Lévy flight

Later researchers have extended the use of the term "Lévy flight" to also include cases where the random walk takes place on a discrete grid rather than on a continuous space.

For jump lengths which have a symmetric probability distribution, the equation takes a simple form in terms of the Riesz fractional derivative.

[citation needed] The definition of a Lévy flight stems from the mathematics related to chaos theory and is useful in stochastic measurement and simulations for random or pseudo-random natural phenomena.

[8][9][10][11] Birds and other animals (including humans)[12] follow paths that have been modeled using Lévy flight (e.g. when searching for food).

When the beetles are hungry and food is scarce, they avoid searching for prey in locations that other individuals of P. melanarius have visited.

[20] Efficient routing in a network can be performed by links having a Levy flight length distribution with specific values of alpha.

Figure 1. An example of 1000 steps of a Lévy flight in two dimensions. The origin of the motion is at [0,0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. stable ) distribution with α = 1 and β = 0 which is a Cauchy distribution . Note the presence of large jumps in location compared to the Brownian motion illustrated in Figure 2.
Figure 2. An example of 1000 steps of an approximation to a Brownian motion type of Lévy flight in two dimensions. The origin of the motion is at [0, 0], the angular direction is uniformly distributed and the step size is distributed according to a Lévy (i.e. stable ) distribution with α = 2 and β = 0 ( i.e., a normal distribution ).