A general random walk model for the leptokurtic distribution of organism movement: Theory and application
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The movement of organisms is usually leptokurtic in which some individuals move long distances while the majority remains at or near the area they are released. There has been extensive research into the origin of such leptokurtic movement, but one important aspect that has been overlooked is that the foraging behaviour of most organisms is not Brownian as assumed in most existing models. In this paper we show that such non-Brownian foraging indeed gives rise to leptokurtic distribution. We first present a general random walk model to describe the organism movement by breaking the foraging of each individual into events of active movement and inactive stationary period; its foraging behaviour is therefore fully characterized by a joint probability of how far the individual can move in each active movement and the duration it remains stationary between two consecutive movements. The spatio-temporal distribution of the organism can be described by a generalized partial differential equation, and the leptokurtic distribution is a special case when the stationary period is not exponentially distributed. Empirical observations of some organisms living in different habitats indicated that their rest time shows a power-law distribution, and we speculate that this is general for other organisms. This leads to a fractional diffusion equation with three parameters to characterize the distributions of stationary period and movement distance. A method to estimate the parameters from empirical data is given, and we apply the model to simulate the movement of two organisms living in different habitats: a stream fish (Cyprinidae: Nocomis leptocephalus) in water, and a root-feeding weevil, Sitona lepidus in the soil. Comparison of the simulations with the measured data shows close agreement. This has an important implication in ecology that the leptokurtic distribution observed at population level does not necessarily mean population heterogeneity as most existing models suggested, in which the population consists of different phenotypes; instead, a homogeneous population moving in homogeneous habitat can also lead to leptokurtic distribution.