Biased Random Walks in Biology

Random walks are used to describe the trajectories of many motile animals and microorganisms. They are a useful tool for both qualitative and quantitative descriptions of the behaviour of such creatures. Simple diffusive random walk models, or position jump processes, are unrealistic as they allow for effectively infinite propagation and do not take into account correlation between steps. Othmer et al. (1988) use a generalised transport equation to model biased and correlated velocity jump processes where the speed of movement is finite. In two dimensions an equation for the underlying spatial distribution of the velocity jump process model cannot be found, so Othmer et al. use a method of calculating moments to derive and solve differential equations for the statistics of interest. We extend the velocity jump process and method of moments used by Othmer et al. to include reorientation models where the mean turning angle is dependent on the previous direction of movement, as observed by Hill & Häder (1997) in experiments on algae. Closure assumptions are made in order to derive and solve a system of differential equations for the higher order moments and statistics of the underlying spatial distribution. Numerical simulations are then used to compare the asymptotic solutions with simulated data, and the fit is good for biologically realistic parameter values. Numerical simulations of velocity jump processes are also used to investigate the method used by Hill & Häder to calculate the reorientation parameters from the angular statistics of a random walk, and also to investigate the effect of spatially dependent parameters or a changing preferred direction on the spatial statistics. We look at the ratio between the root of the mean squared displacement and the mean dispersal distance in both unbiased and biased random walks and demonstrate how this can give us more information about the spatial distribution. We give an application of the velocity jump process model to the movement and recruitment of reef fish larvae. The variability in the movement is found to be important if there is a low survival probability, while in simple reef environments the survival probability appears to be highly sensitive to the reorientation parameters and corresponding swimming behaviour.