simple foraging for simple foragers

frank@math.unimaas.nl

Sex and the Signal: Evolution and Game Theory 1/44

Simple Foraging for Simple Foragers

Frank Thuijsman

joint work with

Bezalel Peleg, Mor Amitai, Avi Shmida

Outline

Two approaches that explain certain

observations of foraging behavior

The Ideal Free Distribution

The Matching Law

…Risk Aversity

Stephen Fretwell & Henry Lucas (1970):

Individual foragers will distribute themselves over various patches proportional to the amounts of resources available in each.

Many foragers

For example: if patch A contains twice as much food as patch B, then there will be twice as many individuals foraging in patch A as in patch B.

The Matching Law

Richard Herrnstein (1961):

The organism allocates its behavior over various activities in proportion to the value derived from each activity.

The Matching Law

Single forager

For example: if the probability of finding food in patch A is twice as much as in patch B, then the foraging individual will visit patch A twice as often as patch B

Simplified Model

?Yellow Blue

p qy b

Two patches

Nectar quantitiesNectar probabilities

One or more bees

Only Yellow …

… And Blue

No Other Colors

Yellow and Blue Patches

IFD and Simplified Model

Yellow Blue

y bnectar quantities:

nY nBnumbers of bees:

two patches:

IFD: nY / nB y / b

Matching Law and Simplified Model

Yellow Blue

p qnectar probabilities:

nY nBvisits by one bee:

two patches:

Matching Law: nY / nB p / q

How to choose where to go?

Alone …

…or with others

No Communication !

bzzz, bzzz, …

ε-sampling orfailures strategy!

The Critical Level cl(t)

cl(t+1) = α·cl(t) + (1- α)·r(t)

0 < α < 1

r(t) reward at time t = 1, 2, 3, …

cl(1) = 0

The ε-Sampling Strategy

Start by choosing a color at randomAt each following stage, with probability:

ε sample other color1 - ε stay at same color.

If sample “at least as good”,then stay at new color,otherwise returnimmediately.

ε > 0

IFD, ε-Sampling, Assumptions

• reward at Y: 0 or 1 with average y/nY

reward at B: 0 or 1 with average b/nB

• no nectar accumulation

• ε very small: only one bee sampling

• At sampling cl is y/nY or b/nB

ε-Sampling gives IFD

Proof:

Let P(nY, nB) = y·(1 + 1/2 + 1/3 + ··· + 1/nY) - b·(1 + 1/2 + 1/3 + ··· + 1/nB)

If bee moves from Y to B,

then we go from (nY, nB) to (nY - 1, nB + 1)

P(nY - 1, nB + 1) - P(nY, nB)

= b/(nB +1) - y/nY > 0

ε-Sampling gives IFD

So P is increasing at each move,until it reaches a maximum

At maximum

b/(nB +1) < y/nY and y/(nY +1) < b/nB

Therefore

y/nY ≈ b/nB and so

y/b ≈ nY/nB

ML, ε-Sampling, Assumptions

• Bernoulli flowers: reward 1 or 0

• with probability p and 1-p resp. at Y

• with probability q and 1-q resp. at B

• ε > 0 small

• at sampling cl is p or q

ML, ε-Sampling, Movementsε

nY/nB = (p + qε)/ (q + pε) ≈ p/q

Markov chain

The Failures Strategy A(r,s)

Start by choosing a color at random

Leave Y after r consecutive failures

Leave B after s consecutive failures

ML, Failures, Assumptions

• Bernoulli flowers: reward 1 or 0

with probability p and 1-p resp. at Y

with probability q and 1-q resp. at B

• ε > 0 small

• “Failure” = “reward 0”

The Failures Strategy A(3,2)

ML and Failures Strategy A(3,2)

Now nY/nB = p/q if and only if

ML and Failures Strategy A(r,s)

Generally: nY/nB = p/q if and only if

This equality holds for many pairs of reals (r, s)

If 0 < δ < p < q < 1 – δ, and M is such that (1 – δ)2 < 4δ (1 – δM), then there are 1 < r, s < Msuch that A(r,s) matches (p, q)

ML and Failures Strategy A(fY,fB)

e.g. If 0 < 0.18 < p < q < 0.82, then there are 1 < r, s < 3such that A(r,s) matches (p, q)

If p < q < 1 – p, then there is x > 1such that A(x, x) matches (p, q)Proof: Ratio of visits Y to B for A(x, x) is

It is bigger than p/q for x = 1,while it goes to 0 as x goes to infinity

IFD 1 and Failures Strategy A(r,s)

Assumptions:•Field of Bernoulli flowers: p on Y, q on B•Finite population of identical A(r,s) bees •Each individual matches (p,q)

Then IFD will appear

Assumptions:•continuum of A(r,s) bees•total nectar supplies y and b•“certain” critical levels at Y and B

If y > b and ys > br, then there exist probabilities p and q and related critical levels on Y and B such that

i.e. IFD will appear

Learning

Attitude Towards Risk

Assuming normal distributions:

If the critical level is less than the mean, then any probability matching forager will favour higher variance

Assuming distributions like below:

If many flowers empty or very low nectar quantities, then any probability matching forager will favour higher variance

Concluding Remarks• A(r,s) focussed on statics of stable situation; no dynamic procedure to

reach it• ε-sampling does not really depend on ε• ε-sampling requires staying in same color for long time• Field data support failures behavior

Simple Foraging?The Truth is in the Field

F. Thuijsman, B. Peleg, M. Amitai, A. Shmida (1995): Automata, matching and foraging behaviour of bees. Journal of Theoretical Biology 175, 301-316.

Questions

simple foraging for simple foragers

Documents

urban foraging guide

behavior - foraging

why do foragers share and sharers forage? explorations...

diurnal cortisol rhythms in tsimaneâ€™ amazonian...

foragers, nearest-neighbour effect draft · draft 2...

foraging to farming

73 optimal foraging

optimal animal foraging in a two-dimensional world · types...

foraging - amherst.edu281+2011... · optimal foraging...

food transfers among hiwi foragers of venezuela: tests of...

foraging - amherst.edu · optimal foraging theory (oft)...

a simple guide to foraging - cold climate garden · a...

species group report card —seabirds · pelagic seabirds...

putting simple hierarchy into ant foraging: cluster-based...

team building -25% of ants work outside the nest -foraging...

foraging ecosystem

cultural resilience among the chabu foragers in … ·...

sharing a resource with randomly arriving foragers -...

15. from foragers to fighters: south africa's … ›...

foraging at mica