Random walk models in biology

Abstract

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

Figure 1

(a,c,e) PDFs and (b,d,f) sample paths of different random walks. (a,b) A lattice BRW with probabilities of moving a distance δ right or left of τ(D/δ²±u/(2δ)) and up or down of τD/δ². (c,d) A non-lattice CRW with probabilities of turning an angle δ_θ clockwise or anticlockwise of $\tau {\sigma }_0^2/(2{\delta }_{\theta }^2)$. (e,f) A non-lattice BCRW with probabilities of turning clockwise or anticlockwise of $\tau \left({\sigma }_0^2/(2{\delta }_{\theta }^2)\pm \theta /(2B{\delta }_{\theta })\right)$ (cf. the linear reorientation model of §3.4). In the BRW and BCRW, the global preferred direction is θ₀=0; in the CRW and BCRW, the initial direction is θ=π/2 and the walker moves with constant speed v. In all cases, the walker starts at (x, y)=(0, 0) at t=0 and is allowed to move until t=10. The PDFs p(x, y, t=10) were calculated from 10⁶ realizations of the walk. In (a), the white lines show the contours of the corresponding theoretical PDF (2.12). In the sample paths for the BRW, at each step the walker either stays still or moves right, left, up or down by a distance δ. In the CRW and BCRW, at each step the walker's direction of motion θ either stays the same or turns clockwise or anticlockwise by an angle δ_θ, and the walker's movement is given by the vector vτ(cos θ, sin θ). Parameter values: D=0.2, u=0.5, ${\sigma }_0^2=0.5$, B=2.5, v=0.5.