Acoustic modes of a spherical thin-walled tank for liquid propellant mass gauging: Theory and experiment.

Acoustic response of a thin-walled spherical flight tank filled with water is explored theoretically and experimentally as a testbed for an application of Weyl's law to the problem of mass-gauging propellants in zero-gravity in space. Weyl's law relates the mode counting function of a resonator to its volume and can be used to infer the volume of liquid in a tank from the tank's acoustic response. One of the challenges of applying Weyl's law to real tanks is to account for the boundary conditions which are neither Neumann nor Dirichlet. We show that the liquid modes in a thin-walled spherical tank correspond to the spectrum of a slightly larger spherical tank with infinitely compliant wall (Dirichlet boundary condition), where Weyl's law can be applied directly. The mass of the liquid enclosed by this "effective" tank's wall is found to equal the actual mass of the liquid plus the mass of the wall. This finding is generalized to thin-walled tanks and liquid configurations of arbitrary shapes and thus provides a calculable correction factor for the propellant mass inferred using Weyl's law with Dirichlet boundary conditions.

Two-strain competition in quasineutral stochastic disease dynamics.

We develop a perturbation method for studying quasineutral competition in a broad class of stochastic competition models and apply it to the analysis of fixation of competing strains in two epidemic models. The first model is a two-strain generalization of the stochastic susceptible-infected-susceptible (SIS) model. Here we extend previous results due to Parsons and Quince [Theor. Popul. Biol. 72, 468 (2007)], Parsons et al. [Theor. Popul. Biol. 74, 302 (2008)], and Lin, Kim, and Doering [J. Stat. Phys. 148, 646 (2012)]. The second model, a two-strain generalization of the stochastic susceptible-infected-recovered (SIR) model with population turnover, has not been studied previously. In each of the two models, when the basic reproduction numbers of the two strains are identical, a system with an infinite population size approaches a point on the deterministic coexistence line (CL): a straight line of fixed points in the phase space of subpopulation sizes. Shot noise drives one of the strain populations to fixation, and the other to extinction, on a time scale proportional to the total population size. Our perturbation method explicitly tracks the dynamics of the probability distribution of the subpopulations in the vicinity of the CL. We argue that, whereas the slow strain has a competitive advantage for mathematically "typical" initial conditions, it is the fast strain that is more likely to win in the important situation when a few infectives of both strains are introduced into a susceptible population.

Spontaneous formation of large clusters in a lattice gas above the critical point.

We consider clustering of particles in the lattice gas model above the critical point. We find the probability for large density fluctuations over scales much larger than the correlation length. This fundamental problem is of interest in various biological contexts such as quorum sensing and clustering of motile, adhesive, cancer cells. In the latter case, it may give a clue to the problem of growth of recurrent tumors. We develop a formalism for the analysis of this rare event employing a phenomenological master equation and measuring the transition rates in numerical simulations. The spontaneous clustering is treated in the framework of the eikonal approximation to the master equation.

Minimizing the population extinction risk by migration.

Many populations in nature are fragmented: they consist of local populations occupying separate patches. A local population is prone to extinction due to the shot noise of birth and death processes. A migrating population from another patch can dramatically delay the extinction. What is the optimal migration rate that minimizes the extinction risk of the whole population? Here, we answer this question for a connected network of model habitat patches with different carrying capacities.

Fast migration and emergent population dynamics.

We consider population dynamics on a network of patches, having the same local dynamics, with different population scales (carrying capacities). It is reasonable to assume that if the patches are coupled by very fast migration the whole system will look like an individual patch with a large effective carrying capacity. This is called a "well-mixed" system. We show that, in general, it is not true that the total population has the same dynamics as each local patch when the migration is fast. Different global dynamics can emerge, and usually must be figured out for each individual case. We give a general condition which must be satisfied for the total population to have the same dynamics as the constituent patches.