Simple Foraging for Simple Foragers

1. Sex and the Signal: Evolution and Game Theory

2. Simple Foraging for Simple Foragers Frank Thuijsman joint work with Bezalel Peleg, Mor Amitai, Avi Shmida Sex and the Signal: Evolution and Game Theory

3. Outline Sex and the Signal: Evolution and Game Theory

4. Outline Two approaches that explain certain observations of foraging behavior The Ideal Free Distribution The Matching Law …Risk Aversity Sex and the Signal: Evolution and Game Theory

5. 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. Sex and the Signal: Evolution and Game Theory

6. 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. Sex and the Signal: Evolution and Game Theory

7. The Matching Law Richard Herrnstein (1961): The organism allocates its behavior over various activities in proportion to the value derived from each activity. Sex and the Signal: Evolution and Game Theory

8. 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 Sex and the Signal: Evolution and Game Theory

9. Simplified Model Two patches One or more bees Yellow Blue ? p y q b Nectar quantities Nectar probabilities Sex and the Signal: Evolution and Game Theory

10. Only Yellow … Sex and the Signal: Evolution and Game Theory

11. … And Blue Sex and the Signal: Evolution and Game Theory

12. No Other Colors Sex and the Signal: Evolution and Game Theory

13. Yellow and Blue Patches Sex and the Signal: Evolution and Game Theory

14. IFD and Simplified Model Yellow Blue two patches: y b nectar quantities: nY nB numbers of bees: IFD: nY / nB y / b Sex and the Signal: Evolution and Game Theory

15. Matching Law and Simplified Model Yellow Blue two patches: p q nectar probabilities: nY nB visits by one bee: nY / nB p / q Matching Law: Sex and the Signal: Evolution and Game Theory

16. How to choose where to go? Alone … Sex and the Signal: Evolution and Game Theory

17. How to choose where to go? …or with others Sex and the Signal: Evolution and Game Theory

18. How to choose where to go? bzzz, bzzz, … No Communication ! Sex and the Signal: Evolution and Game Theory

19. How to choose where to go? ε-sampling orfailures strategy! Sex and the Signal: Evolution and Game Theory

20. 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 Sex and the Signal: Evolution and Game Theory

21. 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 Sex and the Signal: Evolution and Game Theory

22. 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 • εvery small: only one bee sampling • At sampling cl is y/nY or b/nB Sex and the Signal: Evolution and Game Theory

23. 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) and P(nY- 1, nB + 1) - P(nY, nB) = b/(nB +1)-y/nY> 0 Sex and the Signal: Evolution and Game Theory

24. ε-Sampling gives IFD So P is increasing at each move, until it reaches a maximum At maximum b/(nB +1)<y/nYand y/(nY +1)<b/nB Therefore y/nY ≈ b/nB and so y/b≈nY/nB Sex and the Signal: Evolution and Game Theory

25. ML, ε-Sampling, Assumptions • Bernoulli flowers: reward 1or 0 • with probability p and 1-p resp. at Y • with probability q and 1-q resp. at B • no nectar accumulation • ε> 0small • at sampling cl is p or q Sex and the Signal: Evolution and Game Theory

26. ML, ε-Sampling, Movements ε Y1 B2 1- ε 1- p p q Markov chain 1- q B1 Y2 1- ε ε nY/nB = (p + qε)/ (q + pε) ≈ p/q Sex and the Signal: Evolution and Game Theory

27. 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 Sex and the Signal: Evolution and Game Theory

28. ML, Failures, Assumptions • Bernoulli flowers: reward 1or0 with probability p and 1-p resp. at Y with probability q and 1-q resp. at B • no nectar accumulation • ε> 0small • “Failure” = “reward 0” Sex and the Signal: Evolution and Game Theory

29. The Failures Strategy A(3,2) Sex and the Signal: Evolution and Game Theory

30. The Failures Strategy A(3,2) Sex and the Signal: Evolution and Game Theory

31. ML and Failures Strategy A(3,2) Now nY/nB = p/q if and only if Sex and the Signal: Evolution and Game Theory

32. 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) Sex and the Signal: Evolution and Game Theory

33. ML and Failures Strategy A(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 < M such that A(r,s) matches (p, q) Sex and the Signal: Evolution and Game Theory

34. ML and Failures Strategy A(fY,fB) 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) Sex and the Signal: Evolution and Game Theory

35. 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 Sex and the Signal: Evolution and Game Theory

36. 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 Sex and the Signal: Evolution and Game Theory

37. 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 Sex and the Signal: Evolution and Game Theory

38. 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 Sex and the Signal: Evolution and Game Theory

39. Learning Sex and the Signal: Evolution and Game Theory

40. Attitude Towards Risk 2 1 3 2 2 2 ? Sex and the Signal: Evolution and Game Theory

41. Attitude Towards Risk Assuming normal distributions: If the critical level is less than the mean, then any probability matching forager will favour higher variance Sex and the Signal: Evolution and Game Theory

42. 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 Sex and the Signal: Evolution and Game Theory

43. 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 Sex and the Signal: Evolution and Game Theory

44. Questions ? 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. Sex and the Signal: Evolution and Game Theory