Jack D. Dockery

Finger Formation in Biofilm Layers

A simple single substrate limiting model of a growing biofilm layer is presented. One-dimensional moving front solutions are analyzed. Under certain conditions these solutions are shown to be linearly unstable to fingering instabilities. Scaling laws for the biofilm growth rate and length scale are derived. The nonlinear evolution of the fingering instabilities is tracked numerically using a level set method, leading to the observation of mushroom-like structures.

Mathematical Description of Microbial Biofilms

We describe microbial communities denoted biofilms and efforts to model some of their important aspects, including quorum sensing, growth, mechanics, and antimicrobial tolerance mechanisms.

Existence of standing pulse solutions for an excitable activator-inhibitory system

Invariant manifold techniques are used to establish the existence of a standing wave solution for a system of reaction-diffusion equations modeled after excitable media. As a by-product of the method of proof, we obtain a local uniqueness result.

Senescence and antibiotic resistance in an age-structured population model

Different theories have been proposed to understand the growing problem of antibiotic resistance of microbial populations. Here we investigate a model that is based on the hypothesis that senescence is a possible explanation for the existence of so-called persister cells which are resistant to antibiotic treatment. We study a chemostat model with a microbial population which is age-structured and show that if the growth rates of cells in different age classes are sufficiently close to a scalar multiple of a common growth rate, then the population will globally stabilize at a coexistence steady state. This steady state persists under an antibiotic treatment if the level of antibiotics is below a certain threshold; if the level exceeds this threshold, the washout state becomes a globally attracting equilibrium.

Existence of travelling wave solutions for a bistable evolutionary ecology model

The existence of travelling wave solutions for a density-dependent selection migration model in population genetics is proven. A single locus and two alleles are assumed. It is also assumed that the fitnesses of the heterozygotes in the population are below those of the homozygotes. The method of proof is by constructing an isolating neighborhood and computing a connection index.

Numerical Solution of Travelling Waves for Reaction-Diffusion Equations Via the Sinc-Galerkin Method

Travelling waves in reaction—diffusion equations have received the interest of both mathematicians and physical scientists. Indeed, travelling waves of chemical, physical or biological activity are commonly observed in reacting and diffusing systems. The best known example is the neural action potential, a wave of electrical activity that propagates along an axonal membrane [10]. Other examples incLude waves of excitation in the BelousovZhabotinskii [17] reaction, travelling population waves in various genetic models [1, 8], and travelling fronts in hydrodynamic systems [14].

The chemostat with lateral gene transfer

We investigate the standard chemostat model when lateral gene transfer is taken into account. We will show that when the different genotypes have growth rate functions that are sufficiently close to a common growth rate function, and when the yields of the genotypes are sufficiently close to a common value, then the population evolves to a globally stable steady state, at which all genotypes coexist. These results can explain why the antibiotic-resistant strains persist in the pathogen population.

Existence and stability of traveling wave solutions for a population genetic model via singular perturbations

Using singularr perturbation methods, the existence and stability of traveling wave solutions for a density-dependent selection migration model in population genetics is proved. Single locus and two alleles are assumed, and it is also assumed that the fitnesses of the heterozygotes in the population are close to but below those of the homozygotes. Unlike previous models, this paper does not assume that the population is in Hardy–Weinberg equilibrium.

Niche Partitioning Along an Environmental Gradient

Biological systematics studies suggest that species are discretized in niche space. That is, rather than seeing a continuum of organism types with respect to continuous environmental variations, observers instead find discrete species or clumps of species, with one clump separated from another in niche space by a gap. Here, using a simple one dimensional model with a smoothly varying environmental condition, we investigate conditions for a discrete niche partitioning instability of a continuously varying species structure in the context of asexually reproducing microbes. We find that significant perturbation of translational invariance is required for instability, but that conditions for such perturbations might reasonably occur, for example, through influence of boundary conditions.

Symmetry-Breaking Homoclinic Bifurcations in Diffusively Coupled Equations

In this study we examine a symmetry-breaking bifurcation of homoclinic orbits in diffusively coupled ordinary differential equations. We prove that asymmetric homoclinic orbits can bifurcate from a symmetric homoclinic orbit when the equilibria to which the latter is homoclinic undergoes a pitchfork bifurcation. A condition which defines the direction of the bifurcation in a parameter space is given. All hypotheses of the main theorem are verified for a diffusively coupled logistic system and the twistedness of the bifurcating homoclinic orbits is computed for a range of coupling strengths.