# A Levy motion model of microbial dynamics in a pore: Numerical and theoretical results

#### Abstract

Microbial motility is often characterized by 'run and tumble' behavior which consists of a bacteria making sequence of runs followed by tumbles (random changes in direction). A superset of Brownian motion, namely Levy motion, seems to describe such a motility pattern. Levy motion finds application in many other particle transport processes where long-range spatial correlation is preferred. The Eulerian (Fokker-Planck) equation describing these motions is similar to the classical advection-diffusion equation except that the order of highest derivative is given by a fractional number, α ε (0,2]. It is shown analytically that for particle trajectories initially separated by a distance *r* and governed by the symmetric α-stable Levy motions, the finite-size Lyapunov exponent (a measure of dispersive mixing) is proportional to the diffusion coefficient and inversely proportional to * r ^{a}*. This power-law provides an easy method to determine parameters for Levy processes. ^ The Lagrangian equation of Levy motion, driven by a Levy measure with drift, is stochastic and employed to numerically explore the dynamics of microbes in a flow cell with sticky boundaries. The flow cell is the region formed by two infinite parallel plates. The Eulerian equation is used to non-dimensionalize parameters for sensitivity analysis. The amount of sorbed time on the boundaries is modeled as a random variable that can vary over a wide range of values. Salient features of first passage time are studied with respect to scaled parameters. The first passage time densities (breakthrough curves) are further analyzed for decay profiles of their tailing ends. A theoretical formulation, consisting of a set of expressions for iterative computation of mean first passage time, is developed for Levy motion in a slit-pore with sticky boundaries. Comparison between theoretical results and numerical model suggests a good match.^

#### Degree

Ph.D.

#### Advisors

John H. Cushman, Purdue University, Dennis A. Lyn, Purdue University.

#### Subject Area

Engineering, Civil

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