Mathematics of optimal foraging

[shkrobius]

Suppose there is a random, sparse distribution of "food" that needs be gathered (it can be 2D or 3D). It can be either depletable of regrowing, but it is stationary. You move at a constant speed taking random turns (uniform distribution) but the time interval for forward motion is taken from a pre-set distribution. It can either be fixed or

Comments

[chaource]

It seems that Nature has chosen some implementation that is half-way efficient; there is no guarantee that a neurological mechanism could exist that is actually capable of implementing the "mathematically best" strategy (if any), and that this mechanism (if it exists) could evolve.

Experiments were made on humans who were supposed to bet on red or green light, where the frequency of red to green was 3:1 but otherwise the red and the green light appeared completely randomly. Humans figured out that red was more likely, but they did not bet on red all the time (which is the optimal strategy). Humans randomly switch between red and green, of course giving preference to red, but also giving "some chance" to green. This strategy is suboptimal, but it seems that all living beings do this kind of thing.

[shkrobius]

I am not sure it is nonoptimal. One can think about it this way (intuitively). Suppose we have an optimal strategy. Let's rescale (say, zero in). Then it must be optimal again. This consideration suggests that it must be scale invariant. Levy flights are scale invariant. So it seems reasonable that the answer is a Levy flight. On the other hand if you rescale you can rescale your Brownian dispersion. So it all boils down to your ability to rescale during the search. If you can do it, then you can be rescaling, but this requires some kind of intelligence and knwoledge of the surroundings. If you never need to rescale, Brownian motion must be OK. If there is need to rescale but no hard-wired possibility of rescaling, Levy walks could be your answer. I, actually, think that foraging has to be nearly optimal, if only because it is such an important concern for survival. However, it could be constrained both by design and necessity. What surprises me is that there is no mathematical solution to such a simply formulated problem with a

[chaource]

Perhaps, it is far from obvious that the mathematically "best" solution must be scale-invariant. If you add the extra condition that the strategy must be scale-invariant (because the bacteria is too simple to implement a strategy where some parameters vary depending on the length of path already searched, for example), then you get the scale-invariant solution (Levy flights).

Also, it is far from obvious that the foraging strategy implemented in a particular species has to be nearly optimal in the mathematical sense. The strategy just needs to provide enough food for survival, and not be too suboptimal so that other individuals can get significantly ahead of you. Basically, the strategy is about as optimal as you can get given all other constraints imposed on your species (e.g., whether a nervous system or a memory facility is available).

[shkrobius]

I guess you mean that if one retains memory of gathering along the trajectory, there could be a better strategy than simply Levy flights, because, for example one can stay away from already grazed areas. Yes, now I see the difficulty: I tacitly assumed that the organism has no memory or analytical capabilities, whereas in reality it does, and it is difficult to specify how much of it is available. Perhaps the chief difficulty must be formulating the problem in a meaningful way. It might be hard even for the bacteria, because they do have memory of food gradients (about a few seconds) and enough intelligence to use it for biased random walks. So this fuzziness is present from the start

[rshura]

This strategy is only suboptimal if you're 100% sure the 3:1 ratio is set forever. In reality, the rules or rewards change routinely. This is why one should once in a while take a chance and check the "non-winning" outcome.

They have done this with the monkeys: start with 3:1 ratio and slowly change it to drive the winning odds to 1:3 (opposite to initial distribution). The monkeys are coping up with the change *precisely* because they're not dead-set on only sticking with the winning choice, but also explore the non-winning choice, although not as frequently.