1. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory1/44

2. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory2/44 Simple Foraging for Simple Foragers Frank Thuijsman joint work with Bezalel Peleg, Mor Amitai, Avi Shmida

3. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory3/44 Outline

4. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory4/44 Outline Two approaches that explain certain observations of foraging behavior The Ideal Free Distribution The Matching Law …Risk Aversity

5. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory5/44 The Ideal Free Distribution Stephen Fretwell & Henry Lucas (1970): Individual foragers will distribute themselves over various patches proportional to the amounts of resources available in each.

6. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory6/44 The Ideal Free Distribution 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.

7. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory7/44 The Matching Law Richard Herrnstein (1961): The organism allocates its behavior over various activities in proportion to the value derived from each activity.

8. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory8/44 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

9. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory9/44 Simplified Model ? Y ellow B lue p qy b Two patches Nectar quantitiesNectar probabilities One or more bees

10. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory10/44 Only Yellow …

11. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory11/44 … And Blue

12. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory12/44 No Other Colors

13. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory13/44 Yellow and Blue Patches

14. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory14/44 IFD and Simplified Model Y ellow B lue y b nectar quantities: nYnY nBnB numbers of bees: two patches: IFD: n Y / n B y / b

15. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory15/44 Matching Law and Simplified Model Y ellow B lue p q nectar probabilities: nYnY nBnB visits by one bee: two patches: Matching Law: n Y / n B p / q

16. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory16/44 How to choose where to go? Alone …

17. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory17/44 …or with others How to choose where to go?

18. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory18/44 No Communication ! How to choose where to go? bzzz, bzzz, …

19. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory19/44 How to choose where to go? ε-sampling or failures strategy!

20. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory20/44 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

21. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory21/44 The ε-Sampling Strategy Start by choosing a color at random At each following stage, with probability: ε sample other color 1 - ε stay at same color. If sample “at least as good”, then stay at new color, otherwise return immediately. ε > 0

22. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory22/44 IFD, ε-Sampling, Assumptions reward at Y:0 or 1 with average y/n Y reward at B:0 or 1 with average b/n B no nectar accumulation ε very small: only one bee sampling At sampling cl is y/n Y or b/n B

23. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory23/44 ε-Sampling gives IFD Proof: Let P(n Y, n B ) = y·(1 + 1/2 + 1/3 + ··· + 1/n Y ) - b·(1 + 1/2 + 1/3 + ··· + 1/n B ) If bee moves from Y to B, then we go from (n Y, n B ) to (n Y - 1, n B + 1) and P(n Y - 1, n B + 1) - P(n Y, n B ) = b/(n B +1) - y/n Y > 0

24. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory24/44 ε-Sampling gives IFD So P is increasing at each move, until it reaches a maximum At maximum b/(n B +1) < y/n Y and y/(n Y +1) < b/n B Therefore y/n Y ≈ b/n B and so y/b ≈ nY/nBy/b ≈ nY/nB

25. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory25/44 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 no nectar accumulation ε > 0 small at sampling cl is p or q

26. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory26/44 ML, ε-Sampling, Movements ε ε 1- ε 1- p 1- q qp Y1Y1 Y2Y2 B2B2 B1B1 n Y /n B = (p + qε)/ (q + pε) ≈ p/q Markov chain

27. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory27/44 The Failures Strategy A(r,s) Start by choosing a color at random Next: Leave Y after r consecutive failures Leave B after s consecutive failures

28. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory28/44 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 no nectar accumulation ε > 0 small “Failure” = “reward 0”

29. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory29/44 The Failures Strategy A(3,2)

30. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory30/44 The Failures Strategy A(3,2)

31. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory31/44 ML and Failures Strategy A(3,2) Now n Y /n B = p/q if and only if

32. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory32/44 ML and Failures Strategy A(r,s) Generally: n Y /n B = p/q if and only if This equality holds for many pairs of reals (r, s)

33. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory33/44 ML and Failures Strategy A(r,s) If 0 < δ < p < q < 1 – δ, and M is such that (1 – δ) 2 < 4δ (1 – δ M ), then there are 1 < r, s < M such that A(r,s) matches (p, q)

34. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory34/44 ML and Failures Strategy A(f Y,f B ) e.g. If 0 < 0.18 < p < q < 0.82, then there are 1 < r, s < 3 such that A(r,s) matches (p, q)

35. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory35/44 ML and Failures Strategy A(r,s) If p < q < 1 – p, then there is x > 1 such 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

36. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory36/44 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

37. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory37/44 IFD 2 and Failures Strategy A(r,s) Assumptions: continuum of A(r,s) bees total nectar supplies y and b “certain” critical levels at Y and B

38. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory38/44 IFD 2 and Failures Strategy A(r,s) 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

39. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory39/44 Learning

40. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory40/44 Attitude Towards Risk ? 1 3 2 2 2 2

41. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory41/44 Attitude Towards Risk Assuming normal distributions: If the critical level is less than the mean, then any probability matching forager will favour higher variance

42. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory42/44 Attitude Towards Risk Assuming distributions like below: If many flowers empty or very low nectar quantities, then any probability matching forager will favour higher variance

43. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory43/44 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

44. frank@math.unimaas.nlSex and the Signal: Evolution and Game Theory44/44 frank@math.unimaas.nl ? F. Thuijsman, B. Peleg, M. Amitai, A. Shmida (1995): Automata, matching and foraging behaviour of bees. Journal of Theoretical Biology 175, 301-316. Questions