(2dρd) where ρd is the probability a d
dimensional simple random walk does not return to its
starting point.In d=2, λc(θ)/log(θ)→1/4π, while in
d=1, λc(θ)/θ1/2 has lim inf≥1/2√ and lim sup<∞.The
lower bound might be the right answer, but proving this, or
even getting a reasonable upper bound, seems to be a
difficult problem.},
Doi = {10.1214/EJP.v19-3021},
Key = {fds243420}
}

@article{fds243422,
Author = {Shi, F and Mucha, PJ and Durrett, R},
Title = {Multiopinion coevolving voter model with infinitely many
phase transitions},
Journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter
Physics},
Volume = {88},
Number = {6},
Year = {2013},
Month = {December},
ISSN = {1539-3755},
url = {http://dx.doi.org/10.1103/PhysRevE.88.062818},
Doi = {10.1103/PhysRevE.88.062818},
Key = {fds243422}
}

@article{fds243423,
Author = {Varghese, C and Durrett, R},
Title = {Phase transitions in the quadratic contact process on
complex networks},
Journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter
Physics},
Volume = {87},
Number = {6},
Pages = {paper 062819},
Publisher = {American Physical Society (APS)},
Year = {2013},
Month = {June},
ISSN = {1539-3755},
url = {http://dx.doi.org/10.1103/PhysRevE.87.062819},
Abstract = {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 two 1's is required to effect a 0ï·1
change. We extend the study of the QCP, which so far has
been limited to lattices, to complex networks. We define two
versions of the QCP: vertex-centered (VQCP) and
edge-centered (EQCP) with birth events 1-0-1ï·1-1-1 and
1-1-0ï·1-1-1, respectively, where "-" represents an edge.
We investigate the effects of network topology by
considering the QCP on random regular, Erdos-Rényi, and
power-law random graphs. We perform mean-field calculations
as well as simulations to find the steady-state fraction of
occupied vertices as a function of the birth rate. We find
that on the random regular and Erdos-Rényi graphs, there is
a discontinuous phase transition with a region of
bistability, whereas on the heavy-tailed power-law graph,
the transition is continuous. The critical birth rate is
found to be positive in the former but zero in the latter.
© 2013 American Physical Society.},
Doi = {10.1103/PhysRevE.87.062819},
Key = {fds243423}
}

@article{fds243424,
Author = {Mode, CJ and Durrett, R and Klebaner, F and Olofsson,
P},
Title = {Applications of Stochastic Processes in Biology and
Medicine},
Journal = {International Journal of Stochastic Analysis},
Volume = {2013},
Pages = {1-2},
Publisher = {Hindawi Limited},
Year = {2013},
Month = {March},
ISSN = {2090-3332},
url = {http://dx.doi.org/10.1155/2013/790625},
Doi = {10.1155/2013/790625},
Key = {fds243424}
}

@article{fds243425,
Author = {Durrett, R},
Title = {Cancer Modeling: A Personal Perspective},
Journal = {Notices of the American Mathematical Society},
Volume = {60},
Number = {03},
Pages = {304-304},
Publisher = {American Mathematical Society (AMS)},
Year = {2013},
Month = {March},
ISSN = {0002-9920},
url = {http://dx.doi.org/10.1090/noti953},
Doi = {10.1090/noti953},
Key = {fds243425}
}

@article{fds243517,
Author = {Chatterjee, S and Durrett, R},
Title = {A first order phase transition in the threshold
??θ≥2
contact process on random ??r-regular
graphs and ??r-trees},
Journal = {Stochastic Processes and Their Applications},
Volume = {123},
Number = {2},
Pages = {561-578},
Publisher = {Elsevier BV},
Year = {2013},
Month = {February},
url = {http://dx.doi.org/10.1016/j.spa.2012.10.001},
Doi = {10.1016/j.spa.2012.10.001},
Key = {fds243517}
}

@article{fds243526,
Author = {Durrett, R},
Title = {Population genetics of neutral mutations in exponentially
growing cancer cell populations},
Journal = {The Annals of Applied Probability},
Volume = {23},
Number = {1},
Pages = {230-250},
Publisher = {Institute of Mathematical Statistics},
Year = {2013},
Month = {February},
url = {http://dx.doi.org/10.1214/11-AAP824},
Abstract = {In order to analyze data from cancer genome sequencing
projects, we need to be able to distinguish causative, or
"driver," mutations from "passenger" mutations that have no
selective effect. Toward this end, we prove results
concerning the frequency of neutural mutations in
exponentially growing multitype branching processes that
have been widely used in cancer modeling. Our results yield
a simple new population genetics result for the site
frequency spectrum of a sample from an exponentially growing
population. © Institute of Mathematical Statistics,
2013.},
Doi = {10.1214/11-AAP824},
Key = {fds243526}
}

@article{fds243516,
Author = {Danesh, K and Durrett, R and Havrilesky, LJ and Myers,
E},
Title = {A branching process model of ovarian cancer.},
Journal = {Journal of Theoretical Biology},
Volume = {314},
Pages = {10-15},
Year = {2012},
Month = {December},
url = {http://www.ncbi.nlm.nih.gov/pubmed/22959913},
Abstract = {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, we develop a branching process
model for ovarian cancer growth and progression accounting
for three cell populations: Primary (cells in the ovary or
fallopian tube), Peritoneal (viable cells in peritoneal
fluid), and Metastatic (cells implanted on other
intra-abdominal surfaces). Growth and migration parameters
were chosen to match results of clinical studies. Using
these values, our model predicts a window of opportunity of
2.9 years, indicating that one would have to screen at least
every other year to be effective. The model can be used to
inform future efforts in designing improved screening and
treatment strategies.},
Doi = {10.1016/j.jtbi.2012.08.025},
Key = {fds243516}
}

@article{fds243519,
Author = {Durrett, R and Gleeson, JP and Lloyd, AL and Mucha, PJ and Shi, F and Sivakoff, D and Socolar, JES and Varghese, C},
Title = {Graph fission in an evolving voter model.},
Journal = {Proc Natl Acad Sci U S A},
Volume = {109},
Number = {10},
Pages = {3682-3687},
Year = {2012},
Month = {March},
url = {http://www.ncbi.nlm.nih.gov/pubmed/22355142},
Abstract = {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 then, with probability 1 - α, one
imitates the opinion of the other; otherwise (i.e., with
probability α), the link between them is broken and one of
them makes a new connection to an individual chosen at
random (i) from those with the same opinion or (ii) from the
network as a whole. The evolution of the system stops when
there are no longer any discordant edges connecting
individuals with different opinions. Letting ρ be the
fraction of voters holding the minority opinion after the
evolution stops, we are interested in how ρ depends on α
and the initial fraction u of voters with opinion 1. In case
(i), there is a critical value α(c) which does not depend
on u, with ρ ≈ u for α > α(c) and ρ ≈ 0 for
α < α(c). In case (ii), the transition point α(c)(u)
depends on the initial density u. For α > α(c)(u),
ρ ≈ u, but for α < α(c)(u), we have
ρ(α,u) = ρ(α,1/2). Using simulations and approximate
calculations, we explain why these two nearly identical
models have such dramatically different phase
transitions.},
Doi = {10.1073/pnas.1200709109},
Key = {fds243519}
}

@article{fds243518,
Author = {Durrett, R and Remenik, D},
Title = {Evolution of dispersal distance},
Journal = {Journal of Mathematical Biology},
Volume = {64},
Number = {4},
Pages = {657-666},
Year = {2012},
ISSN = {0303-6812},
url = {http://dx.doi.org/10.1007/s00285-011-0444-2},
Abstract = {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 is a trade off between
survivability and dispersal range. Ezoe (J Theor Biol
190:287-293, 1998) and Levin and Muller-Landau (Evol Ecol
Res 2:409-435, 2000) approached this problem using models
that were essentially deterministic, and used calculus to
find optimal dispersal parameters. Here we follow Hiebeler
(Theor Pop Biol 66:205-218, 2004) and use a stochastic
spatial model to study the competition of different
dispersal strategies. Most work on such systems is done by
simulation or nonrigorous methods such as pair
approximation. Here, we use machinery developed by Cox et
al. (Voter model perturbations and reaction diffusion
equations 2011) to rigorously and explicitly compute
evolutionarily stable strategies. © 2011
Springer-Verlag.},
Doi = {10.1007/s00285-011-0444-2},
Key = {fds243518}
}

@article{fds243520,
Author = {Chatterjee, S and Durrett, R},
Title = {Asymptotic behavior of Aldous’ gossip process},
Journal = {The Annals of Applied Probability},
Volume = {21},
Number = {6},
Pages = {2447-2482},
Publisher = {Institute of Mathematical Statistics},
Year = {2011},
Month = {December},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/10-aap750},
Abstract = {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 and at rate N-α to a site chosen
at random from the torus. We will be interested in the case
in which α &lt; 3, where the long range transmission
significantly accelerates the time at which everyone knows
the information. We prove three results that precisely
describe the spread of information in a slightly simplified
model on the real torus. The time until everyone knows the
information is asymptotically T = (2 - 2α/3)Nα/3 logN. If
ρs is the fraction of the population who know the
information at time s and ε is small then, for large N, the
time until ρs reaches ε is T (ε) ~ T + Nα/3 log(3ε/M),
where M is a random variable determined by the early spread
of the information. The value of ρs at time s = T (1/3) +
tNα/3 is almost a deterministic function h(t) which
satisfies an odd looking integro-differential equation. The
last result confirms a heuristic calculation of Aldous. ©
Institute of Mathematical Statistics, 2011.},
Doi = {10.1214/10-aap750},
Key = {fds243520}
}

@article{fds243521,
Author = {Durrett, R and Remenik, D},
Title = {Brunet-derrida particle systems, free boundary problems and
wiener-hopf equations},
Journal = {The Annals of Probability},
Volume = {39},
Number = {6},
Pages = {2043-2078},
Publisher = {Institute of Mathematical Statistics},
Year = {2011},
Month = {November},
ISSN = {0091-1798},
url = {http://dx.doi.org/10.1214/10-AOP601},
Abstract = {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 process associated to the system
converges in distribution to a deterministic measure-valued
process whose densities solve a free boundary
integro-differential equation. We also show that this
equation has a unique traveling wave solution traveling at
speed c or no such solution depending on whether c ≥ a or
c>a,wherea is the asymptotic speed of the branching random
walk obtained by ignoring the removal of the leftmost
particles in our process. The traveling wave solutions
correspond to solutions of Wiener-Hopf equations. © 2011
Institute of Mathematical Statistics.},
Doi = {10.1214/10-AOP601},
Key = {fds243521}
}

@article{fds243523,
Author = {Chatterjee, S and Durrett, R},
Title = {Persistence of activity in threshold contact processes, an
“Annealed approximation” of random Boolean
networks},
Journal = {Random Structures & Algorithms},
Volume = {39},
Number = {2},
Pages = {228-246},
Publisher = {WILEY},
Year = {2011},
Month = {September},
ISSN = {1042-9832},
MRCLASS = {60K35 (05C80)},
MRNUMBER = {2850270},
url = {http://dx.doi.org/10.1002/rsa.20357},
Abstract = {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 $p = {\rm the}$ expected fraction of 1'sin
the Boolean functions at each node. Following a standard
practice in thephysics literature, we use a threshold
contact process on a random graph on n nodes, in which each
node has in degree r, to approximate its dynamics. We show
that if $r\ge 3$ and $r \cdot 2p(1-p)&gt;1$, then the
threshold contact process persists for a long time, which
correspond to chaotic behavior of the Boolean network.
Unfortunately, we are only able to prove the persistence
time is $\ge \exp(cn^{b(p)})$ with $b(p)&gt;0$ when $r\cdot 2p(1-p)&gt; 1$, and $b(p)=1$ when $(r-1)\cdot 2p(1-p)&gt;1$.
© 2011 Wiley Periodicals, Inc..},
Doi = {10.1002/rsa.20357},
Key = {fds243523}
}

@article{fds243529,
Author = {Durrett, R and Foo, J and Leder, K and Mayberry, J and Michor,
F},
Title = {Intratumor heterogeneity in evolutionary models of tumor
progression.},
Journal = {Genetics},
Volume = {188},
Number = {2},
Pages = {461-477},
Year = {2011},
Month = {June},
ISSN = {0016-6731},
url = {http://dx.doi.org/10.1534/genetics.110.125724},
Abstract = {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, at the time of diagnosis most
tumors show a striking amount of intratumor heterogeneity in
all measurable phenotypes; such heterogeneity has
implications for diagnosis, treatment efficacy, and the
identification of drug targets. An understanding of the
extent and evolution of intratumor heterogeneity is
therefore of direct clinical importance. In this article, we
investigate the evolutionary dynamics of heterogeneity
arising during exponential expansion of a tumor cell
population, in which heritable alterations confer random
fitness changes to cells. We obtain analytical estimates for
the extent of heterogeneity and quantify the effects of
system parameters on this tumor trait. Our work contributes
to a mathematical understanding of intratumor heterogeneity
and is also applicable to organisms like bacteria,
agricultural pests, and other microbes.},
Doi = {10.1534/genetics.110.125724},
Key = {fds243529}
}

@article{fds243522,
Author = {Durrett, R and Mayberry, J},
Title = {Traveling waves of selective sweeps},
Journal = {The Annals of Applied Probability},
Volume = {21},
Number = {2},
Pages = {699-744},
Publisher = {Institute of Mathematical Statistics},
Year = {2011},
Month = {April},
ISSN = {1050-5164},
MRCLASS = {60J85 (92D25)},
MRNUMBER = {2807971},
url = {http://dx.doi.org/10.1214/10-AAP721},
Abstract = {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 exponentially growing population. To
better understand this process, Beerenwinkel et al. [PLoS
Comput. Biol. 3 (2007) 2239- 2246] consider a Wright-Fisher
model in which cells from an exponentially growing
population accumulate advantageous mutations. Simulations
show a traveling wave in which the time of the first k-fold
mutant, Tk, is approximately linear in k and heuristics are
used to obtain formulas for ETk. Here, we consider the
analogous problem for the Moran model and prove that as the
mutation rate μ →0, Tk ∼ ck log(1/μ), where the ck can
be computed explicitly. In addition, we derive a limiting
result on a log scale for the size of Xk(t) = the number of
cells with k mutations at time t . © Institute of
Mathematical Statistics, 2011.},
Doi = {10.1214/10-AAP721},
Key = {fds243522}
}

@article{fds243527,
Author = {Cox, JT and Durrett, R and Perkins, E},
Title = {Voter model perturbations and reaction diffusion
equations},
Journal = {Asterique},
Volume = {349},
Pages = {1-113},
Year = {2011},
url = {http://arxiv.org/abs/math/1103.1676},
Key = {fds243527}
}

@article{fds243528,
Author = {Durrett, R and Chatterjee, S},
Title = {Persistence of activity in random Boolean
networks},
Journal = {Random Structures and Algorithms},
Volume = {39},
Pages = {228-246},
Year = {2011},
Key = {fds243528}
}

@article{fds243524,
Author = {Durrett, R and Mayberry, J},
Title = {Evolution in predator–prey systems},
Journal = {Stochastic Processes and Their Applications},
Volume = {120},
Number = {7},
Pages = {1364-1392},
Publisher = {Elsevier BV},
Year = {2010},
Month = {July},
ISSN = {0304-4149},
MRCLASS = {92D25 (60J60 60K35 92D15)},
MRNUMBER = {2639750 (2011d:92087)},
url = {http://dx.doi.org/10.1016/j.spa.2010.03.011},
Abstract = {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 to 0, the system does not
exhibit long term prey coexistence and the trait of the
resident prey type converges to the solution of an ODE. When
only the predators are allowed to evolve, coexistence of
predators occurs. In this case, depending on the parameters
being varied, we see that (i) the number of coexisting
predators remains tight and the differences in traits from a
reference species converge in distribution to a limit, or
(ii) the number of coexisting predators tends to infinity,
and we calculate the asymptotic rate at which the traits of
the least and most "fit" predators in the population
increase. This last result is obtained by comparison with a
branching random walk killed to the left of a linear
boundary and a finite branchingselection particle system. ©
2010 Elsevier B.V. All rights reserved.},
Doi = {10.1016/j.spa.2010.03.011},
Key = {fds243524}
}

@article{fds243557,
Author = {Durrett, R},
Title = {Some features of the spread of epidemics and information on
a random graph.},
Journal = {Proceedings of the National Academy of Sciences of the
United States of America},
Volume = {107},
Number = {10},
Pages = {4491-4498},
Year = {2010},
Month = {March},
ISSN = {0027-8424},
url = {http://dx.doi.org/10.1073/pnas.0914402107},
Abstract = {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 how their behavior on networks differs
from that in homogeneously mixing populations or on regular
lattices of the type commonly used in ecological
models.},
Doi = {10.1073/pnas.0914402107},
Key = {fds243557}
}

@article{fds243525,
Author = {Durrett, R and Moseley, S},
Title = {Evolution of resistance and progression to disease during
clonal expansion of cancer.},
Journal = {Theoretical Population Biology},
Volume = {77},
Number = {1},
Pages = {42-48},
Year = {2010},
Month = {February},
ISSN = {0040-5809},
url = {http://dx.doi.org/10.1016/j.tpb.2009.10.008},
Abstract = {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 mutations. We prove results about the
distribution of the first time when k mutations have
accumulated in some cell, and about the growth of the number
of type-k cells. We show that our results can be used to
derive the previous results about a tumor grown to a fixed
size.},
Doi = {10.1016/j.tpb.2009.10.008},
Key = {fds243525}
}

@article{fds243530,
Author = {Durrett, R and Foo, J and Leder, K and Mayberry, J and Michor,
F},
Title = {Evolutionary dynamics of tumor progression with random
fitness values},
Journal = {Theoretical Population Biology},
Volume = {78},
Number = {1},
Pages = {54-66},
Year = {2010},
ISSN = {0040-5809},
url = {http://dx.doi.org/10.1016/j.tpb.2010.05.001},
Abstract = {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, however, do not change the
phenotype of the cell or even decrease cellular fitness.
While much experimental effort is being devoted to the
identification of the functional effects of individual
mutations, mathematical modeling of tumor progression
generally considers constant fitness increments as mutations
are accumulated. In this paper we study a mathematical model
of tumor progression with random fitness increments. We
analyze a multi-type branching process in which cells
accumulate mutations whose fitness effects are chosen from a
distribution. We determine the effect of the fitness
distribution on the growth kinetics of the tumor. This work
contributes to a quantitative understanding of the
accumulation of mutations leading to cancer. © 2010
Elsevier Inc.},
Doi = {10.1016/j.tpb.2010.05.001},
Key = {fds243530}
}

@article{fds243515,
Author = {Chatterjee, S and Durrett, R},
Title = {Contact processes on random graphs with power law degree
distributions have critical value 0},
Journal = {The Annals of Probability},
Volume = {37},
Number = {6},
Pages = {2332-2356},
Publisher = {Institute of Mathematical Statistics},
Year = {2009},
Month = {November},
ISSN = {0091-1798},
url = {http://dx.doi.org/10.1214/09-AOP471},
Abstract = {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. Physicists seem to regard this as an established
fact, since the result has recently been generalized to
bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl.
Acad. Sci. USA 105 (2008) 1399-1404]. Here, we show that the
critical value λc is zero for any value of α>3, and the
contact process starting from all vertices infected, with a
probability tending to 1 as n →∞, maintains a positive
density of infected sites for time at least exp(n1-δ) for
any δ>0. Using the last result, together with the contact
process duality, we can establish the existence of a
quasi-stationary distribution in which a randomly chosen
vertex is occupied with probability ρ(λ). It is expected
that ρ(λ)~ Cλβ as λ → 0. Here we show that α - 1 ≤
β ≤ 2α - 3, and so β>2 for α>3. Thus even though the
graph is locally tree-like, β does not take the mean field
critical value β = 1. © Institute of Mathematical
Statistics, 2009.},
Doi = {10.1214/09-AOP471},
Key = {fds243515}
}

@article{fds304477,
Author = {Chan, B and Durrett, R and Lanchier, N},
Title = {Coexistence for a multitype contact process with
seasons},
Journal = {The Annals of Applied Probability},
Volume = {19},
Number = {5},
Pages = {1921-1943},
Publisher = {Institute of Mathematical Statistics},
Year = {2009},
Month = {October},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/09-aap599},
Abstract = {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 of the parameters for which both
species can coexist when their dispersal range is large
enough. Numerical simulations also suggest that three
species can coexist in the presence of two seasons. This
contrasts with the long-term behavior of the
time-homogeneous multitype contact process for which the
species with the higher birth rate outcompetes the other
species when the death rates are equal. © Institute of
Mathematical Statistics, 2009.},
Doi = {10.1214/09-aap599},
Key = {fds304477}
}

@article{fds243539,
Author = {Durrett, R and Remenik, D},
Title = {Chaos in a spatial epidemic model},
Journal = {The Annals of Applied Probability},
Volume = {19},
Number = {4},
Pages = {1656-1685},
Publisher = {Institute of Mathematical Statistics},
Year = {2009},
Month = {August},
ISSN = {1050-5164},
MRCLASS = {60K35 (37D45 37N25 60J10 92D30)},
MRNUMBER = {2538084 (2010k:60322)},
url = {http://dx.doi.org/10.1214/08-aap581},
Abstract = {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-dimensional lattice and torus, with d = 2. On
the finite graphs with global dispersal or with a dispersal
radius that grows with the number of sites, we prove
convergence to a dynamical system that is chaotic for some
parameter values. We conjecture that on the infinite lattice
with a fixed finite dispersal distance, distant parts of the
lattice oscillate out of phase so there is a unique
nontrivial stationary distribution. © Institute of
Mathematical Statistics, 2009.},
Doi = {10.1214/08-aap581},
Key = {fds243539}
}

@article{fds243555,
Author = {Durrett, R},
Title = {Coexistence in stochastic spatial models},
Journal = {The Annals of Applied Probability},
Volume = {19},
Number = {2},
Pages = {477-496},
Publisher = {Institute of Mathematical Statistics},
Year = {2009},
Month = {April},
ISSN = {1050-5164},
MRCLASS = {60K35 (92D25)},
MRNUMBER = {MR2521876 (2010g:60213)},
url = {http://dx.doi.org/10.1214/08-aap590},
Abstract = {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 be determined by examining the
mean-field ODE. There are a number of rigorous results in
support of this picture, but we will state nine challenging
and important open problems, most of which date from the
1990's. © Institute of Mathematical Statistics,
2009.},
Doi = {10.1214/08-aap590},
Key = {fds243555}
}

@article{fds243556,
Author = {Durrett, R and Schmidt, D and Schweinsberg, J},
Title = {A waiting time problem arising from the study of multi-stage
carcinogenesis},
Journal = {The Annals of Applied Probability},
Volume = {19},
Number = {2},
Pages = {676-718},
Publisher = {Institute of Mathematical Statistics},
Year = {2009},
Month = {April},
ISSN = {1050-5164},
MRCLASS = {60J80 (60J25 60K40 92C50)},
MRNUMBER = {MR2521885 (2010f:60243)},
url = {http://dx.doi.org/10.1214/08-aap559},
Abstract = {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 larger m are needed for colon cancer and
other diseases where a sequence of mutations leads to cells
with uncontrolled growth. © Institute of Mathematical
Statistics, 2009.},
Doi = {10.1214/08-aap559},
Key = {fds243556}
}

@article{fds243512,
Author = {Durrett, R and Schmidt, D},
Title = {Reply to Michael Behe},
Journal = {Genetics},
Volume = {181},
Number = {2},
Pages = {821-822},
Publisher = {Genetics Society of America},
Year = {2009},
Month = {February},
ISSN = {0016-6731},
url = {http://dx.doi.org/10.1534/genetics.109.100800},
Doi = {10.1534/genetics.109.100800},
Key = {fds243512}
}

@article{fds243538,
Author = {Durrett, R and Popovic, L},
Title = {Degenerate diffusions arising from gene duplication
models},
Journal = {The Annals of Applied Probability},
Volume = {19},
Number = {1},
Pages = {15-48},
Publisher = {Institute of Mathematical Statistics},
Year = {2009},
Month = {February},
ISSN = {1050-5164},
MRNUMBER = {MR2521876 (2010g:60213)},
url = {http://dx.doi.org/10.1214/08-aap530},
Abstract = {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 diffusions are two and
six-dimensional, respectively, we show in each case that the
diffusion stays close to a curve. Using ideas of
Katzenberger [Ann. Probab. 19 (1991) 1587-1628] we show that
one-dimensional projections converge to diffusion processes,
and we obtain asymptotics for the time to loss of one gene
copy. As a corollary we find that the probability of
subfunctionalization decreases exponentially fast as the
population size increases. This rigorously confirms a result
Ward and Durrett [Theor. Pop. Biol. 66 (2004) 93-100] found
by simulation that the likelihood of subfunctionalization
for gene duplicates decays exponentially fast as the
population size increases. © Institute of Mathematical
Statistics, 2009.},
Doi = {10.1214/08-aap530},
Key = {fds243538}
}

@article{fds243514,
Author = {Wu, F and Eannetta, NT and Xu, Y and Durrett, R and Mazourek, M and Jahn,
MM and Tanksley, SD},
Title = {A COSII genetic map of the pepper genome provides a detailed
picture of synteny with tomato and new insights into recent
chromosome evolution in the genus Capsicum},
Journal = {Tag Theoretical and Applied Genetics},
Volume = {118},
Number = {7},
Pages = {1279-1293},
Year = {2009},
ISSN = {0040-5752},
url = {http://dx.doi.org/10.1007/s00122-009-0980-y},
Abstract = {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 was inferred in the pepper map
via pepper-tomato synteny, bringing the total orthologous
markers in the pepper genome to 587. While pepper maps have
been previously reported, this is the first complete map in
the sense that all markers could be placed in 12 linkage
groups corresponding to the 12 chromosomes. The map
presented herein is relevant to the genomes of cultivated C.
annuum and wild C. annuum (as well as related Capsicum
species) which differ by a reciprocal chromosome
translocation. This map is also unique in that it is largely
based on COSII markers, which permits the inference of a
detailed syntenic relationship between the pepper and tomato
genomes-shedding new light on chromosome evolution in the
Solanaceae. Since divergence from their last common ancestor
is approximately 20 million years ago, the two genomes have
become differentiated by a minimum number of 19 inversions
and 6 chromosome translocations, as well as numerous
putative single gene transpositions. Nevertheless, the two
genomes share 35 conserved syntenic segments (CSSs) within
which gene/marker order is well preserved. The high
resolution COSII synteny map described herein provides a
platform for cross-reference of genetic and genomic
information (including the tomato genome sequence) between
pepper and tomato and therefore will facilitate both applied
and basic research in pepper. © 2009 Springer-Verlag.},
Doi = {10.1007/s00122-009-0980-y},
Key = {fds243514}
}

@article{fds243542,
Author = {Chan, B and Durrett, R and Lanchier, N},
Title = {Coexistence in a particle system with seasons},
Journal = {Ann. Appl. Probab.},
Volume = {19},
Number = {5},
Pages = {1921-1943},
Year = {2009},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/09-AAP599},
Abstract = {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 of the parameters for which both
species can coexist when their dispersal range is large
enough. Numerical simulations also suggest that three
species can coexist in the presence of two seasons. This
contrasts with the long-term behavior of the
time-homogeneous multitype contact process for which the
species with the higher birth rate outcompetes the other
species when the death rates are equal. © Institute of
Mathematical Statistics, 2009.},
Doi = {10.1214/09-AAP599},
Key = {fds243542}
}

@article{fds243554,
Author = {Durrett, R and Lanchier, N},
Title = {Coexistence in host–pathogen systems},
Journal = {Stochastic Processes and Their Applications},
Volume = {118},
Number = {6},
Pages = {1004-1021},
Publisher = {Elsevier BV},
Year = {2008},
Month = {June},
ISSN = {0304-4149},
MRCLASS = {60K35 (92D25)},
MRNUMBER = {MR2418255 (2009g:60131)},
url = {http://dx.doi.org/10.1016/j.spa.2007.07.008},
Abstract = {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 specialist pathogen can coexist with
a second species. Our calculations suggest that it is
possible for two hosts with specialist pathogens to coexist
but it is not possible for a host with a specialist
mutualist to coexist with a second species.},
Doi = {10.1016/j.spa.2007.07.008},
Key = {fds243554}
}

@article{fds243553,
Author = {Durrett, R and Restrepo, M},
Title = {One-dimensional stepping stone models, sardine genetics and
Brownian local time},
Journal = {The Annals of Applied Probability},
Volume = {18},
Number = {1},
Pages = {334-358},
Publisher = {Institute of Mathematical Statistics},
Year = {2008},
Month = {February},
ISSN = {1050-5164},
MRCLASS = {60K35 (60J55 92D10)},
MRNUMBER = {MR2380901 (2008j:60229)},
url = {http://dx.doi.org/10.1214/07-aap451},
Abstract = {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 if Mv/L and L/K2 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. © Institute of
Mathematical Statistics, 2008.},
Doi = {10.1214/07-aap451},
Key = {fds243553}
}

@article{fds243540,
Author = {Berestycki, N and Durrett, R},
Title = {Limiting behavior for the distance of a random
walk},
Journal = {Electronic Journal of Probability},
Volume = {13},
Pages = {374-395},
Publisher = {Institute of Mathematical Statistics},
Year = {2008},
Month = {January},
ISSN = {1083-6489},
MRCLASS = {60G50 (60C05 60J10)},
MRNUMBER = {MR2386737 (2009d:60130)},
url = {http://dx.doi.org/10.1214/EJP.v13-490},
Abstract = {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: the distance from the starting
point of the walk has a phase transition from a linear
regime to a sublinear regime at time n/2. Here, we study the
examples of random 3-regular graphs, random adjacent
transpositions, and riffle shuffles. In the case of a random
3-regular graph, there is a phase transition where the speed
changes from 1/3 to 0 at time 3 log2 n. A similar result is
proved for riffle shuffles, where the speed changes from 1
to 0 at time log2 n. Both these changes occur when a
distance equal to the average diameter of the graph is
reached. However in the case of random adjacent
transpositions, the behavior is more complex. We find that
there is no phase transition, even though the distance has
different scalings in three different regimes. © 2008
Applied Probability Trust.},
Doi = {10.1214/EJP.v13-490},
Key = {fds243540}
}

@article{fds323655,
Author = {Chung, KL and Durrett, R},
Title = {Downcrossings and local time},
Pages = {585-587},
Booktitle = {Selected Works of Kai Lai Chung},
Publisher = {World Scientific},
Year = {2008},
Month = {January},
ISBN = {9789812833853},
url = {http://dx.doi.org/10.1142/9789812833860_0037},
Abstract = {© 2008 by World Scientific Publishing Co. Pte. Ltd. All
Rights Reserved. Let formula presented be the standard
Brownian motion with all paths continuous. Let formula
presented be the maximum process and formula presented be
reflecting Brownian motion. If formula presented is the
number of є to 0 times Y crosses down from e to 0 before
time t, then it was Paul Lévy's idea thatformula
presented},
Doi = {10.1142/9789812833860_0037},
Key = {fds323655}
}

@article{fds243536,
Author = {Huerta-Sanchez, E and Durrett, R and Bustamante,
CD},
Title = {Population genetics of polymorphism and divergence under
fluctuating selection},
Journal = {Genetics},
Volume = {178},
Number = {1},
Pages = {325-337},
Year = {2008},
ISSN = {0016-6731},
url = {http://dx.doi.org/10.1534/genetics.107.073361},
Abstract = {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 method uses classical diffusion
approximations to model temporal fluctuations in the
selection coefficients to find the expected distribution of
mutation frequencies in the population. Using the Poisson
random-field setting we derive the site-frequency spectrum
(SFS) for three different models of fluctuating selection.
We find that the general effect of fluctuating selection is
to produce a more "U"-shaped site-frequency spectrum with an
excess of high-frequency derived mutations at the expense of
middle-frequency variants. We present likelihood-ratio
tests, comparing the fluctuating selection models to the
neutral model using SFS data, and use Monte Carlo
simulations to assess their power. We find that we have
sufficient power to reject a neutral hypothesis using
samples on the order of a few hundred SNPs and a sample size
of ∼20 and power to distinguish between selection that
varies in time and constant selection for a sample of size
20. We also find that fluctuating selection increases the
probability of fixation of selected sites even if, on
average, there is no difference in selection among a pair of
alleles segregating at the locus. Fluctuating selection
will, therefore, lead to an increase in the ratio of
divergence to polymorphism similar to that observed under
positive directional selection. Copyright © 2008 by the
Genetics Society of America.},
Doi = {10.1534/genetics.107.073361},
Key = {fds243536}
}

@article{fds243541,
Author = {Durrett, R and Schmidt, D},
Title = {Waiting for two mutations: With applications to regulatory
sequence evolution and the limits of Darwinian
evolution},
Journal = {Genetics},
Volume = {180},
Number = {3},
Pages = {1501-1509},
Year = {2008},
ISSN = {0016-6731},
url = {http://dx.doi.org/10.1534/genetics.107.082610},
Abstract = {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 mutant phenotype becomes fixed in the
population. In this article we apply these results to obtain
insights into regulatory sequence evolution in Drosophila
and humans. In particular, we examine the waiting time for a
pair of mutations, the first of which inactivates an
existing transcription factor binding site and the second of
which creates a new one. Consistent with recent experimental
observations for Drosophila, we find that a few million
years is sufficient, but for humans with a much smaller
effective population size, this type of change would take
&gt;100 million years. In addition, we use these results to
expose flaws in some of Michael Behe's arguments concerning
mathematical limits to Darwinian evolution. Copyright ©
2008 by the Genetics Society of America.},
Doi = {10.1534/genetics.107.082610},
Key = {fds243541}
}

@article{fds243551,
Author = {Durrett, R and Zähle, I},
Title = {On the width of hybrid zones},
Journal = {Stochastic Processes and Their Applications},
Volume = {117},
Number = {12},
Pages = {1751-1763},
Publisher = {Elsevier BV},
Year = {2007},
Month = {December},
ISSN = {0304-4149},
MRCLASS = {60K35 (60J70 92D25)},
MRNUMBER = {MR2437727 (2010d:60215)},
url = {http://dx.doi.org/10.1016/j.spa.2006.05.017},
Abstract = {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 investigate a stochastic spatial
model and prove the existence of a stationary distribution
that represents the hybrid zone in equilibrium. We calculate
the width of the hybrid zone, which agrees with the PDE
formula only in dimensions d ≥ 3. Our results also give
insight into properties of hybrid zones in patchy
environments. © 2007 Elsevier Ltd. All rights
reserved.},
Doi = {10.1016/j.spa.2006.05.017},
Key = {fds243551}
}

@article{fds243552,
Author = {Durrett, R and Jung, P},
Title = {Two phase transitions for the contact process on small
worlds},
Journal = {Stochastic Processes and Their Applications},
Volume = {117},
Number = {12},
Pages = {1910-1927},
Publisher = {Elsevier BV},
Year = {2007},
Month = {December},
ISSN = {0304-4149},
MRCLASS = {60K35 (82C22)},
MRNUMBER = {MR2437735 (2009h:60162)},
url = {http://dx.doi.org/10.1016/j.spa.2007.03.003},
Abstract = {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 individual's interactions at school, at
work, or in social situations introduce long-range
connections. However, this change dramatically alters the
behavior of the contact process, producing two phase
transitions. We establish this by relating the small world
to an infinite "big world" graph where the contact process
behavior is similar to the contact process on a tree. We
then consider the contact process on a slightly modified
small world model in order to show that its behavior is
decidedly different from that of the contact process on a
tree. © 2007 Elsevier Ltd. All rights reserved.},
Doi = {10.1016/j.spa.2007.03.003},
Key = {fds243552}
}

@article{fds243537,
Author = {York, TL and Durrett, R and Nielsen, R},
Title = {Dependence of paracentric inversion rate on tract
length.},
Journal = {Bmc Bioinformatics},
Volume = {8},
Pages = {115},
Year = {2007},
Month = {April},
ISSN = {1471-2105},
url = {http://dx.doi.org/10.1186/1471-2105-8-115},
Abstract = {BACKGROUND:We 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. RESULTS:We apply the method to data from
Drosophila melanogaster and D. yakuba. We find that
pericentric inversions occur at a much lower rate compared
to paracentric inversions. The average paracentric inversion
tract length is approx. 4.8 Mb with small inversions being
more frequent than large inversions. If the two breakpoints
defining a paracentric inversion tract are uniformly and
independently distributed over chromosome arms there will be
more short tract-length inversions than long; we find an
even greater preponderance of short tract lengths than this
would predict. Thus there appears to be a correlation
between the positions of breakpoints which favors shorter
tract lengths. CONCLUSION:The method developed in this paper
provides the first statistical estimator for estimating the
distribution of inversion tract lengths from marker data.
Application of this method for a number of data sets may
help elucidate the relationship between the length of an
inversion and the chance that it will get
accepted.},
Doi = {10.1186/1471-2105-8-115},
Key = {fds243537}
}

@article{fds243550,
Author = {Durrett, R and Schmidt, D},
Title = {Waiting for regulatory sequences to appear},
Journal = {The Annals of Applied Probability},
Volume = {17},
Number = {1},
Pages = {1-32},
Publisher = {Institute of Mathematical Statistics},
Year = {2007},
Month = {February},
ISSN = {1050-5164},
MRCLASS = {92D10 (60C05 60F05 92D20)},
MRNUMBER = {MR2292578 (2007j:92034)},
url = {http://dx.doi.org/10.1214/105051606000000619},
Abstract = {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. © Institute of
Mathematical Statistics, 2007.},
Doi = {10.1214/105051606000000619},
Key = {fds243550}
}

@article{fds183998,
Author = {R. Durrett and Iljana Zahle},
Title = {On the width of hybrid zones},
Journal = {Stochastic Processes and their Applications},
Volume = {117},
Pages = {1751--1763},
Year = {2007},
ISSN = {0304-4149},
MRCLASS = {60K35 (60J70 92D25)},
MRNUMBER = {2437727 (2010d:60215)},
url = {http://www.ams.org/mathscinet-getitem?mr=2010d:60215},
Key = {fds183998}
}

@article{fds243531,
Author = {Huerta-Sanchez, E and Durrett, R},
Title = {Wagner's canalization model},
Journal = {Theoretical Population Biology},
Volume = {71},
Number = {2},
Pages = {121-130},
Year = {2007},
ISSN = {0040-5809},
url = {http://dx.doi.org/10.1016/j.tpb.2006.10.006},
Abstract = {Wagner (1996, Does evolutionary plasticity evolve? Evolution
50, 1008-1023.) and Siegal and Bergman (2002, Waddington's
canalization revisited: Developmental stability and
evolution. Proc. Natl. Acad. Sci. USA 99, 10528-10532.) have
studied a simple model of the evolution of a network of N
genes, in order to explain the observed phenomenon that
systems evolve to be robust. These authors primarily
considered the case N = 10 and used simulations to reach
their conclusions. Here we investigate this model in more
detail, considering systems of different sizes with and
without recombination, and with selection for convergence
instead of to a specified limit. For the simpler
evolutionary model lacking recombination, we analyze the
system as a neutral network. This allows us to describe the
equilibrium distribution networks within genotype space. Our
results show that, given a sufficiently large population
size, the qualitative observation that systems evolve to be
robust, is itself robust, as it does not depend on the
details of the model. In simple terms, robust systems have
more viable offspring, so the evolution of robustness is
merely selection for increased fecundity, an observation
that is well known in the theory of neutral networks. ©
2006 Elsevier Inc. All rights reserved.},
Doi = {10.1016/j.tpb.2006.10.006},
Key = {fds243531}
}

@article{fds243533,
Author = {De, A and Durrett, R},
Title = {Spatial structure of the human population contributes to the
slow decay of linkage diseqeuilibrium and shifts the site
frequency spectrum},
Journal = {Genetics},
Volume = {176},
Number = {2},
Pages = {969-981},
Year = {2007},
ISSN = {0016-6731},
url = {http://dx.doi.org/10.1534/genetics.107.071464},
Abstract = {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 samples from a small geographical
region commonly used in genetic studies of humans and
Drosophila, there is a shift of the frequency spectrum that
decreases the number of low-frequency-derived alleles and
skews the distribution of statistics of Tajima, Fu and Li,
and Fay and Wu. Stepping-stone spatial structure also
changes the two-locus sampling distribution and increases
both linkage disequilibrium and the probability that two
sites are perfectly correlated. This may cause a false
prediction of cold spots of recombination and may confuse
haplotype tests that compute probabilities on the basis of a
homogeneously mixing population. Copyright © 2007 by the
Genetics Society of America.},
Doi = {10.1534/genetics.107.071464},
Key = {fds243533}
}

@article{fds243535,
Author = {Sainudiin, R and Clark, AG and Durrett, RT},
Title = {Simple models of genomic variation in human SNP
density},
Journal = {Bmc Genomics},
Volume = {8},
Year = {2007},
ISSN = {1471-2164},
url = {http://dx.doi.org/10.1186/1471-2164-8-146},
Abstract = {Background: Descriptive 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. Results: Using empirical estimates of
recombination rate across the human genome and the observed
SNP density distribution, we produce a maximum likelihood
estimate of the genomic heterogeneity in the scaled mutation
rate θ. Such models produce significantly better fits to
the observed SNP density distribution than those that ignore
the empirically observed recombinational heterogeneities.
Conclusion: Accounting for mutational and recombinational
heterogeneities can allow for empirically sound null
distributions in genome scans for "outliers", when the
alternative hypotheses include fundamentally historical and
unobserved phenomena. © 2007 Sainudiin et al; licensee
BioMed Central Ltd.},
Doi = {10.1186/1471-2164-8-146},
Key = {fds243535}
}

@article{fds304476,
Author = {De, A and Durrett, R},
Title = {Stepping-stone spatial structure causes slow decay of
linkage disequilibrium and shifts the site frequency
spectrum},
Journal = {Genetics},
Volume = {176},
Number = {2},
Pages = {969-981},
Year = {2007},
ISSN = {0016-6731},
url = {http://dx.doi.org/10.1534/genetics.107.071464},
Abstract = {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 samples from a small geographical
region commonly used in genetic studies of humans and
Drosophila, there is a shift of the frequency spectrum that
decreases the number of low-frequency-derived alleles and
skews the distribution of statistics of Tajima, Fu and Li,
and Fay and Wu. Stepping-stone spatial structure also
changes the two-locus sampling distribution and increases
both linkage disequilibrium and the probability that two
sites are perfectly correlated. This may cause a false
prediction of cold spots of recombination and may confuse
haplotype tests that compute probabilities on the basis of a
homogeneously mixing population. Copyright © 2007 by the
Genetics Society of America.},
Doi = {10.1534/genetics.107.071464},
Key = {fds304476}
}

@article{fds243548,
Author = {Berestycki, N and Durrett, R},
Title = {A phase transition in the random transposition random
walk},
Journal = {Probability Theory and Related Fields},
Volume = {136},
Number = {2},
Pages = {203-233},
Publisher = {Springer Nature},
Year = {2006},
Month = {October},
ISSN = {0178-8051},
MRCLASS = {60C05 (60G50)},
MRNUMBER = {MR2240787 (2007i:60009)},
url = {http://dx.doi.org/10.1007/s00440-005-0479-7},
Abstract = {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 elements. Consider this walk in
continuous time starting at the identity and let D t be the
minimum number of transpositions needed to go back to the
identity from the location at time t. D t undergoes a phase
transition: the distance D cn/2̃ u(c)n, where u is an
explicit function satisfying u(c)=c/2 for c ≤ 1 and
u(c)&lt;c/2 for c&gt;1. In addition, we describe the
fluctuations of D cn/2 about its mean in each of the three
regimes (subcritical, critical and supercritical). The
techniques used involve viewing the cycles in the random
permutation as a coagulation-fragmentation process and
relating the behavior to the Erdos-Renyi random graph
model.},
Doi = {10.1007/s00440-005-0479-7},
Key = {fds243548}
}

@article{fds243549,
Author = {Chan, B and Durrett, R},
Title = {A new coexistence result for competing contact
processes},
Journal = {The Annals of Applied Probability},
Volume = {16},
Number = {3},
Pages = {1155-1165},
Publisher = {Institute of Mathematical Statistics},
Year = {2006},
Month = {August},
ISSN = {1050-5164},
MRCLASS = {60K35},
MRNUMBER = {MR2260060 (2008h:60400)},
url = {http://dx.doi.org/10.1214/105051606000000132},
Abstract = {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 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."
© Institute of Mathematical Statistics,
2006.},
Doi = {10.1214/105051606000000132},
Key = {fds243549}
}

@article{fds243412,
Author = {Durrett, R},
Title = {Random graph dynamics},
Journal = {Random Graph Dynamics},
Series = {Cambridge Series in Statistical and Probabilistic
Mathematics},
Pages = {1-212},
Publisher = {Cambridge University Press},
Address = {Cambridge},
Year = {2006},
Month = {January},
ISBN = {978-0-521-86656-9; 0-521-86656-1},
MRCLASS = {05C80 (05-02 60-02 60C05 60G50 60K35 82C41)},
MRNUMBER = {MR2271734 (2008c:05167)},
url = {http://dx.doi.org/10.1017/CBO9780511546594},
Abstract = {© Rick Durrett 2007 and Cambridge University Press, 2009.
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 each other, inspired Strogatz
and Watts to define the small world random graph in which
each site is connected to k close neighbors, but also has
long-range connections. At about the same time, it was
observed in human social and sexual networks and on the
Internet that the number of neighbors of an individual or
computer has a power law distribution. This inspired
Barabasi and Albert to define the preferential attachment
model, which has these properties. These two papers have led
to an explosion of research. While this literature is
extensive, many of the papers are based on simulations and
nonrigorous arguments. The purpose of this book is to use a
wide variety of mathematical argument to obtain insights
into the properties of these graphs. A unique feature of
this book is the interest in the dynamics of process taking
place on the graph in addition to their geometric
properties, such as connectedness and diameter.},
Doi = {10.1017/CBO9780511546594},
Key = {fds243412}
}

@article{fds243534,
Author = {Durrett, R and Schweinsberg, J},
Title = {Power laws for family sizes in a duplication
model},
Journal = {The Annals of Probability},
Volume = {33},
Number = {6},
Pages = {2094-2126},
Publisher = {Institute of Mathematical Statistics},
Year = {2005},
Month = {November},
ISSN = {0091-1798},
MRCLASS = {60J85 (60J80 92D15 92D20)},
MRNUMBER = {2184092 (2006j:60092)},
url = {http://dx.doi.org/10.1214/009117905000000369},
Abstract = {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 no deaths and each individual
gives birth to a new individual at rate 1. When a new
individual is born, it has the same type as its parent with
probability 1 - r and is a new type, different from all
previously observed types, with probability r. We refer to
individuals with the same type as families and provide an
approximation to the joint distribution of family sizes when
the population size reaches N. We also show that if 1 ≪ S
≪ N 1-r, then the number of families of size at least 5 is
approximately CNS -1/(1-r), while if N 1-r ≪ S the
distribution decays more rapidly than any power. ©
Institute of Mathematical Statistics, 2005.},
Doi = {10.1214/009117905000000369},
Key = {fds243534}
}

@article{fds243547,
Author = {Durrett, R and Schweinsberg, J},
Title = {A coalescent model for the effect of advantageous mutations
on the genealogy of a population},
Journal = {Stochastic Processes and Their Applications},
Volume = {115},
Number = {10},
Pages = {1628-1657},
Publisher = {Elsevier BV},
Year = {2005},
Month = {October},
ISSN = {0304-4149},
MRCLASS = {92D15 (60J27 92D10 92D20)},
MRNUMBER = {MR2165337 (2006h:92026)},
url = {http://dx.doi.org/10.1016/j.spa.2005.04.009},
Abstract = {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 backwards in time, many lineages
may coalesce almost instantaneously at the time of a
selective sweep. We show that as the population size goes to
infinity, this process converges to a coalescent process
called a coalescent with multiple collisions. A better
approximation for finite populations can be obtained using a
coalescent with simultaneous multiple collisions. We also
show how these coalescent approximations can be used to get
insight into how beneficial mutations affect the behavior of
statistics that have been used to detect departures from the
usual Kingman's coalescent. © 2005 Elsevier B.V. All rights
reserved.},
Doi = {10.1016/j.spa.2005.04.009},
Key = {fds243547}
}

@article{fds243544,
Author = {Blasiak, J and Durrett, R},
Title = {Random Oxford graphs},
Journal = {Stochastic Processes and Their Applications},
Volume = {115},
Number = {8},
Pages = {1257-1278},
Publisher = {Elsevier BV},
Year = {2005},
Month = {August},
ISSN = {0304-4149},
MRCLASS = {60C05 (05C80)},
MRNUMBER = {MR2152374 (2006j:60008)},
url = {http://dx.doi.org/10.1016/j.spa.2005.03.008},
Abstract = {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 asymptotic results for the number of such
graphs and the number of (i, j) trees they contain. We
compute the thresholds for the emergence of a giant
component and for the graph to be connected. © 2005
Elsevier B.V. All rights reserved.},
Doi = {10.1016/j.spa.2005.03.008},
Key = {fds243544}
}

@article{fds243545,
Author = {Schweinsberg, J and Durrett, R},
Title = {Random partitions approximating the coalescence of lineages
during a selective sweep},
Journal = {The Annals of Applied Probability},
Volume = {15},
Number = {3},
Pages = {1591-1651},
Publisher = {Institute of Mathematical Statistics},
Year = {2005},
Month = {August},
ISSN = {1050-5164},
MRCLASS = {92D10 (05A18 60C05 60J80 60J85 92D15)},
MRNUMBER = {MR2152239 (2006c:92012)},
url = {http://dx.doi.org/10.1214/105051605000000430},
Abstract = {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 chromosome that is close to the location of
the beneficial mutation, then many of the lineages will
likely be descended from the individual that had the
beneficial mutation, while others will be descended from a
different individual because of recombination between the
two sites. We introduce two approximations for the effect of
a selective sweep. The first one is simple but not very
accurate: flip n independent coins with probability p of
heads and say that the lineages whose coins come up heads
are those that are descended from the individual with the
beneficial mutation. A second approximation, which is
related to Kingman's paintbox construction, replaces the
coin flips by integer-valued random variables and leads to
very accurate results. © Institute of Mathematical
Statistics. 2005.},
Doi = {10.1214/105051605000000430},
Key = {fds243545}
}

@article{fds243510,
Author = {Durrett, R and Levin, SA},
Title = {Can stable social groups be maintained by homophilous
imitation alone?},
Journal = {Journal of Economic Behavior and Organization},
Volume = {57},
Number = {3},
Pages = {267-286},
Publisher = {Elsevier BV},
Year = {2005},
Month = {July},
url = {http://dx.doi.org/10.1016/j.jebo.2003.09.017},
Abstract = {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 or punishment. Here, we show that
such payoffs are unnecessary, and that stable social groups
can sometimes be maintained provided simply that agents are
more likely to imitate others who are like them (homophily).
In contrast to other studies, to sustain multiple types we
need not impose the restriction that agents also choose to
make their opinions different from those in other groups. ©
2004 Elsevier B.V. All rights reserved.},
Doi = {10.1016/j.jebo.2003.09.017},
Key = {fds243510}
}

@article{fds243543,
Author = {Z�hle, I and Cox, JT and Durrett, R},
Title = {The stepping stone model. II: Genealogies and the infinite
sites model},
Journal = {The Annals of Applied Probability},
Volume = {15},
Number = {1B},
Pages = {671-699},
Publisher = {Institute of Mathematical Statistics},
Year = {2005},
Month = {February},
ISSN = {1050-5164},
MRCLASS = {60K35 (92D10)},
MRNUMBER = {MR2114986 (2006d:60157)},
url = {http://dx.doi.org/10.1214/105051604000000701},
Abstract = {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. ©
Institute of Mathematical Statistics, 2005.},
Doi = {10.1214/105051604000000701},
Key = {fds243543}
}

@article{fds177893,
Author = {R. Durrett},
Title = {Genome rearrangement},
Series = {Stat. Biol. Health},
Pages = {307--323},
Booktitle = {Statistical methods in molecular evolution},
Publisher = {Springer},
Address = {New York},
Year = {2005},
MRCLASS = {92D10 (05A05 60K25 60K30 62F15 62P10)},
MRNUMBER = {MR2161835 (2006f:92021)},
url = {http://www.ams.org/mathscinet-getitem?mr=2161835},
Key = {fds177893}
}

@article{fds243532,
Author = {York, TL and Durrett, RT and Tanksley, S and Nielsen,
R},
Title = {Bayesian and maximum likelihood estimation of genetic
maps},
Journal = {Genetical Research},
Volume = {85},
Number = {2},
Pages = {159-168},
Year = {2005},
url = {http://dx.doi.org/10.1017/S0016672305007494},
Abstract = {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 present an extension of the
previous methods that makes the Bayesian method applicable
to large data sets. We present an extensive simulation study
examining the statistical properties of the method and
comparing it with the likelihood method implemented in
Mapmaker. We show that the Maximum A Posteriori (MAP)
estimator of the genetic distances, corresponding to the
maximum likelihood estimator, performs better than
estimators based on the posterior expectation. We also show
that while the performance is similar between Mapmaker and
the MCMC-based method in the absence of genotyping errors,
the MCMC-based method has a distinct advantage in the
presence of genotyping errors. A similar advantage of the
Bayesian method was not observed for missing data. We also
re-analyse a recently published set of data from the
eggplant and show that the use of the MCMC-based method
leads to smaller estimates of genetic distances. © 2005
Cambridge University Press.},
Doi = {10.1017/S0016672305007494},
Key = {fds243532}
}

@article{fds243546,
Author = {Durrett, R and Mytnik, L and Perkins, E},
Title = {Competing super-Brownian motions as limits of interacting
particle systems},
Journal = {Electronic Journal of Probability},
Volume = {10},
Pages = {1147-1220},
Publisher = {Institute of Mathematical Statistics},
Year = {2005},
ISSN = {1083-6489},
MRCLASS = {60G57 (60G17)},
MRNUMBER = {MR2164042 (2006f:60052)},
url = {http://dx.doi.org/10.1214/EJP.v10-229},
Abstract = {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 colicin model as special cases. We
verify in some cases convergence of scaling limits of these
models to a pair of super-Brownian motions interacting
through their collision local times, constructed by Evans
and Perkins.},
Doi = {10.1214/EJP.v10-229},
Key = {fds243546}
}

@article{fds243505,
Author = {Durrett, R and Schweinsberg, J},
Title = {Approximating selective sweeps},
Journal = {Theoretical Population Biology},
Volume = {66},
Number = {2},
Pages = {129-138},
Year = {2004},
url = {http://dx.doi.org/10.1016/j.tpb.2004.04.002},
Abstract = {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, most subsequent work has simplified
their approach by assuming that the number of individuals
with the advantageous allele follows the logistic
differential equation. We show that the impact of a
selective sweep can be accurately approximated by a random
partition created by a stick-breaking process. Our
simulation results show that ignoring the randomness when
the number of individuals with the advantageous allele is
small can lead to substantial errors. © 2004 Elsevier Inc.
All rights reserved.},
Doi = {10.1016/j.tpb.2004.04.002},
Key = {fds243505}
}

@article{fds243506,
Author = {Ward, R and Durrett, R},
Title = {Subfunctionalization: How often does it occur? How long does
it take?},
Journal = {Theoretical Population Biology},
Volume = {66},
Number = {2},
Pages = {93-100},
Year = {2004},
url = {http://dx.doi.org/10.1016/j.tpb.2004.03.004},
Abstract = {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 complementary functions. We
investigate the probability that subfunctionalization
occurs, the amount of time after duplication that it takes
for the outcome to be resolved, and the relationship of
these quantities to the population size and mutation rates.
© 2004 Elsevier Inc. All rights reserved.},
Doi = {10.1016/j.tpb.2004.03.004},
Key = {fds243506}
}

@article{fds243507,
Author = {Durrett, R and Nielsen, R and York, TL},
Title = {Bayesian Estimation of Genomic Distance},
Journal = {Genetics},
Volume = {166},
Number = {1},
Pages = {621-629},
Year = {2004},
url = {http://dx.doi.org/10.1534/genetics.166.1.621},
Abstract = {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 the distribution of inversion track
lengths or rate constancy among lineages. Here, we apply
these methods to comparative maps of eggplant and tomato,
human and cat, and human and cattle with 170, 269, and 422
markers, respectively. In the first case the most likely
number of events is larger than the parsimony value. In the
last two cases the parsimony solutions have very small
probability.},
Doi = {10.1534/genetics.166.1.621},
Key = {fds243507}
}

@article{fds243508,
Author = {Schmidt, D and Durrett, R},
Title = {Adaptive evolution drives the diversification of zinc-finger
binding domains},
Journal = {Molecular Biology and Evolution},
Volume = {21},
Number = {12},
Pages = {2326-2339},
Year = {2004},
url = {http://dx.doi.org/10.1093/molbev/msh246},
Abstract = {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 well as the clustered
organization of these genes in humans, suggests that gene
duplication has played an important role in expanding this
family of genes. Using likelihood methods developed by Yang
and parsimony methods introduced by Suzuki and Gojobori, we
have investigated four clusters of zinc-finger genes on
human chromosome 19 and found evidence that positive
selection was involved in diversifying the family of
zinc-finger binding motifs.},
Doi = {10.1093/molbev/msh246},
Key = {fds243508}
}

@article{fds243509,
Author = {Sainudiin, R and Durrett, RT and Aquadro, CF and Nielsen,
R},
Title = {Microsatellite mutation models: Insights from a comparison
of humans and chimpanzees},
Journal = {Genetics},
Volume = {168},
Number = {1},
Pages = {383-395},
Year = {2004},
url = {http://dx.doi.org/10.1534/genetics.103.022665},
Abstract = {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, linear-biased, one-phase model emerges
as the best model. A focal length toward which the
mutational and/or substitutional process is linearly biased
is a crucial feature of microsatellite evolution. We find
that two-phase models do not lead to a significantly better
fit than their one-phase counterparts. The performance of
models based on the fit of their stationary distributions to
the empirical distribution of microsatellite lengths in the
human genome is consistent with that based on the
human-chimp comparison. Microsatellites interrupted by even
a single point mutation exhibit a twofold decrease in their
mutation rate when compared to pure AC repeats. In general,
models that allow chimps to have a larger per-repeat unit
slippage rate and/or a shorter focal length compared to
humans give a better fit to the human-chimp data as well as
the human genomic data.},
Doi = {10.1534/genetics.103.022665},
Key = {fds243509}
}

@article{fds243503,
Author = {Durrett, R and Limic, V},
Title = {Rigorous results for the N K model},
Journal = {The Annals of Probability},
Volume = {31},
Number = {4},
Pages = {1713-1753},
Publisher = {Institute of Mathematical Statistics},
Year = {2003},
Month = {October},
ISSN = {0091-1798},
url = {http://dx.doi.org/10.1214/aop/1068646364},
Abstract = {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 landscape is quite complicated with many
local maxima. Its properties have been extensively
investigated by simulation but until our work and the
independent investigations of Evans and Steinsaltz little
was known rigorously about its properties except in the case
K = N - 1. Here, we prove results about the number of local
maxima, their heights and the height of the global maximum.
Our main tool is the theory of (substochastic) Harris
chains.},
Doi = {10.1214/aop/1068646364},
Key = {fds243503}
}

@article{fds243501,
Author = {Calabrese, P and Durrett, R},
Title = {Dinucleotide repeats in the drosophila and human genomes
have complex, length-dependent mutation processes},
Journal = {Molecular Biology and Evolution},
Volume = {20},
Number = {5},
Pages = {715-725},
Year = {2003},
url = {http://dx.doi.org/10.1093/molbev/msg084},
Abstract = {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, one with quadratic and
another with piecewise linear slippage rates, have the best
fits and agree with recent experimental studies by
predicting that long microsatellites have a bias toward
contractions.},
Doi = {10.1093/molbev/msg084},
Key = {fds243501}
}

@article{fds243502,
Author = {Cox, T and Durrett, R},
Title = {Erratum: The stepping stone model: New formulas expose old
myths (The Annals of Applied Probability (2002) 12
(1348-1377))},
Journal = {The Annals of Applied Probability},
Volume = {13},
Number = {2},
Pages = {816-},
Year = {2003},
ISSN = {1050-5164},
Key = {fds243502}
}

@article{fds243504,
Author = {Durrett, R},
Title = {Shuffling Chromosomes},
Journal = {Journal of Theoretical Probability},
Volume = {16},
Number = {3},
Pages = {725-750},
Year = {2003},
ISSN = {0894-9840},
url = {http://dx.doi.org/10.1023/A:1025676617383},
Abstract = {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 all reversals are equally
likely and when the segments reversed are never more than L
genes. In studying this question we prove some new results
about the convergence to equilibrium of shuffling by
transposition and the one dimensional simple exclusion
process.},
Doi = {10.1023/A:1025676617383},
Key = {fds243504}
}

@article{fds243498,
Author = {Durrett, R and Limic, V},
Title = {A surprising Poisson process arising from a species
competition model},
Journal = {Stochastic Processes and Their Applications},
Volume = {102},
Number = {2},
Pages = {301-309},
Publisher = {Elsevier BV},
Year = {2002},
Month = {December},
ISSN = {0304-4149},
url = {http://dx.doi.org/10.1016/s0304-4149(02)00209-0},
Abstract = {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 probability a. Intuitively this
dynamic will create a negative dependence between the number
of points in adjacent intervals. However, we show that the
ensemble of points converges to a Poisson process with
intensity 1/(a(1 - x)), and the number of points at time t
grows like (log t)/a. © 2002 Elsevier Science B.V. All
rights reserved.},
Doi = {10.1016/s0304-4149(02)00209-0},
Key = {fds243498}
}

@article{fds243500,
Author = {Cox, JT and Durrett, R},
Title = {The stepping stone model: New formulas expose old
myths},
Journal = {The Annals of Applied Probability},
Volume = {12},
Number = {4},
Pages = {1348-1377},
Publisher = {Institute of Mathematical Statistics},
Year = {2002},
Month = {November},
url = {http://dx.doi.org/10.1214/aoap/1037125866},
Abstract = {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 separation of the colonies and for
Wright's genetic distance FST. These results confirm a
result of Crow and Aoki (1984) found by simulation: in the
usual biological range of parameters FST grows like the log
of the number of colonies. In the other direction, our
formulas show that there is significant spatial structure in
parts of parameter space where Maruyama and Nei (1971) and
Slatkin and Barton (1989) have called the stepping model
"effectively panmictic".},
Doi = {10.1214/aoap/1037125866},
Key = {fds243500}
}

@article{fds243497,
Author = {Durrett, R and Kesten, H and Limic, V},
Title = {Once edge-reinforced random walk on a tree},
Journal = {Probability Theory and Related Fields},
Volume = {122},
Number = {4},
Pages = {567-592},
Publisher = {Springer Nature},
Year = {2002},
Month = {April},
url = {http://dx.doi.org/10.1007/s004400100179},
Abstract = {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 &gt; 1 when
the edge is traversed for the first time. After such a
change the weight of an edge stays at c forever. We show
that such a walk is transient for all values of c ≥ 1, and
that the walk moves off to infinity at a linear rate. We
also prove an invariance principle for the height of the
walk.},
Doi = {10.1007/s004400100179},
Key = {fds243497}
}

@article{fds243496,
Author = {Balding, DJ and Carothers, AD and Marchini, JL and Cardon, LR and Vetta,
A and Griffiths, B and Weir, BS and Hill, WG and Goldstein, D and Strimmer,
K and Myers, S and Beaumont, MA and Glasbey, CA and Mayer, CD and Richardson, S and Marshall, C and Durrett, R and Nielsen, R and Visscher, PM and Knott, SA and Haley, CS and Ball, RD and Hackett, CA and Holmes, S and Husmeier, D and Jansen, RC and Ter Braak and CJF and Maliepaard, CA and Boer, MP and Joyce, P and Li, N and Stephens, M and Marcoulides, GA and Drezner, Z and Mardia, K and McVean, G and Meng, XL and Ochs, MF and Pagel, M and Sha, N and Vannucci, M and Sillanpää, MJ and Sisson, S and Yandell, BS and Jin, C and Satagopan, JM and Gaffney, PJ and Zeng, ZB and Broman, KW and Speed, TP and Fearnhead, P and Donnelly, P and Larget, B and Simon, DL and Kadane, JB and Nicholson, G and Smith, AV and Jónsson, F and Gústafsson, O and Stefánsson, K and Parmigiani, G and Garrett, ES and Anbazhagan, R and Gabrielson, E},
Title = {Discussion on the meeting on 'statistical modelling and
analysis of genetic data'},
Journal = {Journal of the Royal Statistical Society: Series B
(Statistical Methodology)},
Volume = {64},
Number = {4},
Pages = {737-775},
Publisher = {WILEY},
Year = {2002},
Month = {January},
ISSN = {1369-7412},
url = {http://dx.doi.org/10.1111/1467-9868.00359},
Doi = {10.1111/1467-9868.00359},
Key = {fds243496}
}

@article{fds243492,
Author = {Durrett, RT and Chen, K-Y and Tanksley, SD},
Title = {A simple formula useful for positional cloning},
Journal = {Genetics},
Volume = {160},
Number = {1},
Pages = {353-355},
Year = {2002},
ISSN = {0016-6731},
Abstract = {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 to narrow a target gene down to a
specific interval size and hence should be useful for
planning positional cloning experiments. The predictions of
this formula agree well with the results from a number of
published experiments in Arabidopsis.},
Key = {fds243492}
}

@article{fds243494,
Author = {Durrett, R},
Title = {Mutual invadability implies coexistence in spatial
models},
Journal = {Memoirs of the American Mathematical Society},
Number = {740},
Year = {2002},
Abstract = {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 always independent. Here we prove a
general result in support of that picture. We give a
condition on an ordinary differential equation which implies
that densities stay bounded away from 0 in the associated
reaction-diffusion equation, and that coexistence occurs in
the stochastic spatial model with fast stirring. Then using
biologists' notion of invadability as a guide, we show how
this condition can be checked in a wide variety of examples
that involve two or three species: epidemics, diploid
genetics models, predator-prey systems, and various
competition models.},
Key = {fds243494}
}

@article{fds243495,
Author = {York, TL and Durrett, R and Nielsen, R},
Title = {Bayesian estimation of the number of inversions in the
history of two chromosomes},
Journal = {Journal of Computational Biology},
Volume = {9},
Number = {6},
Pages = {805-818},
Year = {2002},
url = {http://dx.doi.org/10.1089/10665270260518281},
Abstract = {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 marker order. We apply the method both to
simulated data and to two real data sets. For the simulated
data, we show that the MCMC method provides accurate
estimates of the true posterior distributions and in the
analysis of the real data we show that the most likely
number of inversions in some cases is considerably larger
than estimates obtained based on the parsimony inferred
number of inversions. Indeed, in the case of the Drosophila
repleta-D. melanogaster comparison, the lower boundary of a
95% highest posterior density credible interval for the
number of inversions is considerably larger than the most
parsimonious number of inversions.},
Doi = {10.1089/10665270260518281},
Key = {fds243495}
}

@article{fds243499,
Author = {Buttel, LA and Durrett, R and Levin, SA},
Title = {Competition and Species Packing in Patchy
Environments},
Journal = {Theoretical Population Biology},
Volume = {61},
Number = {3},
Pages = {265-276},
Year = {2002},
url = {http://dx.doi.org/10.1006/tpbi.2001.1569},
Abstract = {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, depend on the model used to ask the
question. In the Tilman-May-Nowak ordinary differential
equation model, the number of species is asymptotically C
log N with most species packed in at the upper end of the
competitive hierarchy. In contrast, for metapopulation
models with discrete individuals and stochastic spatial
systems with various competition neighborhoods, we find a
traditional species area relationship CNa, with no species
clumping along the phenotypic gradient. The exponent a is
larger by a factor of 2 for spatially explicit models. In
words, a spatial distribution of competitors allows for
greater diversity than a metapopulation model due to the
effects of recruitment limitation in their competition. ©
2002 Elsevier Science (USA).},
Doi = {10.1006/tpbi.2001.1569},
Key = {fds243499}
}

@article{fds243411,
Author = {Calabrese, PP and Durrett, RT and Aquadro, CF},
Title = {Dynamics of microsatellite divergence under stepwise
mutation and proportional slippage/point mutation
models.},
Journal = {Genetics},
Volume = {159},
Number = {2},
Pages = {839-852},
Year = {2001},
Month = {October},
ISSN = {0016-6731},
Abstract = {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 a perfect repeat and a simplification
that ignores point mutations. We develop a detailed
mathematical theory for these models that exhibits
properties of microsatellite distributions, such as positive
skewness of allele lengths, that are consistent with data
but are inconsistent with the predictions of the stepwise
mutation model. We use our theoretical results to analyze
the successes and failures of the genetic distances
(delta(mu))(2) and D(SW) when used to date four divergences:
African vs. non-African human populations, humans vs.
chimpanzees, Drosophila melanogaster vs. D. simulans, and
sheep vs. cattle. The influence of point mutations explains
some of the problems with the last two examples, as does the
fact that these genetic distances have large stochastic
variance. However, we find that these two features are not
enough to explain the problems of dating the
human-chimpanzee split. One possible explanation of this
phenomenon is that long microsatellites have a mutational
bias that favors contractions over expansions.},
Key = {fds243411}
}

@article{fds243493,
Author = {Arkendra, DE and Ferguson, M and Sindi, S and Durrett,
R},
Title = {The equilibrium distribution for a generalized
Sankoff-Ferretti model accurately predicts chromosome size
distributions in a wide variety of species},
Journal = {Journal of Applied Probability},
Volume = {38},
Number = {2},
Pages = {324-334},
Publisher = {Cambridge University Press (CUP)},
Year = {2001},
Month = {June},
ISSN = {0021-9002},
url = {http://dx.doi.org/10.1239/jap/996986747},
Abstract = {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 found that their short chromosomes
were too short, and their long chromosomes were too long.
Here, we consider a generalization of their proportional
model with explicit chromosome centromeres and introduce
fitness functions based on recombination probabilities and
on the length of the longest chromosome arm. We find a
simple formula for the stationary distribution for our model
which fits the data on chromosome lengths in many, but not
all, species.},
Doi = {10.1239/jap/996986747},
Key = {fds243493}
}

@article{fds243491,
Author = {Durrett, R and Limic, V},
Title = {On the quantity and quality of single nucleotide
polymorphisms in the human genome},
Journal = {Stochastic Processes and Their Applications},
Volume = {93},
Number = {1},
Pages = {1-24},
Publisher = {Elsevier BV},
Year = {2001},
Month = {May},
url = {http://dx.doi.org/10.1016/S0304-4149(00)00090-9},
Abstract = {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 uses Kruglyak's (Nature Gen. 12
(1999) 139-144) model of the growth of the human population
and one assumes an average mutation rate of 1×10-8per
nucleotide per generation then there are about 5.7 million
SNP's in the human genome, or one every 526 base pairs. We
also obtain results for the number of SNPs that will be
found in samples of sizes n≥2 to gain insight into the
number that will be found by various experimental
procedures. © 2001 Elsevier Science B.V.},
Doi = {10.1016/S0304-4149(00)00090-9},
Key = {fds243491}
}

@article{fds243489,
Author = {Diaconis, P and Durrett, R},
Title = {Chutes and Ladders in Markov Chains},
Journal = {Journal of Theoretical Probability},
Volume = {14},
Number = {3},
Pages = {899-926},
Year = {2001},
url = {http://dx.doi.org/10.1023/A:1017509611178},
Abstract = {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 three changes the stationary
distribution linearly, logarithmically, or only locally.
Related results are derived for birth and death chains
approximating Bessel diffusions and for random walk on the
Sierpinski gasket.},
Doi = {10.1023/A:1017509611178},
Key = {fds243489}
}

@article{fds243490,
Author = {Sundell, NM and Durrett, RT},
Title = {Exponential distance statistics to detect the effects of
population subdivision},
Journal = {Theoretical Population Biology},
Volume = {60},
Number = {2},
Pages = {107-116},
Year = {2001},
ISSN = {0040-5809},
url = {http://dx.doi.org/10.1006/tpbi.2001.1522},
Abstract = {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 between individuals, which can be
used with any genetic distance (e.g., nucleotide
differences, number of nonshared alleles, or separation on a
phylogenetic tree). The power of the tests to detect genetic
differentiation in Wright-Fisher island models and stepping
stone models was calculated for various sample sizes, rates
of migration and mutation, and definitions of spatial
neighborhoods. We found that our new test was in some cases
more powerful than the K S* statistic of Hudson et al. (Mol.
Biol. Evol. 9, 138-151, 1992), but in all cases was slightly
less powerful than both a traditional X 2 test without
lumping of rare haplotypes and the S nn test of Hudson
(Genetics 155, 2011-2014, 2000). However, when we applied
our new tests to three data sets, we found in some cases
highly significant results that were missed by the other
tests. © 2001 Academic Press.},
Doi = {10.1006/tpbi.2001.1522},
Key = {fds243490}
}

@article{fds243481,
Author = {Kruglyak, S and Durrett, R and Schug, MD and Aquadro,
CF},
Title = {Distribution and abundance of microsatellites in the yeast
genome can Be explained by a balance between slippage events
and point mutations.},
Journal = {Molecular Biology and Evolution},
Volume = {17},
Number = {8},
Pages = {1210-1219},
Year = {2000},
Month = {August},
ISSN = {0737-4038},
url = {http://dx.doi.org/10.1093/oxfordjournals.molbev.a026404},
Abstract = {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 microsatellites can be explained by this
simple model, which incorporates a competition between
slippage events and base pair substitutions, with no need to
invoke selection or constraints on the lengths. Our results
provide some new information on slippage rates for
individual repeat motifs, which suggest that AT-rich
trinucleotide repeats have higher slippage rates. As our
model predicts, we found that many repeats were adjacent to
shorter repeats of the same motif. However, we also found a
significant tendency of microsatellites of different motifs
to cluster.},
Doi = {10.1093/oxfordjournals.molbev.a026404},
Key = {fds243481}
}

@article{fds243485,
Author = {Vision, TJ and Brown, DG and Shmoys, DB and Durrett, RT and Tanksley,
SD},
Title = {Selective mapping: a strategy for optimizing the
construction of high-density linkage maps.},
Journal = {Genetics},
Volume = {155},
Number = {1},
Pages = {407-420},
Year = {2000},
Month = {May},
Abstract = {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 is neither necessary nor desirable to
genotype all markers in every individual of a large mapping
population. Instead, a reduced sample of individuals bearing
complementary recombinational or radiation-induced
breakpoints may be selected for genotyping subsequent
markers from a large, but sparsely genotyped, mapping
population. Choosing such a sample can be reduced to a
discrete stochastic optimization problem for which the goal
is a sample with breakpoints spaced evenly throughout the
genome. We have developed several different methods for
selecting such samples and have evaluated their performance
on simulated and actual mapping populations, including the
Lister and Dean Arabidopsis thaliana recombinant inbred
population and the GeneBridge 4 human radiation hybrid
panel. Our methods quickly and consistently find
much-reduced samples with map resolution approaching that of
the larger populations from which they are derived. This
approach, which we have termed selective mapping, can
facilitate the production of high-quality, high-density
genome-wide linkage maps.},
Key = {fds243485}
}

@article{fds243484,
Author = {Liu, YC and Durrett, R and Milgroom, MG},
Title = {A spatially-structured stochastic model to simulate
heterogenous transmission of viruses in fungal
populations},
Journal = {Ecological Modelling},
Volume = {127},
Number = {2-3},
Pages = {291-308},
Publisher = {Elsevier BV},
Year = {2000},
Month = {March},
url = {http://dx.doi.org/10.1016/S0304-3800(99)00216-1},
Abstract = {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 debilitate their hosts and function as
biological control agents. An important characteristic of
this system is that virus transmission occurs freely between
individuals in the same genetically defined vegetative
compatibility (vc) type, but is restricted among individuals
in different vc types, resulting in heterogeneous
transmission. An additional source of heterogeneity is
spatial structure in host populations; viruses are dispersed
by fungal spores which disperse relatively short distances.
The model showed that vc type diversity is highly correlated
to the horizontal transmission rate and therefore
significantly affects virus invasion. The probability of
virus invasion decreased as the diversity of vc types
increased. We also demonstrated that virus transmission
would be overestimated if we assumed virus transmission was
homogeneous, ignoring both genetic and spatial
heterogeneity. Genetic and spatial heterogeneity are not
independent because both are affected by the reproductive
biology of the fungus. In asexual populations, restricted
fungus dispersal resulted in nonrandom spatial patterns of
vc types, increasing the chance of contact between
vegetatively compatible individuals, and promoting virus
transmission. In contrast, virus transmission was poor in
sexual populations due to spatial randomization of vc types
by long distance dispersed sexual spores. Finally, this
model was used to evaluate the release of genetically
engineered virus-infected strains for disease management.
The release of transgenic strains resulted in only
marginally greater virus establishment than for
non-transgenic strains. Virus invasion was still restricted
by vc type diversity in the resident fungus population.
Simulation of inundative releases of transgenic
virus-infected strains slightly improved virus
establishment, but viruses did not persist after treatment
was terminated. (C) 2000 Elsevier Science
B.V.},
Doi = {10.1016/S0304-3800(99)00216-1},
Key = {fds243484}
}

@article{fds243483,
Author = {Cox, JT and Durrett, R and Perkins, EA},
Title = {Rescaled voter models converge to super-Brownian
motion},
Journal = {The Annals of Probability},
Volume = {28},
Number = {1},
Pages = {185-234},
Publisher = {Institute of Mathematical Statistics},
Year = {2000},
Month = {January},
url = {http://dx.doi.org/10.1214/aop/1019160117},
Abstract = {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 dimension.},
Doi = {10.1214/aop/1019160117},
Key = {fds243483}
}

@article{fds243482,
Author = {Durrett, R and Buttel, L and Harrison, R},
Title = {Spatial models for hybrid zones},
Journal = {Heredity},
Volume = {84},
Number = {1},
Pages = {9-19},
Year = {2000},
ISSN = {0018-067X},
url = {http://dx.doi.org/10.1046/j.1365-2540.2000.00566.x},
Abstract = {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, a locus under selection in a
two-patch environment, and a linked neutral locus. We
consider several possible starting conditions that represent
explicit realizations of two alternative scenarios for
hybrid zone origins: primary intergradation and secondary
contact. Our results indicate that in some circumstances,
differences in hybrid zone origins will result in
substantially different patterns of variation that may
persist for thousands of generations. Our conclusions are
generally similar to those previously derived from partial
differential equations, but there are also some important
differences.},
Doi = {10.1046/j.1365-2540.2000.00566.x},
Key = {fds243482}
}

@article{fds243486,
Author = {Durrett, R and Schinazi, RB},
Title = {Boundary Modified Contact Processes},
Journal = {Journal of Theoretical Probability},
Volume = {13},
Number = {2},
Pages = {575-594},
Year = {2000},
Abstract = {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 internal). Deaths occur at rate 1. The case λe = λi is
the well known basic contact process for which there is a
critical value λc &gt; 1 such that if the birth rate is
larger than λc the process has a positive probability of
surviving. Our main motivation here is to understand the
relative importance of the external birth rates. We show
that if λe ≤ 1 then the process always dies out while if
λe &gt; 1 and if λi is large enough then the process may
survive. We also show that if λi &lt; λc the process dies
out for all λe. To extend this notion to d &gt; 1 we
introduce a second process that has an epidemiological
interpretation. For this process each site can be in one of
three states: infected, a susceptible that has never been
infected, or a susceptible that has been infected
previously. Furthermore, the rates at which the two types of
susceptible become infected are different. We obtain some
information about the phase diagram about this case as
well.},
Key = {fds243486}
}

@article{fds243487,
Author = {Durrett, R and Levin, S},
Title = {Lessons on pattern formation from planet
WATOR},
Journal = {Journal of Theoretical Biology},
Volume = {205},
Number = {2},
Pages = {201-214},
Year = {2000},
ISSN = {0022-5193},
url = {http://dx.doi.org/10.1006/jtbi.2000.2061},
Abstract = {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 pattern formation mechanisms that
operate even when the diffusion rates are equal. In
particular, we can say that when the mean-field ODE has an
attracting periodic orbit then the stochastic spatial model
will have large-scale spatial structures in equilibrium. We
explore this mechanism in depth through the dynamics of the
simulator WATOR. (C) 2000 Academic Press.},
Doi = {10.1006/jtbi.2000.2061},
Key = {fds243487}
}

@article{fds243488,
Author = {Broughton, RE and Stanley, SE and Durrett, RT},
Title = {Quantification of homoplasy for nucleotide transitions and
transversions and a reexamination of assumptions in weighted
phylogenetic analysis},
Journal = {Systematic Biology},
Volume = {49},
Number = {4},
Pages = {617-627},
Year = {2000},
Abstract = {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 reliable phylogenetic characters. Relative
amounts of homoplastic and consistent transition and
transversion changes in mitochondrial protein coding genes
were determined from character-state reconstructions on a
highly corroborated phylogeny of mammals. We found that
although homoplasy was related to evolutionary rates and was
greater for transitions, the absolute number of consistent
transitions greatly exceeded the number of consistent
transversions. Consequently, transitions provided
substantially more useful phylogenetic information than
transversions. These results suggest that down-weighting
transitions may be unwarranted in many cases. This
conclusion was supported by the fact that a range of
transition: transversion weighting schemes applied to
various mitochondrial genes and genomic partitions rarely
provided improvement in phylogenetic estimates relative to
equal weighting, and in some cases weighting transitions
more heavily than transversions was most
effective.},
Key = {fds243488}
}

@article{fds243477,
Author = {Durrett, R and Perkins, EA},
Title = {Rescaled contact processes converge to super-Brownian motion
in two or more dimensions},
Journal = {Probability Theory and Related Fields},
Volume = {114},
Number = {3},
Pages = {309-399},
Publisher = {Springer Nature},
Year = {1999},
Month = {June},
url = {http://dx.doi.org/10.1007/s004400050228},
Abstract = {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 Swindle (1989) by replacing their
order of magnitude estimates of how close the critical value
is to 1 with sharp asymptotics.},
Doi = {10.1007/s004400050228},
Key = {fds243477}
}

@article{fds243478,
Author = {Molofsky, J and Durrett, R and Dushoff, J and Griffeath, D and Levin,
S},
Title = {Local frequency dependence and global coexistence.},
Journal = {Theoretical Population Biology},
Volume = {55},
Number = {3},
Pages = {270-282},
Year = {1999},
Month = {June},
ISSN = {0040-5809},
url = {http://dx.doi.org/10.1006/tpbi.1998.1404},
Abstract = {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-dependent competition influenced whether two
species could coexist. Individuals of the two species were
randomly placed on a grid and allowed to interact according
to local frequency-dependent rules. For four different
frequency-dependent scenarios, the results indicated that
over a broad parameter range the two species could coexist.
Comparisons between explicit spatial simulations and the
mean-field approximation indicate that coexistence occurs
over a broader region in the explicit spatial
simulation.},
Doi = {10.1006/tpbi.1998.1404},
Key = {fds243478}
}

@article{fds243476,
Author = {Durrett, R},
Title = {Stochastic spatial models},
Journal = {Siam Review},
Volume = {41},
Number = {4},
Pages = {677-718},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1137/S0036144599354707},
Abstract = {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 Levin (1994): the behavior of these
models can be predicted from the properties of the mean
field ODE, i.e., the equations for the densities of the
various types that result from pretending that all sites are
always independent. We will illustrate this picture through
a discussion of eight families of examples from statistical
mechanics, genetics, population biology, epidemiology, and
ecology. Some of our findings are only conjectures based on
simulation, but in a number of cases we are able to prove
results for systems with fast stirring' by exploiting
connections between the spatial model and an associated
reaction diffusion equation.},
Doi = {10.1137/S0036144599354707},
Key = {fds243476}
}

@article{fds243479,
Author = {Durrett, R and Granovsky, BL and Gueron, S},
Title = {The Equilibrium Behavior of Reversible Coagulation-Fragmentation
Processes},
Journal = {Journal of Theoretical Probability},
Volume = {12},
Number = {2},
Pages = {447-474},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1023/A:1021682212351},
Abstract = {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 studied for a long time. As a result,
different approximations to the model were suggested. Our
paper deals with the exact model which is viewed as a
time-homogeneous interacting particle system on the state
space ΩN, the set of all partitions of N. We obtain the
stationary distribution (invariant measure) on ΩN for the
whole class of reversible coagulation-fragmentation
processes, and derive explicit expressions for important
functionals of this measure, in particular, the expected
numbers of groups of all sizes at the steady state. We also
establish a characterization of the transition rates that
guarantee the reversibility of the process. Finally, we make
a comparative study of our exact solution and the
approximation given by the steady-state solution of the
coagulation-fragmentation integral equation, which is known
in the literature. We show that in some cases the latter
approximation can considerably deviate from the exact
solution.},
Doi = {10.1023/A:1021682212351},
Key = {fds243479}
}

@article{fds243480,
Author = {Durrett, R and Kruglyak, S},
Title = {A new stochastic model of microsatellite
evolution},
Journal = {Journal of Applied Probability},
Volume = {36},
Number = {3},
Pages = {621-631},
Publisher = {Cambridge University Press (CUP)},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1017/S0021900200017447},
Abstract = {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 base pairs of DNA from fruit flies, mice,
and humans. The slippage rates from the best fit for our
model are consistent with experimental findings.},
Doi = {10.1017/S0021900200017447},
Key = {fds243480}
}

@article{fds243475,
Author = {Kruglyak, S and Durrett, RT and Schug, MD and Aquadro,
CF},
Title = {Equilibrium distributions of microsatellite repeat length
resulting from a balance between slippage events and point
mutations.},
Journal = {Proceedings of the National Academy of Sciences of the
United States of America},
Volume = {95},
Number = {18},
Pages = {10774-10778},
Year = {1998},
Month = {September},
ISSN = {0027-8424},
url = {http://dx.doi.org/10.1073/pnas.95.18.10774},
Abstract = {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 on microsatellite length and a mutation
process that includes both strand slippage and point
mutation events. We compute the stationary distribution of
allele lengths under this model and use it to fit DNA data
for di-, tri-, and tetranucleotide repeats in humans, mice,
fruit flies, and yeast. The best fit results lead to
slippage rate estimates that are highest in mice, followed
by humans, then yeast, and then fruit flies. Within each
organism, the estimates are highest in di-, then tri-, and
then tetranucleotide repeats. Our estimates are consistent
with experimentally determined mutation rates from other
studies. The results suggest that the different length
distributions among organisms and repeat motifs can be
explained by a simple difference in slippage rates and that
selective constraints on length need not be
imposed.},
Doi = {10.1073/pnas.95.18.10774},
Key = {fds243475}
}

@article{fds243470,
Author = {Durrett, R and Levin, S},
Title = {ERRATUM},
Journal = {Theoretical Population Biology},
Volume = {53},
Number = {3},
Pages = {284-284},
Publisher = {Elsevier BV},
Year = {1998},
Month = {June},
url = {http://dx.doi.org/10.1006/tpbi.1998.1374},
Doi = {10.1006/tpbi.1998.1374},
Key = {fds243470}
}

@article{fds243473,
Author = {Bramson, M and Cox, JT and Durrett, R},
Title = {A spatial model for the abundance of species},
Journal = {The Annals of Probability},
Volume = {26},
Number = {2},
Pages = {658-709},
Publisher = {Institute of Mathematical Statistics},
Year = {1998},
Month = {April},
url = {http://dx.doi.org/10.1214/aop/1022855647},
Abstract = {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 right triangular in d = 2 and
uniform in d > 3. We also obtain more detailed results for
the histograms that biologists use to estimate the
underlying density functions.},
Doi = {10.1214/aop/1022855647},
Key = {fds243473}
}

@article{fds243474,
Author = {Durrett, R and Levin, S},
Title = {Spatial aspects of interspecific competition},
Journal = {Theoretical Population Biology},
Volume = {53},
Number = {1},
Pages = {30-43},
Year = {1998},
ISSN = {0040-5809},
url = {http://dx.doi.org/10.1006/tpbi.1997.1338},
Abstract = {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 over each of the others, then
eventually it takes over all the sites in the system.
Second, we examine tradeoffs between competition and
dispersal distance in a two-species system. Third, we
consider a cyclic competitive relationship between three
types. In this case, a nonspatial treatment leads to
densities that follow neutrally stable cycles or even
unstable spiral solutions, while a spatial model yields a
stationary distribution with an interesting spatial
structure.},
Doi = {10.1006/tpbi.1997.1338},
Key = {fds243474}
}

@article{fds243471,
Author = {Durrett, R and Levin, S},
Title = {Allelopathy in Spatially Distributed Populations},
Journal = {Journal of Theoretical Biology},
Volume = {185},
Number = {2},
Pages = {165-171},
Publisher = {Elsevier BV},
Year = {1997},
Month = {March},
url = {http://dx.doi.org/10.1006/jtbi.1996.0292},
Abstract = {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 spatially-structured population, there is
a unique ESS for any given set of parameters; the outcome is
determined by how effective allelopathy is in relation to
its costs. Furthermore, in a spatially-structured
environment, a dynamic equilibrium may be sustained among a
colicin-sensitive type, a high colicin-producing type, and a
'cheater' that expends less on colicin production but is
resistant.},
Doi = {10.1006/jtbi.1996.0292},
Key = {fds243471}
}

@article{fds243469,
Author = {Durrett, R and Neuhauser, C},
Title = {Coexistence results for some competition
models},
Journal = {The Annals of Applied Probability},
Volume = {7},
Number = {1},
Pages = {10-45},
Publisher = {Institute of Mathematical Statistics},
Year = {1997},
Month = {February},
url = {http://dx.doi.org/10.1214/aoap/1034625251},
Abstract = {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 in the dominant strain
occurred between 1957 and 1976. Motivated by this
phenomenon, we develop a model for the competition between
different strains of the barley yellow dwarf virus. Our main
goal is to understand the phase diagram of the model, that
is, to identify parameter values where one strain
competitively excludes the other strain and where both
strains coexist. Our analysis applies to a number of other
systems as well, for example to a model of competition of
water flea species studied by Hanski and Ranta and
Bengtsson.},
Doi = {10.1214/aoap/1034625251},
Key = {fds243469}
}

@article{fds243472,
Author = {Chen, ZQ and Durrett, R and Ma, G},
Title = {Holomorphic diffusions and boundary behavior of harmonic
functions},
Journal = {The Annals of Probability},
Volume = {25},
Number = {3},
Pages = {1103-1134},
Publisher = {Institute of Mathematical Statistics},
Year = {1997},
Month = {January},
url = {http://dx.doi.org/10.1214/aop/1024404507},
Abstract = {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
metric in D). Using the diffusion processes associated with
these (degenerate) differential operators, the boundary
behavior of Lα-harmonic functions is studied in a unified
way for 0 ≤ α ≤ 1/2. More specifically, we show that a
bounded Lα-harmonic function in D has boundary limits in
approaching regions at almost every boundary point and the
boundary approaching region increases from the Stolz cone to
the Korányi admissible region as α runs from 0 to 1/2. A
local version for this Fatou-type result is also
established.},
Doi = {10.1214/aop/1024404507},
Key = {fds243472}
}

@article{fds243468,
Author = {Allouba, H and Durrett, R and Hawkes, J and Perkins,
E},
Title = {Super-Tree Random Measures},
Journal = {Journal of Theoretical Probability},
Volume = {10},
Number = {3},
Pages = {773-794},
Year = {1997},
Abstract = {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 cases can produce measures with
very smooth densities in any dimension.},
Key = {fds243468}
}

@article{fds243466,
Author = {Durrett, R and Levin, S},
Title = {Spatial Models for Species-Area Curves},
Journal = {Journal of Theoretical Biology},
Volume = {179},
Number = {2},
Pages = {119-127},
Publisher = {Elsevier BV},
Year = {1996},
Month = {March},
url = {http://dx.doi.org/10.1006/jtbi.1996.0053},
Abstract = {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 range of slopes of species-area curves is due
to the differences in the rates at which new species enter
this system. However, two other predictions, that the slope
increases with increasing migration/mutation and that the
curves for remote islands are flatter than those for near
islands, are at odds with some interpretations of data. This
suggests either that the data have been misinterpreted, or
that the model is not sufficient to explain
them.},
Doi = {10.1006/jtbi.1996.0053},
Key = {fds243466}
}

@article{fds243467,
Author = {Bramson, M and Cox, JT and Durrett, R},
Title = {Spatial models for species area curves},
Journal = {The Annals of Probability},
Volume = {24},
Number = {4},
Pages = {1727-1751},
Publisher = {Institute of Mathematical Statistics},
Year = {1996},
url = {http://dx.doi.org/10.1214/aop/1041903204},
Abstract = {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 this model as the mutation rate α
tends to zero. We obtain two basic types of behavior
depending on the size of the spatial region under
consideration. If the region is a square with area α-r, r
&gt; 1, then, for small α, the number of species is of
order α1r(log α)2, whereas if r &lt; 1, the number of
species is bounded.},
Doi = {10.1214/aop/1041903204},
Key = {fds243467}
}

@article{fds323656,
Author = {Cox, JT and Durrett, R},
Title = {Hybrid zones and voter model interfaces},
Journal = {Bernoulli},
Volume = {1},
Number = {4},
Pages = {343-370},
Publisher = {Bernoulli Society for Mathematical Statistics and
Probability},
Year = {1995},
Month = {December},
url = {http://dx.doi.org/10.3150/bj/1193758711},
Abstract = {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-dimensional particle systems. © 1995 Chapman
& Hall.},
Doi = {10.3150/bj/1193758711},
Key = {fds323656}
}

@article{fds243463,
Author = {Durrett, R and Swindle, G},
Title = {Coexistence results for catalysts},
Journal = {Probability Theory and Related Fields},
Volume = {98},
Number = {4},
Pages = {489-515},
Publisher = {Springer Nature},
Year = {1994},
Month = {December},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/bf01192836},
Abstract = {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 occurs (the catalyst surface becomes
completely occupied by one type of atom) and another in
which there is a translation invariant stationary
distribution in which the two atoms have positive density.
The last result is proved by exploiting a connection between
the particle system with fast stirring and a limiting system
of reaction diffusion equations. © 1994
Springer-Verlag.},
Doi = {10.1007/bf01192836},
Key = {fds243463}
}

@article{fds243464,
Author = {Durrett, R and Levin, S},
Title = {The importance of being discrete (and spatial)},
Journal = {Theoretical Population Biology},
Volume = {46},
Number = {3},
Pages = {363-394},
Publisher = {Elsevier BV},
Year = {1994},
Month = {January},
url = {http://dx.doi.org/10.1006/tpbi.1994.1032},
Abstract = {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 every other individual;
patch models that group discrete individuals into patches
without additional spatial structure; reaction-diffusion
equations, in which infinitesimal individuals are
distributed in space; and interacting particle systems, in
which individuals are discrete and space is treated
explicitly. We apply these four approaches to three examples
of species interactions in spatially distributed populations
and compare their predictions. Each represents different
assumptions about the biology and hence a comparison among
them has biological as well as modeling implications. In the
first case all four approaches agree, in the second the
spatial models disagree with the nonspatial ones, while in
the third the stochastic models with discrete individuals
disagree with the ones based on differential equations. We
show further that the limiting reaction-diffusion equations
associated with particle systems can have different
qualitative behavior from those obtained by simply adding
diffusion terms to mean field equations. © 1994 Academic
Press. All rights reserved.},
Doi = {10.1006/tpbi.1994.1032},
Key = {fds243464}
}

@article{fds243465,
Author = {Durrett, R and Levin, SA},
Title = {Stochastic spatial models: A user's guide to ecological
applications},
Journal = {Philosophical Transactions of the Royal Society of London.
Series B, Biological Sciences},
Volume = {343},
Number = {1305},
Pages = {329-350},
Publisher = {The Royal Society},
Year = {1994},
Month = {January},
ISSN = {0962-8436},
url = {http://dx.doi.org/10.1098/rstb.1994.0028},
Doi = {10.1098/rstb.1994.0028},
Key = {fds243465}
}

@article{fds323657,
Author = {Durrett, R and Griffeath, D},
Title = {Asymptotic Behavior of Excitable Cellular
Automata},
Journal = {Experimental Mathematics},
Volume = {2},
Number = {3},
Pages = {183-208},
Publisher = {Informa UK Limited},
Year = {1993},
Month = {January},
url = {http://dx.doi.org/10.1080/10586458.1993.10504277},
Doi = {10.1080/10586458.1993.10504277},
Key = {fds323657}
}

@article{fds243462,
Author = {Durrett, RT and Rogers, LCG},
Title = {Asymptotic behavior of Brownian polymers},
Journal = {Probability Theory and Related Fields},
Volume = {92},
Number = {3},
Pages = {337-349},
Publisher = {Springer Nature},
Year = {1992},
Month = {September},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/BF01300560},
Abstract = {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
that d=1 and the interaction function f(x)≥0. If, in
addition, f(x) behaves for large x like Cx-β with β<1 we
obtain a strong law that gives the exact growth rate. ©
1992 Springer-Verlag.},
Doi = {10.1007/BF01300560},
Key = {fds243462}
}

@article{fds243460,
Author = {Durrett, R},
Title = {Multicolor particle systems with large threshold and
range},
Journal = {Journal of Theoretical Probability},
Volume = {5},
Number = {1},
Pages = {127-152},
Publisher = {Springer Nature},
Year = {1992},
Month = {January},
ISSN = {0894-9840},
url = {http://dx.doi.org/10.1007/bf01046781},
Abstract = {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-Hastings model either dies out, or
has equilibria that consist of "debris" or "fire fronts".
The phase diagram for the cyclic color models is more
complicated. The main result of this paper, Theorem 1,
proves that the debris phase exists for both systems. ©
1992 Plenum Publishing Corporation.},
Doi = {10.1007/bf01046781},
Key = {fds243460}
}

@article{fds243461,
Author = {Durrett, R and Steif, JE},
Title = {Some rigorous results for the Greenberg-Hastings
Model},
Journal = {Journal of Theoretical Probability},
Volume = {4},
Number = {4},
Pages = {669-690},
Publisher = {Springer Nature},
Year = {1991},
Month = {October},
ISSN = {0894-9840},
url = {http://dx.doi.org/10.1007/bf01259549},
Abstract = {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 changes to 0, and a site which is 0
changes to a 1 if one of its 2 d neighbors is a 1. In one
dimension, we compute the exact asymptotic rate at which the
system dies out when started at random and compute the
topological entropy. In two or more dimensions we show that
starting from a nontrivial product measure, the limit exists
as 3 m→∞ and is Bernoulli shift. Finally, we investigate
the behavior of the system on a large finite box. © 1991
Plenum Publishing Corporation.},
Doi = {10.1007/bf01259549},
Key = {fds243461}
}

@article{fds243457,
Author = {Cox, JT and Durrett, R and Schinazi, R},
Title = {The critical contact process seen from the right
edge},
Journal = {Probability Theory and Related Fields},
Volume = {87},
Number = {3},
Pages = {325-332},
Publisher = {Springer Nature},
Year = {1991},
Month = {September},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/bf01312213},
Abstract = {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 it held starting in any semi-infinite
initial state. We prove the same for the critical contact
process. We also prove that the process starting with one
particle, conditioned to survive until time t, converges to
the unique invariant measure as t→∞. © 1991
Springer-Verlag.},
Doi = {10.1007/bf01312213},
Key = {fds243457}
}

@article{fds243459,
Author = {Durrett, R and M�ller, AM},
Title = {Complete convergence theorem for a competition
model},
Journal = {Probability Theory and Related Fields},
Volume = {88},
Number = {1},
Pages = {121-136},
Publisher = {Springer Nature},
Year = {1991},
Month = {March},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/bf01193585},
Abstract = {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 theorem holds and hence there is a unique
nontrivial stationary distribution. © 1991
Springer-Verlag.},
Doi = {10.1007/bf01193585},
Key = {fds243459}
}

@article{fds243455,
Author = {Durrett, R and Swindle, G},
Title = {Are there bushes in a forest?},
Journal = {Stochastic Processes and Their Applications},
Volume = {37},
Number = {1},
Pages = {19-31},
Publisher = {Elsevier BV},
Year = {1991},
Month = {January},
ISSN = {0304-4149},
url = {http://dx.doi.org/10.1016/0304-4149(91)90057-J},
Abstract = {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. © 1991.},
Doi = {10.1016/0304-4149(91)90057-J},
Key = {fds243455}
}

@article{fds243456,
Author = {Durrett, R and Kesten, H and Waymire, E},
Title = {On weighted heights of random trees},
Journal = {Journal of Theoretical Probability},
Volume = {4},
Number = {1},
Pages = {223-237},
Publisher = {Springer Nature},
Year = {1991},
Month = {January},
ISSN = {0894-9840},
url = {http://dx.doi.org/10.1007/bf01047004},
Abstract = {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 &lt;e&gt; we associate a
weight W(e). The weights are i.i.d. random variables and
independent of T. The weighted height of a self-avoiding
path in T starting at the root is the sum of the weights
associated with the path. We are interested in the
asymptotic distribution of the maximum weighted path height
in the limit as v=n→∞. Depending on the tail of the
weight distribution, we obtain the limit in three cases. In
particular if y2P(W(e)&gt; y)→0, then the limit
distribution depends strongly on the tree and, in fact, is
the distribution of the maximum of a Brownian excursion. If
the tail of the weight distribution is regularly varying
with exponent 0≤α&lt;2, then the weight swamps the tree
and the answer is the asymptotic distribution of the maximum
edge weight in the tree. There is a borderline case, namely,
P(W(e)&gt; y)∼cy-2 as y→∞, in which the limit
distribution exists but involves both the tree and the
weights in a more complicated way. © 1991 Plenum Publishing
Corporation.},
Doi = {10.1007/bf01047004},
Key = {fds243456}
}

@article{fds243458,
Author = {Bramson, M and Wan-ding, D and Durrett, R},
Title = {Annihilating branching processes},
Journal = {Stochastic Processes and Their Applications},
Volume = {37},
Number = {1},
Pages = {1-17},
Publisher = {Elsevier BV},
Year = {1991},
Month = {January},
ISSN = {0304-4149},
url = {http://dx.doi.org/10.1016/0304-4149(91)90056-I},
Abstract = {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 vacant site results. When δ = 0 product measure
with density 1 2 is a stationary distribution; we show it is
the limit whenever P(η0≠ ø) = 1. We also show that if δ
is small there is a nontrivial stationary distribution, and
that for any δ there are most two extremal translation
invariant stationary distributions. © 1991.},
Doi = {10.1016/0304-4149(91)90056-I},
Key = {fds243458}
}

@article{fds243453,
Author = {Jinwen, C and Durrett, R and Xiufang, L},
Title = {Exponential convergence for one dimensional contact
processes},
Journal = {Acta Mathematica Sinica, English Series},
Volume = {6},
Number = {4},
Pages = {349-353},
Publisher = {Springer Nature},
Year = {1990},
Month = {December},
ISSN = {1439-8516},
url = {http://dx.doi.org/10.1007/BF02107968},
Abstract = {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 to occur with exponential
rapidity. © 1990 Springer-Verlag.},
Doi = {10.1007/BF02107968},
Key = {fds243453}
}

@article{fds243452,
Author = {Ding, W-D and Durrett, R and Liggett, TM},
Title = {Ergodicity of reversible reaction diffusion
processes},
Journal = {Probability Theory and Related Fields},
Volume = {85},
Number = {1},
Pages = {13-26},
Publisher = {Springer Nature},
Year = {1990},
Month = {March},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/bf01377624},
Abstract = {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 measure. We show that the stationary
distribution is unique and is the limit starting from any
initial distribution. © 1990 Springer-Verlag.},
Doi = {10.1007/bf01377624},
Key = {fds243452}
}

@article{fds243454,
Author = {Cox, JT and Durrett, R},
Title = {Large deviations for independent random walks},
Journal = {Probability Theory and Related Fields},
Volume = {84},
Number = {1},
Pages = {67-82},
Publisher = {Springer Nature},
Year = {1990},
Month = {March},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/bf01288559},
Abstract = {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 large deviations of
Ln(0)-Ln(1). The behavior we find is much different from
that of Ln(0). We investigate the limiting behavior when the
initial configurations has asymptotic density 1 and when
ξ0(x) are i.i.d Poisson mean 1, finding that the
asymptotics are different in these two cases. © 1990
Springer-Verlag.},
Doi = {10.1007/bf01288559},
Key = {fds243454}
}

@article{fds243450,
Author = {Durrett, R and Tanaka, NI},
Title = {Scaling inequalities for oriented percolation},
Journal = {Journal of Statistical Physics},
Volume = {55},
Number = {5-6},
Pages = {981-995},
Publisher = {Springer Nature},
Year = {1989},
Month = {June},
ISSN = {0022-4715},
url = {http://dx.doi.org/10.1007/bf01041075},
Abstract = {We look at seven critical exponents associated with
two-dimensional oriented percolation. Scaling theory implies
that these quantities satisfy four equalities. We prove five
related inequalitites. © 1989 Plenum Publishing
Corporation.},
Doi = {10.1007/bf01041075},
Key = {fds243450}
}

@article{fds243451,
Author = {Durrett, R and Schonmann, RH and Tanaka, NI},
Title = {Correlation lengths for oriented percolation},
Journal = {Journal of Statistical Physics},
Volume = {55},
Number = {5-6},
Pages = {965-979},
Publisher = {Springer Nature},
Year = {1989},
Month = {June},
ISSN = {0022-4715},
url = {http://dx.doi.org/10.1007/bf01041074},
Abstract = {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 relationship between the various
definitions. The definitions are used in a companion paper
to prove inequalities between critical exponents. © 1989
Plenum Publishing Corporation.},
Doi = {10.1007/bf01041074},
Key = {fds243451}
}

@article{fds243447,
Author = {Bramson, M and Durrett, R},
Title = {A simple proof of the stability criterion of Gray and
Griffeath},
Journal = {Probability Theory and Related Fields},
Volume = {80},
Number = {2},
Pages = {293-298},
Publisher = {Springer Nature America, Inc},
Year = {1988},
Month = {December},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/BF00356107},
Abstract = {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 under small perturbations. In this
paper we use a renormalized site construction to give a much
simpler proof. Our new approach can be used in many
situations as a substitute for the contour method. © 1988
Springer-Verlag.},
Doi = {10.1007/BF00356107},
Key = {fds243447}
}

@article{fds243448,
Author = {Cox, JT and Durrett, R},
Title = {Limit theorems for the spread of epidemics and forest
fires},
Journal = {Stochastic Processes and Their Applications},
Volume = {30},
Number = {2},
Pages = {171-191},
Publisher = {Elsevier BV},
Year = {1988},
Month = {December},
ISSN = {0304-4149},
url = {http://dx.doi.org/10.1016/0304-4149(88)90083-x},
Abstract = {We prove that the "spatial epidemic with removal" grows
linearly and has an asymptotic shape on the set of
nonextinction. © 1988.},
Doi = {10.1016/0304-4149(88)90083-x},
Key = {fds243448}
}

@article{fds243446,
Author = {Bramson, M and Durrett, R},
Title = {Random walk in random environment: A counterexample?},
Journal = {Communications in Mathematical Physics},
Volume = {119},
Number = {2},
Pages = {199-211},
Publisher = {Springer Nature},
Year = {1988},
Month = {June},
ISSN = {0010-3616},
url = {http://dx.doi.org/10.1007/bf01217738},
Abstract = {We describe a family of random walks in random environments
which have exponentially decaying correlations, nearest
neighbor transition probabilities which are bounded away
from 0, and yet are subdiffusive in any dimension d&lt;∞.
© 1988 Springer-Verlag.},
Doi = {10.1007/bf01217738},
Key = {fds243446}
}

@article{fds323658,
Author = {Durrett, R},
Title = {Crabgrass, measles and gypsy moths: An introduction to
modern probability},
Journal = {Bulletin of the American Mathematical Society},
Volume = {18},
Number = {2},
Pages = {117-144},
Publisher = {American Mathematical Society (AMS)},
Year = {1988},
Month = {April},
url = {http://dx.doi.org/10.1090/s0273-0979-1988-15625-x},
Doi = {10.1090/s0273-0979-1988-15625-x},
Key = {fds323658}
}

@article{fds243444,
Author = {Durrett, R},
Title = {Crabgrass, measles, and gypsy moths: An introduction to
interacting particle systems},
Journal = {The Mathematical Intelligencer},
Volume = {10},
Number = {2},
Pages = {37-47},
Publisher = {Springer Nature},
Year = {1988},
Month = {March},
ISSN = {0343-6993},
url = {http://dx.doi.org/10.1007/bf03028355},
Doi = {10.1007/bf03028355},
Key = {fds243444}
}

@article{fds243445,
Author = {Chayes, JT and Chayes, L and Durrett, R},
Title = {Connectivity properties of Mandelbrot's percolation
process},
Journal = {Probability Theory and Related Fields},
Volume = {77},
Number = {3},
Pages = {307-324},
Publisher = {Springer Nature America, Inc},
Year = {1988},
Month = {March},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/BF00319291},
Abstract = {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 connectivity or "percolation" properties of these sets,
proving all of the claims he made in Sect. 23 of the
"Fractal Geometry of Nature" and a new one that he did not
anticipate: There is a probability pc∈(0,1) so that if
p<pc then the set is "duslike" i.e., the largest connected
component is a point, whereas if p≧pc (notice the =)
opposing sides are connected with positive probability and
furthermore if we tile the plane with independent copies of
the system then there is with probability one a unique
unbounded connected component which intersects a positive
fraction of the tiles. More succinctly put the system has a
first order phase transition. © 1988 Springer-Verlag.},
Doi = {10.1007/BF00319291},
Key = {fds243445}
}

@article{fds243449,
Author = {Durrett, R and Schonmann, RH},
Title = {Large deviations for the contact process and two dimensional
percolation},
Journal = {Probability Theory and Related Fields},
Volume = {77},
Number = {4},
Pages = {583-603},
Publisher = {Springer Nature},
Year = {1988},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/bf00959619},
Abstract = {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 number of sites considered. 2) For
supercritical two-dimensional nearest neighbor site (or
bond) percolation the density YΛ of sites inside a square
Λ which belong to the infinite cluster has the following
large deviation properties. The probability that YΛ
deviates from its expected value by a positive amount decays
exponentially with the area of Λ, while the probability
that it deviates from its expected value by a negative
amount decays exponentially with the perimeter of Λ. These
two problems are treated together in this paper because
similar techniques (renormalization) are used for both. ©
1988 Springer-Verlag.},
Doi = {10.1007/bf00959619},
Key = {fds243449}
}

@article{fds243442,
Author = {Brennan, MD and Durrett, R},
Title = {Splitting intervals II: Limit laws for lengths},
Journal = {Probability Theory and Related Fields},
Volume = {75},
Number = {1},
Pages = {109-127},
Publisher = {Springer Nature America, Inc},
Year = {1987},
Month = {May},
ISSN = {0178-8051},
url = {http://dx.doi.org/10.1007/BF00320085},
Abstract = {In the processes under consideration, a particle of size L
splits with exponential rate L α , 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 Z t be the number of particles at time t and L t the
size of a randomly chosen particle. If α=0, it is well
known how the system evolves: e -t Z t converges a.s. to an
exponential r.v. and -L t ≈t + Ct 1/2 X where X is a
standard normal t.v. Our results for α>0 are in sharp
contrast. In "Splitting Intervals" we showed that t -1/α Z
t converges a.s. to a constant K>0, and in this paper we
show {Mathematical expression}. We show that the empirical
d.f. of the rescaled lengths, {Mathematical expression},
converges a.s. to a non-degenerate limit depending on F that
we explicitly describe. Our results with α=2/3 are relevant
to polymer degradation. © 1987 Springer-Verlag.},
Doi = {10.1007/BF00320085},
Key = {fds243442}
}

@article{fds243443,
Author = {Chayes, JT and Chayes, L and Durrett, R},
Title = {Inhomogeneous percolation problems and incipient infinite
clusters},
Journal = {Journal of Physics A: Mathematical and General},
Volume = {20},
Number = {6},
Pages = {1521-1530},
Publisher = {IOP Publishing},
Year = {1987},
Month = {April},
ISSN = {0305-4470},
url = {http://dx.doi.org/10.1088/0305-4470/20/6/034},
Abstract = {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 length
exponent v for the homogeneous percolation model, they prove
that in d=2, the borderline value of lambda is lambda b=1/v.
If lambda &gt;1/v then, with probability one, there is no
infinite cluster, while if lambda &lt;1/v then, with
positive probability, the origin is part of an infinite
cluster. This result sheds some light on numerical and
theoretical predictions of certain properties of incipient
infinite clusters. Furthermore, for d&gt;2, the models
studied suggest what sort of 'incipient objects' should be
examined in random surface models.},
Doi = {10.1088/0305-4470/20/6/034},
Key = {fds243443}
}

@article{fds243440,
Author = {Chayes, JT and Chayes, L and Durrett, R},
Title = {Critical behavior of the two-dimensional first passage
time},
Journal = {Journal of Statistical Physics},
Volume = {45},
Number = {5-6},
Pages = {933-951},
Publisher = {Springer Nature},
Year = {1986},
Month = {December},
ISSN = {0022-4715},
url = {http://dx.doi.org/10.1007/bf01020583},
Abstract = {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 exhibits the same critical behavior as the correlation
function of the underlying percolation problem. In
particular, if the correlation length obeys ξ(p)
∼|p-pc|-v, then the first passage time constant satisfies
μ(p)∼|p-pc|v. At pc, where it has been asserted that the
first passage time from 0 to x scales as |x| to a power ψ
with 0&lt;ψ&lt;1, we show that the passage times grow like
log |x|, i.e., the fluid spreads exponentially rapidly. ©
1986 Plenum Publishing Corporation.},
Doi = {10.1007/bf01020583},
Key = {fds243440}
}

@article{fds243441,
Author = {Durrett, R},
Title = {Multidimensional random walks in random environments with
subclassical limiting behavior},
Journal = {Communications in Mathematical Physics},
Volume = {104},
Number = {1},
Pages = {87-102},
Publisher = {Springer Nature},
Year = {1986},
Month = {March},
ISSN = {0010-3616},
url = {http://dx.doi.org/10.1007/bf01210794},
Abstract = {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. Our results make rigorous a
heuristic argument of Marinari et al. (1983), and show that
in any d&lt;∞ we can have (a)Xn is recurrent and
(b)Xn∼(log n)2. © 1986 Springer-Verlag.},
Doi = {10.1007/bf01210794},
Key = {fds243441}
}

@article{fds243439,
Author = {Durrett, R},
Title = {Some general results concerning the critical exponents of
percolation processes},
Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
Gebiete},
Volume = {69},
Number = {3},
Pages = {421-437},
Publisher = {Springer Nature America, Inc},
Year = {1985},
Month = {September},
ISSN = {0044-3719},
url = {http://dx.doi.org/10.1007/BF00532742},
Abstract = {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 one side for what happens in general.
These results and their proofs are closely related to their
analogues for the Ising model. © 1985 Springer-Verlag.},
Doi = {10.1007/BF00532742},
Key = {fds243439}
}

@article{fds243438,
Author = {Durrett, R and Nguyen, B},
Title = {Thermodynamic inequalities for percolation},
Journal = {Communications in Mathematical Physics},
Volume = {99},
Number = {2},
Pages = {253-269},
Publisher = {Springer Nature},
Year = {1985},
Month = {June},
ISSN = {0010-3616},
url = {http://dx.doi.org/10.1007/bf01212282},
Abstract = {In this paper we describe the percolation analogues of the
Gibbs and Helmholtz potentials and use these quantities to
prove some general inequalities concerning the critical
exponents of percolation processes. © 1985
Springer-Verlag.},
Doi = {10.1007/bf01212282},
Key = {fds243438}
}

@article{fds243437,
Author = {Durrett, R and Liggett, TM},
Title = {Fixed points of the smoothing transformation},
Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
Gebiete},
Volume = {64},
Number = {3},
Pages = {275-301},
Publisher = {Springer Nature},
Year = {1983},
Month = {September},
ISSN = {0044-3719},
url = {http://dx.doi.org/10.1007/BF00532962},
Abstract = {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 1X1+ ... + W N X N, where the X i are independent
random variables with distribution μ, which are independent
of W 1,..., W N as well. In earlier work, first Kahane and
Peyriere, and then Holley and Liggett, obtained necessary
and sufficient conditions for T to have a nontrivial fixed
point of finite mean in the special cases that the W i are
independent and identically distributed, or are fixed
multiples of one random variable. In this paper we study the
transformation in general. Assuming only that for some γ>1,
EW i γ <∞ for all i, we determine exactly when T has a
nontrivial fixed point (of finite or infinite mean). When it
does, we find all fixed points and prove a convergence
result. In particular, it turns out that in the previously
considered cases, T always has a nontrivial fixed point. Our
results were motivated by a number of open problems in
infinite particle systems. The basic question is: in those
cases in which an infinite particle system has no invariant
measures of finite mean, does it have invariant measures of
infinite mean? Our results suggest possible answers to this
question for the generalized potlatch and smoothing
processes studied by Holley and Liggett. © 1983
Springer-Verlag.},
Doi = {10.1007/BF00532962},
Key = {fds243437}
}

@article{fds243435,
Author = {Durrett, R},
Title = {Maxima of branching random walks},
Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
Gebiete},
Volume = {62},
Number = {2},
Pages = {165-170},
Publisher = {Springer Nature America, Inc},
Year = {1983},
Month = {June},
ISSN = {0044-3719},
url = {http://dx.doi.org/10.1007/BF00538794},
Abstract = {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 has φ{symbol}(θ) = ∝ exp (θx)dF(x)<∞ for
some θ>0. In this paper we investigate what happens when
there is a slowly varying function K so that
1-F(x)∼x}-qK(x) as x → ∞ and log (-x)F(x)→0 as
x→-∞. In this case we find that there is a sequence of
constants an, which grow exponentially, so that Ln/an
converges weakly to a nondegenerate distribution. This
result is in sharp contrast to the linear growth of Ln
observed in the case φ{symbol}(θ)<∞. © 1983
Springer-Verlag.},
Doi = {10.1007/BF00538794},
Key = {fds243435}
}

@article{fds243436,
Author = {Chung, KL and Durrett, R and Zhao, Z},
Title = {Extension of domains with finite gauge},
Journal = {Mathematische Annalen},
Volume = {264},
Number = {1},
Pages = {73-79},
Publisher = {Springer Nature},
Year = {1983},
Month = {March},
ISSN = {0025-5831},
url = {http://dx.doi.org/10.1007/bf01458051},
Doi = {10.1007/bf01458051},
Key = {fds243436}
}

@article{fds323659,
Author = {Cox, JT and Durrett, R},
Title = {Oriented percolation in dimensions d ≥ 4: bounds and
asymptotic formulas},
Journal = {Mathematical Proceedings of the Cambridge Philosophical
Society},
Volume = {93},
Number = {01},
Pages = {151-151},
Publisher = {Cambridge University Press (CUP)},
Year = {1983},
Month = {January},
url = {http://dx.doi.org/10.1017/s0305004100060436},
Doi = {10.1017/s0305004100060436},
Key = {fds323659}
}

@article{fds243433,
Author = {Durrett, R and Griffeath, D},
Title = {Contact processes in several dimensions},
Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
Gebiete},
Volume = {59},
Number = {4},
Pages = {535-552},
Publisher = {Springer Nature America, Inc},
Year = {1982},
Month = {December},
ISSN = {0044-3719},
url = {http://dx.doi.org/10.1007/BF00532808},
Doi = {10.1007/BF00532808},
Key = {fds243433}
}

@article{fds243434,
Author = {Durrett, R},
Title = {An introduction to infinite particle systems},
Journal = {Stochastic Processes and Their Applications},
Volume = {11},
Number = {2},
Pages = {109-150},
Publisher = {Elsevier BV},
Year = {1981},
Month = {May},
ISSN = {0304-4149},
url = {http://dx.doi.org/10.1016/0304-4149(81)90001-6},
Abstract = {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 object of much research in
the last ten years. In this paper we will survey some of the
results which have been obtained and some of the open
problems, concentrating on six overlapping classes of
processes: the voter model, additive processes, the
exponential family, one dimensional systems, attractive
systems, and the Ising model. © 1981.},
Doi = {10.1016/0304-4149(81)90001-6},
Key = {fds243434}
}

@article{fds243432,
Author = {Durrett, R},
Title = {Conditioned limit theorems for random walks with negative
drift},
Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
Gebiete},
Volume = {52},
Number = {3},
Pages = {277-287},
Publisher = {Springer Nature},
Year = {1980},
ISSN = {0044-3719},
url = {http://dx.doi.org/10.1007/bf00538892},
Abstract = {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&gt;n) converges weakly to a
process which he called the Brownian excursion. It will be
shown that his conjecture is false or, more precisely, that
if ES1=-a&lt;0, ES12&lt;∞, and there is a slowly varying
function L so that P(S1&gt;x)∼x-q L(x) as x→∞ then
(S[n.]/n|Sn&gt;0) and (S[n.]/n|N&gt;n) converge weakly to
nondegenerate limits. The limit processes have sample paths
which have a single jump (with d.f. (1-(x/a)-q)+) and are
otherwise linear with slope -a. The jump occurs at a
uniformly distributed time in the first case and at t=0 in
the second. © 1980 Springer-Verlag.},
Doi = {10.1007/bf00538892},
Key = {fds243432}
}

@article{fds243430,
Author = {Durrett, R},
Title = {Maxima of branching random walks vs. independent random
walks},
Journal = {Stochastic Processes and Their Applications},
Volume = {9},
Number = {2},
Pages = {117-135},
Publisher = {Elsevier BV},
Year = {1979},
Month = {November},
ISSN = {0304-4149},
url = {http://dx.doi.org/10.1016/0304-4149(79)90024-3},
Abstract = {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 walks. A comparison of our
results with the known results of branching random walk then
identifies the limit behaviors which are due to the number
of particles and those which are determined by the branching
structure. © 1979.},
Doi = {10.1016/0304-4149(79)90024-3},
Key = {fds243430}
}

@article{fds243431,
Author = {Anthony, PF and Durrett, R and Pulec, JL and Hartstone,
JL},
Title = {A new parameter in brain stem evoked response: Component
wave areas},
Journal = {Laryngoscope},
Volume = {89},
Number = {10 I},
Pages = {1569-1578},
Year = {1979},
Abstract = {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 breaking down the complex
BSER curve, using a successive approximation technique, into
its individual component curves. Each component curve is
thought to represent the isolated electrical activity of one
generator site. The component curves are the shape of normal
distribution curves. The area under each individual
component curve is the new parameter which the authors feel
represents the isolated electrical activity of one generator
site. Using this parameter a clinical trial was performed.
Constant ipsilateral pure tone masking was superimposed upon
the stimulus clicks in the test ears of two subjects. The
constant ipsilateral masking was superimposed at 500 Hz,
1000 Hz and 2000 Hz. A statistically significant decrease in
the area of component wave II (one masking sound caused an
increased area) was seen when the pure tone masking sound
became at least 15-25 db louder than the patient's threshold
at that individual frequency. These preliminary results give
reason to think that a method of quantification of BSER
responses has been found. More importantly, a method of
identifying individual audiometric thresholds from 500 Hz
throughout 2000 Hz to within 15-25 db has been found. These
findings need extensive further testing.},
Key = {fds243431}
}

@article{fds243428,
Author = {Durrett, R},
Title = {The genealogy of critical branching processes},
Journal = {Stochastic Processes and Their Applications},
Volume = {8},
Number = {1},
Pages = {101-116},
Publisher = {Elsevier BV},
Year = {1978},
Month = {November},
ISSN = {0304-4149},
url = {http://dx.doi.org/10.1016/0304-4149(78)90071-6},
Abstract = {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 branching random walks. ©
1978.},
Doi = {10.1016/0304-4149(78)90071-6},
Key = {fds243428}
}

@article{fds243429,
Author = {Durrett, RT and Resnick, SI},
Title = {Weak convergence with random indices},
Journal = {Stochastic Processes and Their Applications},
Volume = {5},
Number = {3},
Pages = {213-220},
Publisher = {Elsevier BV},
Year = {1977},
Month = {January},
ISSN = {0304-4149},
url = {http://dx.doi.org/10.1016/0304-4149(77)90031-X},
Abstract = {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 in
specific ways and the process Y possesses a scaling
property. If {Nn} are positive integer valued random
variables we discuss when YNn → Y and Y'n=(X[Nn]-bn)/an
⇒ Y'. Results given subsume random index limit theorems
for convergence to Brownian motion, stable processes and
extremal processes. © 1977.},
Doi = {10.1016/0304-4149(77)90031-X},
Key = {fds243429}
}

@article{fds243426,
Author = {Chung, KL and Durrett, R},
Title = {Downcrossings and local time},
Journal = {Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte
Gebiete},
Volume = {35},
Number = {2},
Pages = {147-149},
Publisher = {Springer Nature America, Inc},
Year = {1976},
Month = {June},
ISSN = {0044-3719},
url = {http://dx.doi.org/10.1007/BF00533319},
Doi = {10.1007/BF00533319},
Key = {fds243426}
}

@article{fds243427,
Author = {Durrett, RT and Ghurye, SG},
Title = {WAITING TIMES WITHOUT MEMORY.},
Journal = {J Appl Probab},
Volume = {13},
Number = {1},
Pages = {65-75},
Year = {1976},
Abstract = {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 as the unconditional probability
of T greater than y; in other words, the physical process
operates as if it has no memory concerning the successive
occurrences of a certain event. The paper investigates the
consequences of defining the property of lack of memory on
more general time-domains than the positive reals. As a side
issue, there is discussion of a stochastic variation of
Cauchy's functional equation.},
Key = {fds243427}
}

%% Papers Accepted
@article{fds220503,
Author = {R. Durrett and S. Moseley},
Title = {Spatial Moran Models. I. Tunneling in the Neutral
Case},
Journal = {Annals Applied Probability},
Year = {2013},
Key = {fds220503}
}

%% Papers Submitted
@article{fds220504,
Author = {S. Magura and V. Pong and D. Sivakoff and R. Durrett},
Title = {Two evolving social network models},
Year = {2013},
Key = {fds220504}
}

@article{fds211319,
Author = {R. Durrett},
Title = {Phase transition in a meta-population version of Schelling's
model},
Year = {2012},
Key = {fds211319}
}

@article{fds211320,
Author = {R. Durrett and J. Foo and K. Leder},
Title = {Spatial Moran Models II. Tumor growth and
progression},
Year = {2012},
Key = {fds211320}
}

`

dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320