Don A. Jones

Effects of random motility on microbial growth and competition in a flow reactor

The authors investigate the effects of random motility on the ability of a microbial population to survive in pure culture and to be a good competitor for scarce nutrient in mixed culture in a flow reactor model consisting of a nonlinear parabolic system of partial differential equations. For pure culture (*a single population), a sharp condition is derived which distinguishes between the two outcomes: (1) washout of the population from the reactor or (2) persistence of the population and the existence of a unique single-population steady state. The simulations suggest that this steady state is globally attracting. For the case of two populations competing for scarce nutrient, they obtain sufficient conditions for the uniform persistence of the two populations, for the existence of a coexistence steady state, and for the ability of one population to competitively exclude a rival. Extensive simulations are reported which suggest that (1) all solutions approach some steady state solution, (2) all possible outcomes exhibited by the classical competitive Lotka-Volterra ODE model can occur in the model, and (3) the outcome of competition between two bacterial strains can depend rather subtly on their respective random motility coefficients.

Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems

We show that the long-time behavior of the projection of the exact solutions to the Navier-Stokes equations and other dissipative evolution equations on the finite-dimensional space of interpolant polynomials determines the long-time behavior of the solution itself provided that the spatial mesh is fine enough. We also provide an explicit estimate on the size of the mesh. Moreover, we show that if the evolution equation has an inertial manifold, then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite-dimensional space spanned by the approximating polynomials. Our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation.

Determining finite volume elements for the 2D Navier-Stokes equations

We consider the 2D Navier-Stokes equations on a square with periodic boundary conditions. Dividing the square into N equal subsquares, we show that if the asymptotic behavior of the average of solutions on these subsquares (finite volume elements) is known, then the large time behavior of the solution itself is completely determined, provided N is large enough. We also establish a rigorous upper bound for N needed to determine the solutions to the Navier-Stokes equation in terms of the physical parameters of the problem.

On the Number of Determining Nodes for the 2D Navier-Stokes Equations

Abstract It is known that the solutions of the 2D Navier-Stokes equations, in bounded domains, are determined by a finite discrete set of nodal values. That is if the large time behavior of the solutions to the Navier-Stokes equations is known on an appropriate finite discrete set, then the large time behavior of the solution itself is totally determined. Here, an upper-bound is rigorously established for the number of nodes needed to determine the solutions of the Navier-Stokes equations in two dimensions with periodic boundary conditions.

On the effectiveness of the approximate inertial manifold-a computational study

We present a computational study evaluating the effectiveness of the nonlinear Galerkin method for dissipative evolution equations. We begin by reviewing the theoretical estimates of the rate of convergence for both the standard spectral Galerkin and the nonlinear Galerkin methods. We show that the rate of convergence in both cases depends mainly on how well the basis functions of the spectral method approximate the elements in the space of solutions. This in turn depends on the degree of smoothness of the basis functions, the smoothness of the solutions, and on the level of compatibility at the boundary between the basis functions of the spectral method and the solutions. When the solutions are very smooth inside the domain and very compatible with the basis functions at the boundary, there may be little advantage in using the nonlinear Galerkin method. On the other hand, for less smooth solutions or when there is less compatibility at the boundary with the basis functions, there is a significant improvement in the rate of convergence when using the nonlinear Galerkin method. We demonstrate the validity of our assertions with numerical simulations of the forced dissipative Burgers equation and of the forced Kuramoto-Sivashinsky equation. These simulations also demonstrate that the analytical upper bounds derived for the rates of convergence of both the standard Galerkin and the nonlinear Galerkin are nearly sharp.

Microbial Competition in Reactors with Wall Attachment: A Mathematical Comparison of Chemostat and Plug Flow Models

Competition for nutrient and the ability of bacteria to colonize the gut wall are factors believed to play a role in the observed stability of the indigenous microbiota of the mammalian large intestine. These factors were incorporated into the two-strain continuous-stirred tank reactor (CSTR) model formulated and numerically investigated by Freter et al. In their model simulations, the reactor is parameterized using data for the mouse intestine. An invading bacterial strain is introduced into a CSTR that has already been colonized by a resident strain. The two strains compete for a single growth-limiting nutrient and for limited adhesion sites on the wall of the reactor. The mathematical model described in this paper is motivated in part by the CSTR model, but is based on the plug flow reactor (PFR). Parameter values and initial conditions are chosen so that the numerical performance of the PFR can be compared to that of the CSTR. In simulations bearing a remarkable qualitative and quantitative resemblance to those of the CSTR it is found that the invader is virtually eliminated, despite the fact that it has uptake rate and affinity for the wall identical to those of the resident. The PFR model is then parametrized using data for the human large intestine, and the two-strain simulations are repeated. Though obvious quantitative differences are noted, the more important qualitative outcome is preserved. It is also found that when three strains compete for a single nutrient and for adhesion sites there exists a steady-state solution characterized by the segregation of the bacterial strains into separate nonoverlapping segments along the wall of the reactor.

Microbial competition for nutrient and wall sites in plug flow

A mathematical model of microbial competition for limiting nutrients and for a limited set of available wall-attachment sites in an advection-dominated tubular reactor is formulated as a limiting case of the more general model considered in (M. Ballyk and H. L. Smith, A flow reactor with wall growth, in Mathematical Models in Medical and Health Sciences, M. Horn, G. Simonett, and G. Webb, eds., Vanderbilt University Press, Nashville, TN, 1998). The model consists of a system of hyperbolic PDE. The existence and stability properties of its steady state solutions are investigated, both analytically and numerically. A surprising finding in the case of two-strain competition is the existence of a steady state solution characterized by the segregation of the two bacterial strains to separate nonoverlapping segments along the tubular reactor.

Bacterial Wall Attachment in a Flow Reactor

A mathematical model of microbial growth for limiting nutrients in a fully three dimensional flow reactor which accounts for the colonization of the reactor wall surface by the microbes is studied analytically. It can be viewed as a model of the large intestine or of the fouling of a commercial bioreactor or pipe flow. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each. The effects of adding an antimicrobial agent to the reactor are examined with and without wall growth.

Enslaved finite difference approximations for quasigeostrophic shallow flows

Abstract We describe a procedure to improve both the accuracy and computational efficiency of finite difference schemes used to simulate the nonlinear PDEs that govern barotropic two-dimensional geophysical fluid dynamics. Our underlying strategy is to reduce the truncation error of a given difference scheme in such a way that the time step restrictions are not changed. To accomplish this reduction, we use information from the governing equations in the case of vanishing time derivatives. Our method is based on a change of variables from fine to coarse grids, which allows us to order the various terms that appear and justify further approximations. These approximations lead to algebraic closures for the new higher-order variables, and finally to a new, enslaved scheme. We demonstrate the utility of the procedure for the shallow water equations in both periodic and closed basins. In the latter case we present results that demonstrate the ability of the enslaved scheme to capture dynamics on scales smaller than those resolved by the original scheme.

Local existence results for the generalized inverse of the vorticity equation in the plane

We prove the finite-time existence of a solution to the Euler - Lagrange equations corresponding to the necessary conditions for minimization of a functional defining variational assimilation of observational data into the two-dimensional, incompressible Euler equations. The data are given by linear functionals acting on the space of functions representing vorticity. The data are sparse and available on a fixed space - time domain. The objective of the data assimilation is to obtain an estimate of the vorticity which minimizes a cost functional and is analogous to a distributed parameter control problem. The cost functional is the sum of a weighted squared error in the dynamics, the initial condition, and in the misfit to the observed data. Vorticity estimates which minimize the cost functional are obtained by solving the corresponding system of Euler - Lagrange equations. The Euler - Lagrange system is a coupled two-point boundary value problem in time. An application of the Schauder fixed-point theorem establishes the existence of a least one solution to the system. Iterative methods for generating solutions have proven useful in applications in meteorology and oceanography.