Random walk in nonhomogeneous environments: A possible approach to human and animal mobility

The random walk process in a nonhomogeneous medium, characterized by a Lévy stable distribution of jump length, is discussed. The width depends on a position: either before the jump or after that. In the latter case, the density slope is affected by the variable width and the variance may be finite; then all kinds of the anomalous diffusion are predicted. In the former case, only the time characteristics are sensitive to the variable width. The corresponding Langevin equation with different interpretations of the multiplicative noise is discussed. The dependence of the distribution width on position after jump is interpreted in terms of cognitive abilities and related to such problems as migration in a human population and foraging habits of animals.

Figure 1

Distributions corresponding to Eq. (3) evaluated from trajectory simulations with $\alpha =1.5, \eta =0$, and $\theta =-0.5,0,0.5$ (from top to bottom) at $t=10$. Right inset: The tail of the distributions at $t=1$ for $\theta =-0.5$, 0.5, 1, 1.5, and 2 (from left to right). The initial condition: $p(x,0)=\delta (x-0.1)$. Left inset: Time dependence of densities corresponding to Eq. (3) evaluated from trajectory simulations for $\alpha =1.5$ and two values of $\theta$: 0.6 (squares) and $-0.6$ (stars). Dots refer to the case of Eq. (9) with $\theta =0.6$ and $\eta =0$. The red straight lines mark the analytical results.

Figure 2

(a) Time evolution of densities corresponding to Eq. (9) evaluated from trajectory simulations with $\alpha =1.5, \theta =0.3$, and $\eta =0$ for $t=1$, 10, 50 (from top to bottom). Inset: The evolution for $\theta =0.6 t=$ 1, 10, 100, and 500 (from left to right). (b) The tails for $\theta =$ 0.9, 0.6, 0.4, 0.2, and $-0.4$ (from left to right). The red straight lines mark the dependence $x^{-1-\alpha -\alpha \theta }$.

Figure 3

Variance as a function of time evaluated from CTRW trajectory simulations for $\alpha =1.5$ and $\theta =0.6$ (points). The slope of straight lines obeys the dependence (22) with $\mu =1.33$, 1, and 0.833. Stars mark rescaled results of the numerical solving of Eq. (23) where $\nu \left(x\right)$ was evaluated before the random kick and $f\left(x\right)$ after that (anti-Itô interpretation).