128 research outputs found
Coexistence in stochastic spatial models
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.Comment: Published in at http://dx.doi.org/10.1214/08-AAP590 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Population genetics of neutral mutations in exponentially growing cancer cell populations
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.Comment: Published in at http://dx.doi.org/10.1214/11-AAP824 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Coexistence of grass, saplings and trees in the Staver-Levin forest model
In this paper, we consider two attractive stochastic spatial models in which
each site can be in state 0, 1 or 2: Krone's model in which 0vacant,
1juvenile and 2a mature individual capable of giving birth, and
the Staver-Levin forest model in which 0grass, 1sapling and
2tree. Our first result shows that if is an unstable fixed point
of the mean-field ODE for densities of 1's and 2's then when the range of
interaction is large, there is positive probability of survival starting from a
finite set and a stationary distribution in which all three types are present.
The result we obtain in this way is asymptotically sharp for Krone's model.
However, in the Staver-Levin forest model, if is attracting then there
may also be another stable fixed point for the ODE, and in some of these cases
there is a nontrivial stationary distribution.Comment: Published at http://dx.doi.org/10.1214/14-AAP1079 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations
We consider a branching-selection system in with 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 , 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
or no such solution depending on whether or , where 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.Comment: Published in at http://dx.doi.org/10.1214/10-AOP601 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Random Oxford Graphs
Inspired by a concept in comparative genomics, we investigate properties of
randomly chosen members of G_1(m,n,t), the set of bipartite graphs with
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
trees they contain. We compute the thresholds for the emergence of a
giant component and for the graph to be connected
Contact processes on random graphs with power law degree distributions have critical value 0
If we consider the contact process with infection rate on a random
graph on vertices with power law degree distributions, mean field
calculations suggest that the critical value of the infection rate
is positive if the power . Physicists seem to regard this as an
established fact, since the result has recently been generalized to bipartite
graphs by G\'{o}mez-Garde\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008)
1399--1404]. Here, we show that the critical value is zero for any
value of , and the contact process starting from all vertices
infected, with a probability tending to 1 as , maintains a positive
density of infected sites for time at least for any
. 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 . It is
expected that as . Here we
show that , and so for . Thus
even though the graph is locally tree-like, does not take the mean
field critical value .Comment: Published in at http://dx.doi.org/10.1214/09-AOP471 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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