Spatial effects in random walks

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.