Browsing by Author "Durrett, Richard T"

Interacting particle systems have been applied to model the spread of infectious diseases and opinions, interactions between competing species, and evolution of forest landscapes. In this thesis, we study three spatial models arising from from ecology and social sciences. First, in a model introduced by Schelling in 1971, in which families move if they have too many neighbors of the opposite type, we study the phase transition between a randomly distributed and a segregated equilibrium. Second, we consider a combination of the contact process and the voter model and study the asymptotics of the critical value of the contact part as the rate of the voting term goes to infinity. Third, we consider a family of attractive stochastic spatial models, one of which is introduced by Staver and Levin to describe the coverage of forest. We prove that the mean-field ODE gives the asymptotically sharp phase diagram for existence of stationary distributions, while for Staver and Levin model there can still be non-trivial stationary distributions even when the absorbing fixed point of the mean-field ODE is stable.

We study the contact process on random graphs and trees.

\medskipIn Chapter \ref{periodictree} we study the asymptotics for the critical values $\lambda_1$ and $\lambda_2$ on a general class of periodic trees. A little over 25 years ago Pemantle \cite{Pemantle} pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values $\lambda_1$ and $\lambda_2$ for global and local survival were different. He also considered trees with periodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generations is $(n,a_1,\ldots, a_k)$ with $\max_i a_i \le Cn^{1-\delta}$ and $\log(a_1 \cdots a_k)/\log n \to b$ as $n\to\infty$. We show that the critical value for local survival is asymptotically $\sqrt{c (\log n)/n}$ where $c=(k-b)/2$. This supports Pemantle's claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.

\medskipIn Chapter \ref{CPonGW} we study the contact process on Galton-Watson trees and configuration models. The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $\lambda_2=0$ and (ii) when it is Geometric($p$) we have $\lambda_2 \le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $\lambda_c(n)$ for ``prolonged persistence'' on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $\lambda_c(n) \to 0$ we give estimates on the rate of convergence. It was predicted in physics papers that $\lambda_c(n) \sim 1/\Lambda(n)$ where $\Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is accurate for graphs with power-law degree distributions, but not for stretched exponentials.

\medskipIn Chapter \ref{expCP} we study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $\lambda_1$ for weak survival, and the survival probability $p(\lambda)$ is continuous with respect to the infection rate $\lambda$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $\lambda_1<\lambda_2$, which confirms a conjecture of Stacey's \cite{Stacey}. We also prove that if the contact process survives strongly at $\lambda$ then it survives strongly at a $\lambda'<\lambda$, which implies that the process does not survive strongly at the critical value $\lambda_2$ for strong survival.

The evoSIR model is a modification of the usual SIR process on a graph $G$ in which $S$-$I$ connections are broken at rate $\rho$ and the $S$ connects to a randomly chosen vertex. The evoSI model is the same as evoSI but recovery is impossible. In \cite{DOMath} the critical value for evoSIR was computed and simulations showed that when $G$ is an Erd\H os-R\'enyi graph with mean degree 5 the system has a discontinuous phase transition, i.e., as the infection rate $\lambda$ decreases to $\lambda_c$, the final fraction of once infected individuals does not converge to 0. In this paper we study evoSI dynamics on graphs generated by the configuration model. We show that there is a quantity $\Delta$ determined by the first three moments of the degree distribution, so that the transition is discontinuous if $\Delta>0$ and continuous if $\Delta<0$.

We study three types of interacting particle systems arising from biology.

In Chapter 2, 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 $x + {\cal N} _ c$, while a cheater at $x$ gives birth at rate $\lambda> 1$ times the fraction of producers in $x + {\cal N}_d$ and sends its offspring to a randomly chosen point in $x + {\cal N}_c$. We first study this game on the $d$-dimensional torus $\TL$ with ${\cal N}_d = \TL$ and ${\cal N} _ c$ = the $2d$ nearest neighbors. If we let $L \to \infty$ then $t \to \infty$ the fraction of producers converges to $1/\lambda$. In $d \ge 3$ the limiting finite dimensional distributions converge as $t \to \infty$ to the voter model equilibrium with density $1/\lambda$ . We next reformulate the system as an evolutionary game with ``birth-death'' updating and take ${\cal N}_c = {\cal N}_d = \cal N}$. Using results for voter model perturbations we show that in $d = 3$ with ${\cal N} =$ the six nearest neighbors, the density of producers converges to $(2/\lambda)-0.5$ for $4/3 < \lambda < 4$. Producers take over the system when $\lambda < 4/3$ and die out when $\lambda >4$. In $d=2$ with ${\cal N} = [-c\sqrt{\log N},c\sqrt{\log N}]^2$ there are similar phase transitions, with coexistence occurring when $(1+2\theta)/(1+\theta) < \lambda < (1+2\theta) / \theta$ where $\theta = (e^{3/(\pi c^2)}-1)/2$.

In Chapter 3, we formulate a nonhomogeneous spatial model of the competition between forest and savanna. In work with a variety of co-authors, Staver and Levin have argued that savanna and forest coexist as alternative stable states with discontinuous changes in density of trees at the boundary. We prove this claim and that coexistence occurs for a time that is exponential in the size of the system, and that after an initial transient, boundaries between the alternative equilibria remain stable.

In Chapter 4, we study a two-level contact process. We think of fleas living on a species of animals. We let the animals follow the law of a supercritical contact process in $\mathbb{ Z }^d$ with parameter $ \lambda $. The contact process acts as the random environment where neutral symbionts (``fleas'') grow. The fleas give birth at rate $\mu$ when they are living on a host animal, and the fleas die at rate $\delta$ when they do not have a host animal. Using a block construction, we show that if the contact process is supercritical, the fleas survive with a positive probability for any sufficiently large $\mu$. The main result of this chapter is the complete convergence theorem of the model.

This dissertation consists of two projects in mathematical biology. The first project studies tumor heterogeneity through the site frequency spectrum, the expected number of mutations with frequency greater than $f$. Recent work of Sottoriva, Graham, and collaborators have led to the controversial claim that exponentially growing tumors have a site frequency spectrum that follows the $1/f$ law consistent with neutral evolution. This conclusion has been criticized based on data quality issues, statistical considerations, and simulation results. Here, we use rigorous mathematical arguments to investigate the site frequency spectrum in the two-type model of clonal evolution. If the fitnesses of the two types are $\lambda_0<\lambda_1$, then the site frequency spectrum is $c/f^\alpha$ where $\alpha=\lambda_0/\lambda_1$. This deviation from the $1/f$ law is due to the advantageous mutations that produce the founders of the type 1 population; mutations within the growing type 0 and type 1 populations still follow the $1/f$ law. Our results show that, in contrast to published criticisms, neutral evolution in an exponentially growing tumor can be distinguished from the two-type model using the site frequency spectrum.

The second project considers whether three species can coexist in a resource competition model with two seasons. Investigating how temporal variation in environment affects species coexistence has been of longstanding interest. The competitive exclusion principle states that $n$ niches can support at most $n$ species, but what constitutes a niche is not always clear. For example, Hutchinson in 1961 drew attention to the diversity of phytoplankton coexisting despite the small number of resources in ocean water. Hutchinson then suggested that this could be explained by a changing environment; times when different species are favored would be considered different niches. In this paper, we examine a model where three species interact with each other solely through the consumption of one resource. The growth per resource rates, death rates, resource rates, and methods of resource consumption vary periodically through time. We give a necessary and sufficient condition for the coexistence of all three species. In particular, this condition rules out coexistence for the mean field limit of a three species two seasons model studied by Chan, Durrett, and Lanchier in 2009.