Levy walk with multiple internal states

TL;DR

This paper develops a Levy walk model with multiple internal states to better represent complex animal foraging behaviors and other systems, deriving governing equations and analyzing diffusion types and first passage times.

Contribution

It introduces a novel Levy walk framework with multiple internal states, deriving its governing equations and exploring its diffusion properties and first passage time distributions.

Findings

Uncovered diffusion types for non-immediately-repeating Levy walks.

Derived governing equations for position distribution with multiple internal states.

Numerically simulated first passage time and distribution.

Abstract

Levy walk is a fundamental model with applications ranging from quantum physics to paths of animal foraging. Taking animal foraging as an example, a natural idea that comes to one's mind is to introduce the multiple internal states for dealing with the dependence of the PDF of waiting time on the energy of the animal and richness of the food at a particular location, etc; the framework can also be used to model the moving trajectories of smart animals without returning to the directions or locations which they come from immediately. After building the Levy walk model with multiple internal states and deriving the governing equation of the distribution of the positions of the particles, some applications are discussed with specific transition matrices. The type of diffusion for non-immediately-repeating L\'{e}vy walk is uncovered, and the distribution and average of first passage time are numerically simulated.