## Friday, February 27, 2009

### Powerlaw Foraging

Many foraging animals try to find their prey by random walks, i.e. chains of straight segments (steps) separated by random turns. As measurements show, these walks are not Brownian, but correspond to Levy flights with powerlaw-distributed lenghts L of the straight segments:
P(L)\propto L^{-m}.
Here, the exponents m range from 1 to 3 and the resulting stationary PDF of visited sites P(x,y) is not Gaussian.

In a 1999 Nature paper, G.M. Viswanathan shows that such powerlaw foraging (with exponent m=2) can be an optimum search strategy if target sites are sparse and can be visited repeatedly.

For an intuitive understanding, he notes:
• In a Levi flight, irrespective of the exponent, the probability of returning to a previously visited site is smaller than for a Brownian walk with Gaussian PDF.
• Also, N Levy walkers visit much more new sites than N Brownian walkers.