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

TL;DR

The paper addresses how to model organism movements by unifying comoving-frame representations with Cartesian Langevin dynamics. It develops a general framework to transform stochastic dynamics across Cartesian, polar, and comoving frames and shows an exact transformation of the two-dimensional Cartesian Ornstein-Uhlenbeck process into a self-consistent comoving-frame process. A key contribution is the derivation of an exact turning-angle distribution in the comoving frame and a self-contained comoving OU description, validated numerically. The approach clarifies when exact transformations are possible (for Markovian Cartesian dynamics) and points to practical applications in movement ecology and autonomous robotics.

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.

Figures (13)

• Figure 1: Time-discrete trajectory (blue) of an organism (the pictures show an eagle) in a fixed Cartesian frame (black horizontal and vertical lines). Also shown is the velocity of the eagle at some point along the trajectory in Cartesian coordinates $v_x,v_y$. In contrast, at the final point of the trajectory we represent the movement of the eagle in comoving frame coordinates (red) in terms of speed $s$ and turning angle $\alpha$, where the latter is defined as the angle between subsequent velocity directions (dashed, respectively bold red lines).
• Figure 2: Conceptual illustration of the interplay between three different reference frames for a process over four discrete time steps $n=0,1,2,3$: Cartesian frame described by positions $x_n$ and $y_n$ (black), polar frame described by the orientation angle $\beta_n$ (blue) and the speed $s_n$ (red), and comoving frame described by the turning angle $\alpha_n$ and speed $s_n$ (red).
• Figure 3: Full transformation rules between the three different frames of reference and the respective inverse transfromation.
• Figure 4: Top: Simulation results for the probability density of the velocity component $v_x$ for the simple random walk formulated in three different frames, yielding three different random walk models, compared with the Gaussian distribution. Bottom: The autocorrelation function of the random walk velocities are all uncorrelated .
• Figure 5: Same as Fig. \ref{['velxpdfrw']} for the velocity component $v_y$