Alexander M. Davie

Individual Path Uniqueness of Solutions of Stochastic Differential Equations

We consider the stochastic differential equation dx(t) = f(t, x(t))dt + b(t, x(t))dW(t), x(0) = x0 for t ≥ 0, where x(t) ∈ ℝ d , W is a standard d-dimensional Brownian motion, f is a bounded Borel function from [0, ∞) ×ℝ d to ℝ d , and b is an invertible matrix-valued function satisfying some regularity conditions. We show that, for almost all Brownian paths W(t), there is a unique x(t) satisfying this equation, interpreted in a “rough path” sense.

KMT Theory Applied to Approximations of SDE

The dyadic method of Komlos, Major and Tusnady is a powerful way of constructing simultaneous normal approximations to a sequence of partial sums of i.i.d. random variables. We use a version of this KMT method to obtain order 1 approximation in a Vaserstein metric to solutions of vector SDEs under a mild non-degeneracy condition using an easily implemented numerical scheme.

MAXIMIZING MEASURES ON METRIZABLE NON-COMPACT SPACES

We prove the existence and uniqueness of maximizing measures for various classes of continuous integrands on metrizable (non-compact) spaces and close subsets of Borel probability measures. We apply these results to various dynamical contexts, especially to hyperbolic mappings of the form fλ(z) = λez , λ = 0, and associated canonical maps Fλ of an infinite cylinder. It is then shown that, for all hyperbolic maps Fλ, all dynamically maximizing measures have compact supports and, for all 0+-potentials φ, the set of (weak) limit points of equilibrium states of potentials tφ, t ↗ +∞, is nonempty and consists of dynamically maximizing measures.

Introduction to Competition Between Continuous Cultures of Phaeodactylum tricornutum and Dunaliella tertiolecta

When the diatom, Phaeodactylum tricornutum, and the microalga, Dunaliella tertiolecta, are cultured together in a chemostat at dilution factors of ∼ 0.5 day− 1, the diatom develops the higher population density. At dilution factors above 1.2 day− 1 the inability of the diatom to assimilate nutrient as fast as it flows into the chemostat results in the microalga generating the larger population. This change in population densities is accompanied by an increase in the chlorophyll content of the diatom and a decrease in the chlorophyll content of the microalga. Two species of phytoplankton can coexist when they compete for nutrient in a chemostat providing they do not otherwise interact. When the species do interact coexistence in a stable steady state is possible providing intraspecies interactions exceed the interactions between the species. Both species adjust their consumption to minimise the concentration of nutrient in the chemostat and their growth is modified to match the dilution factor of the flow.

Group Testing, the Pooled Hypergeometric Distribution, and Estimating the Number of Defectives in Small Populations

The testing of combined bacteriological samples – or “group testing” – was introduced to reduce the cost of identifying defective individuals in populations containing small proportions of defectives. It may also be applied to plants, animals, or food samples to estimate proportions infected, or to accept or reject populations. Given the proportion defective in the population, the number of positive combined samples is approximately binomial when the population is large: we find the exact distribution when groups include the same number of samples. We derive some properties of this distribution, and consider maximum-likelihood and Bayesian estimation of the number defective.

Continuous cultures of phytoplankton

The development of cultures of phytoplankton adapting throughout several days in an axenic, continuous-flow chemostat to yield a steady kinetic state of competing species is described mathematically. The adaptation of the growth rate to the chemostat environment inhibits integration of the equation of conservation of phytoplankton populations, though eventually when a steady state is reached the growth rate becomes equal to the rate of flow through the chemostat. Representation of species growth rates by a Verhuls formulation utilising experimentally determinable intra- and interspecies interaction constants permits the rapid prediction of the adaptation and alteration in the populations of competing phytoplankton species with changes in the chemostat environment. Illustrations of the behaviour of two and three competing species are extended to consideration of the stabilities of cultures of many competing species. Stable steady states of phytoplankton in a continuous-flow chemostat comprise a classic thermodynamic system and consequently the utilisation of light energy by the cells varies inversely with their growth rate. It is probable that when growth is nutrient limited, intra-and interspecies interaction parameters diminish as the demands of consumption are more nearly matched by the ratios of the limiting nutrients.