Lévy-flight in foraging:
When I was young … er, all right, when I was a child, we played in a rather wild environment.  There were delicious lawns, and if the owners had children we could occasionally play ball in lush grass.  Commonly however we played in vacant lots with a lot of weeds and brush.  So of course we lost the ball a lot.  When that happened we would poke around a bit and then lose heart.

My father, frugal man, did not care to see something valuable lost, and he would sometimes help us look.  Methodically and patiently he would ask us for the general area and then pace though it back and forth like a man plowing a field until he came to it.  I never knew him to fail.  There is a small part of my mind that recalls that he was amazingly observant and possibly had noticed the ball at the outset but went though the motions to show us how to do it, but we never did. 

In Mark Twain’s Tom Sawyer there is a similar episode in which a small object has been lost.  Tom’s solution was to take a similar object and say to it, “Brother, go find your brother,” and toss it in the same general area.  Then he would go to the second object and make a meticulous search of its surroundings.  Then he repeated the process until he succeeded.  I do not remember Twain’s numbers, but it was pretty clear that Tom found what he was looking for by chance. 

Foraging animals use yet another strategy.  They wander around aimlessly, frequently revisiting the same site, like Tom without his ritual.  This is called a Brownian search strategy because the path is somewhat like the random path a smoke particle takes under the influence of Brownian motion, as it gets knocked about at random by air molecules. 

There is yet another strategy that has been observed recently although it was predicted before.  (Fish in Lévy-flight, Gandhimohan M. Viswanathan NATURE vol. 465 no. 7301 June 24, 2010 page 1018 and Environmental Context Explains Lévy and Brownian Movement Patterns of Marine Predators Nicolas E. Humphries et al, in the same issue page 1066)  In this case, a foraging animal, having spent some time in random search, will take a long leg to a new area and then resume its random pattern.  It seems inherently logical.  Ideally one might expect that spiraling out from a center and then occasionally make a Lévy-flight move would be a trifle better.  However fish are no doubt less adept at following a geometric pattern than are the crafters of crop circles. 

What was observed was that if food was abundant, then the Brownian strategy was adequate.  But if food was scarce, the alternate strategy was adopted. 

As I have mentioned before, there is a lesson here.  If the species doing the foraging had increased its population size to the carrying power of its environment, then it would always be forced to use the more complex pattern. 

But no, populations do not do that.  The population size is limited by some heritable effect of kinship on population size. 

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