A generalized distribution interpolated between the exponential and power law distributions and applied to the walking data of the pill bug (Armadillidium vulgare)

Abstract

The Levy walk, a type of random walk in which the frequency of linear step-lengths follows a power-law distribution, can be observed in the migratory behavior of organisms at various levels, from bacteria and T cells to humans. Compared to the Brownian walk, which is also a type of random walk (characterized by an exponential distribution of the frequency of occurrence of step-length), the Levy walk is characterized by the occasional appearance of linear movements over very long distances. In this paper, we propose a general distribution that includes the power-law and exponential distributions as special cases. This distribution has two parameters: the first parameter represents the exponent, similar to the power-law and exponential distributions and the second is a shape parameter representing the shape of the distribution. By introducing this distribution, an intermediate distribution model can be interpolated between the power-law and exponential distributions. The shape parameter measures whether a distribution assimilates a power-law or exponential distribution. In this study, the proposed distribution was fitted to the frequency distribution of the step-length calculated from the walking data of pill bugs. The autocorrelation coefficients were also calculated from the time-series data of the step-length, and the relationship between the shape parameter and time dependency was investigated. The results showed a significant negative correlation between the two (r=-0.61, n=30, t=4.04, and p=0.00037). This means that pill bugs with gait patterns closer to Levy than Brownian walks have a stronger time dependence with respect to step-length changes.