Pole-Seeking Brownian Motion and Bird Navigation

It turns out that some formulae for the brownian theory can be used to graduate the results obtained from actual or simulated observations, and this is its sole ornithological relevance, and there may be a general need for a shift of emphasis in thinking, away from weak convergence theorems towards weak convergence approximations.

Abstract

Summary This paper is not concerned with identifying the mechanism of bird navigation, but rather with assessing the degree of success possible using very bad navigational aids of whatever sort. Flight-paths are simulated using virtually no hypotheses, and all numerical parameters are determined ornithologically. The results are compared with the performance of actual birds, and also with a brownian model which is analytically convenient. It turns out that some formulae for the brownian theory can be used to graduate the results obtained from actual or simulated observations, and this is its sole ornithological relevance. The data imply a surprising numerical coincidence which allows of a Darwinian explanation. It seems likely that under suitable asymptotic hypotheses, the ornithological model will converge weakly towards the brownian one, but that question is deliberately not investigated here. Indeed it is ornithologically of little interest because (i) we are certainly not at the limit and (ii) weak convergence theorems do not normally have anything useful to say about the rate of convergence. This suggests that there may be a general need for a shift of emphasis in our thinking, away from weak convergence theorems towards weak convergence approximations (validated by simulation).