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 0${}={}$vacant, 1${}={}$juvenile and 2${}={}$a mature individual capable of giving birth, and the Staver-Levin forest model in which 0${}={}$grass, 1${}={}$sapling and 2${}={}$tree. Our first result shows that if $(0,0)$ 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 $(0,0)$ 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

Contact processes on random graphs with power law degree distributions have critical value 0

If we consider the contact process with infection rate $\lambda$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $\lambda_c$ of the infection rate is positive if the power $\alpha>3$. 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 $\lambda_c$ is zero for any value of $\alpha>3$, and the contact process starting from all vertices infected, with a probability tending to 1 as $n\to\infty$, maintains a positive density of infected sites for time at least $\exp(n^{1-\delta})$ for any $\delta>0$. 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 $\rho(\lambda)$. It is expected that $\rho(\lambda)\sim C\lambda^{\beta}$ as $\lambda \to0$. Here we show that $\alpha-1\le\beta\le2\alpha-3$, and so $\beta>2$ for $\alpha>3$. Thus even though the graph is locally tree-like, $\beta$ does not take the mean field critical value $\beta=1$.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

Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations

We consider a branching-selection system in $\mathbb {R}$ 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 $N\to\infty$, 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 $c\geq a$ or $c<a$, where $a$ 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 $m$ 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