Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.Comment: 23 pages, 1 figur

On Recent Progress for the Stochastic Navier Stokes Equations

We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations. Since our desire is to make the core ideas clear, we will mostly work with a specific example: the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. A few general theorems are given. Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and hypoellipticity are all discussed.Comment: Corrected version of Journees Equations aux derivees partielles paper(June 2003). Original at http://www.math.sciences.univ-nantes.fr/edpa/2003

A practical criterion for positivity of transition densities

We establish a simple criterion for locating points where the transition density of a degenerate diffusion is strictly positive. Throughout, we assume that the diffusion satisfies a stochastic differential equation (SDE) on $\mathbf{R}^d$ with additive noise and polynomial drift. In this setting, we will see that it is often that case that local information of the flow, e.g. the Lie algebra generated by the vector fields defining the SDE at a point $x\in \mathbf{R}^d$, determines where the transition density is strictly positive. This is surprising in that positivity is a more global property of the diffusion. This work primarily builds on and combines the ideas of Ben Arous and L\'eandre (1991) and Jurdjevic and Kupka (1981, 1985).Comment: 24 page

Noise-Induced Stabilization of Planar Flows I

We show that the complex-valued ODE \begin{equation*} \dot z_t = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0, \end{equation*} which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II of this paper extends the main results to the general setting.Comment: Part one of a two part pape

An Elementary Proof of the Existence and Uniqueness Theorem for the Navier-Stokes Equations

We give a geometric approach to proving know regularity and existence theorems for the 2D Navier-Stokes Equations. We feel this point of view is instructive in better understanding the dynamics. The technique is inspired by constructions in the Dynamical Systems.Comment: 15 Page

Evaluating Partisan Gerrymandering in Wisconsin

We examine the extent of gerrymandering for the 2010 General Assembly district map of Wisconsin. We find that there is substantial variability in the election outcome depending on what maps are used. We also found robust evidence that the district maps are highly gerrymandered and that this gerrymandering likely altered the partisan make up of the Wisconsin General Assembly in some elections. Compared to the distribution of possible redistricting plans for the General Assembly, Wisconsin's chosen plan is an outlier in that it yields results that are highly skewed to the Republicans when the statewide proportion of Democratic votes comprises more than 50-52% of the overall vote (with the precise threshold depending on the election considered). Wisconsin's plan acts to preserve the Republican majority by providing extra Republican seats even when the Democratic vote increases into the range when the balance of power would shift for the vast majority of redistricting plans.Comment: Slightly updated version of initially released report dated September 2, 2017. Typos were corrected and some wording improve