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Authors:Krasky, Don Anthony
Contributors:Takagi, Daisuke (advisor)
Mathematics (department)
Applied mathematics
Brownian Motion
Chemical Signalling
show 1 moreStochastic Processes
show less
Date Issued:2020
Publisher:University of Hawai'i at Manoa
Abstract:In this dissertation we draw inspiration from natural systems, and discuss some mathematical models
inspired by them. With the goal of describing the motility of copepods, a model typically used to
describe bacterial dispersion is extended. From a static diffusion constant and some statistics on
velocity, the model provides an effective diffusion constant in the n-dimensional Cartesian space Rn.
We then explore interactions between organisms. Chemical signals are described mathematically
for stationary and moving sources. The results describe how detecting and signaling organisms are
likely to pair up based on motility. Both sections apply mathematical ideas to biological systems,
taking a few simple assumptions and discussing their consequences.
We introduce a model for dispersion of independent swimmers jumping randomly between multiple
translational velocities in arbitrary dimensions. Sample trajectories of the individual swimmers
are simulated using the governing stochastic differential equations. The associated Fokker-Planck
equations are derived and an analytic prediction is obtained for the effective diffusion constant, which
is shown to be consistent with simulations. We show adaptability of the model by fitting to three
previous models of swimmers having two or three preferred velocities. We explore how stochastic
vs. deterministic velocity changes and restricting certain velocity jumps result in different rates of
Chemical signals are present over a wide range of scales in nature, from as small as being inhabited
by bacteria, all the way up to salmon tracking their home stream in the open ocean. These signals
can alert organisms in the presence of predators, help locate food resources, or aid in the effort to find
mates. Evolution has had the chance to optimize the motility and sensitivity of organisms to best
exploit chemical signals. However, detection is still only possible down to a certain concentration
threshold. Motivated by bacteria searching for food patches in a heterogeneous environment we
provide calculations to describe the detectable region of a diffusing nutrient patch. From this, a
critical chemotactic velocity necessary to reach the origin of the patch can be obtained. Motility
is seen to provide additional information only in a donut-shaped region around the source. We
then change focus to a moving source. Our calculations show that in ideal conditions access to
concentration gradient and sufficient mobility guarantee the ability to find the source once it is
detected. Descriptive formulae are provided for the dimensions and shape of the trailing plume
from a given release rate and minimum detectable concentration. We discuss optimum types of
chaser/source match-ups and provide relevant descriptive calculations. We then apply our results
to real world scenarios, demonstrating the usefulness and coherence of our work.
Pages/Duration:56 pages
Rights:All UHM dissertations and theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission from the copyright owner.
Appears in Collections: Ph.D. - Mathematics

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