Spatial effects in random walks
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Saudi Digital Library
Abstract
In this thesis we investigate spatial effects in random walks by applying tech-
niques from statistical physics. We first study a chase-capture model on a rectan-
gular L1xL2 lattice with n hiding places. We derive an exact formula calculate the
prey survival probability. We investigate the effects of the shape of the home range
or the natural habitat of the prey (the lattice) on the prey survival probability and
explore the optimal arrangement of hiding places. We show that the shape of home
range does not have an effect on the prey survival probability unless the ratio L2=L1
( the width to the length of the home range) less than 1
n. We give an approximation
of the optimal arrangements of three, four and five holes inside different shapes of
home range. We then investigate the average shape of the spatial memory of a
foraging animal. The spatial memory is obtained by evaluating a two-dimensional
Brownian motion at the arrival times of a non-homogeneous Poisson process. We
obtain an analytic formula that measures how spherical this spatial memory is. We
verify our analytic result by simulation, and show that a slower decaying memory
leads to a more spherical shape. We then extend our work to the case where the
walker repeatedly returns to a particular place. Rather thinking in the terms of
memory, we consider the path to be tracked by an independent party at a time
varying rate called the "tracking strategy". We derive an analytic expression which
gives us the average size and elongation of the tracked Brownian bridge. We confirm
our exact result by simulation and we give examples of different tracking strategies.