Correlated random walk equations of animal dispersal resolved by simulation

Animal movement and dispersal can be described as a correlated random walk dependent on three parameters: number of steps, step size, and distribution of random turning angles. Equations of Kareiva and Shigesada use the parameters to predict the mean square displacement distance (MSDD), but this is less meaningful than the mean dispersal distance (MDD) about which the population would be distributed. I found that the MDD can be estimated by multiplying the square root of the MSDD by a three-dimensional surface correction factor obtained from simulations. The correction factors ranged from 0.89 to 1 depending on the number of steps and the variation in random turns, expressed as the standard deviation of the turning angles (SDA) about 0 degrees (straight ahead). Corrected equations were used to predict MDDs for bark beetles, butterflies, ants, and beetles (based on parameters from the literature) and the nematode Steinernema carocapasae (Weiser). Another equation from the literature finds the MDD directly, and this agreed with the MDD obtained by simulation at some combinations of SDA and numbers of steps. However. the equation has an error that increases as a power function when the standard deviation of turning angles becomes smaller (e.g., <6 degrees at 1000 steps or < 13 degrees at 250 steps). Lower numbers of steps also increase the error. Equivalent values of AMT tangle of maximum turn) in uniform random models and of SDA in normal random models were found that allowed these two models to yield similar MDD values. The step size and turning angle variation of animal paths during dispersal and host and mate starching were investigated and found to be correlated; thus, use of different measured step sizes gives consistent estimates of the MDD.