Modelling the movements of organisms by stochastic theory in a comoving frame

Abstract

Imagine you walk in a plane. You move by making a step of a certain length per time interval in a chosen direction. Repeating this process by randomly sampling step length and turning angle defines a two-dimensional random walk in what we call comoving frame coordinates. This is precisely how Ross and Pearson proposed to model the movements of organisms more than a century ago. Decades later their concept was generalised by including persistence leading to a correlated random walk, which became a popular model in Movement Ecology. In contrast, Langevin equations describing cell migration and used in active matter theory are typically formulated by position and velocity in a fixed Cartesian frame. In this article, we explore the transformation of stochastic Langevin dynamics from the Cartesian into the comoving frame. We show that the Ornstein-Uhlenbeck process for the Cartesian velocity of a walker can be transformed exactly into a stochastic process that is defined self-consistently in the comoving frame, thereby profoundly generalising correlated random walk models. This approach yields a general conceptual framework how to transform stochastic processes from the Cartesian into the comoving frame. Our theory paves the way to derive, invent and explore novel stochastic processes in the comoving frame for modelling the movements of organisms. It can also be applied to design novel stochastic dynamics for autonomously moving robots and drones.