1. The Prisoner’s Dilemma Applied to the Interaction of Black Flies and Their Residents Maria Byrne – Math & Stats John McCreadie – Biology University of South Alabama MAA Local Meeting University of West Florida Friday, November 18 th

2. Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) Game Theory: Analysis of decisions made by rational agents in a hypothetical situation with fixed rules (game) where each agent has options that affect themselves and the group (different payoffs). When will cooperative or altruistic behavior be the winning strategy? (Verses uncooperative or ‘cheating’ behavior.)

3. Prisoner’s Dilemma (Axelrod, 1984) Prisoner’s Dilemma –Two Prisoners –Police do not have enough evidence for a conviction. Prisoner Options (Silence, Defection) –The prisoners can stay silent, in which case they will be sentenced for 1 month on a minor charge. –A prisoner can inform on the other prisoner (defect) in which case that prisoner goes free and the other serves a year in jail. –If both prisoners defect, they both serve 3 months in jail. Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)

4. Payoff Matrix Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

5. Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

6. Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

7. Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” 1 year jail “ not symmetric” Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

8. Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” 1 year jail “ not symmetric” Defect 1 year jail “ not symmetric” A: 3 months jail B: 3 months jail Prisoner A Prisoner B

9. Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” 1 year jail “ not symmetric” Defect 1 year jail “ not symmetric” 6 months jail “fair” Prisoner A Prisoner B

10. Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” 1 year jail “ not symmetric” Defect 1 year jail “ not symmetric” 6 months jail “fair” Prisoner A Prisoner B

11. Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

12. Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

13. Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

14. Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

15. Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B Either way – should defect!

16. Conundrum: Rational strategy is for both prisoners is to defect, however this leads to an outcome where the outcome is worse than if they had cooperated! Solution: Extended game play. The players gain information over time regarding whether the other is trustworthy, and have motive to cooperate so the other will too. Extended game play  Evolutionary Timescales Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)

17. Prisoner’s Dilemma Applied to Interaction of Black Flies and their Resident Fungi How to characterize their relationship? –Black flies spend larval stage in moving water, where they may encounter fungi that take residence in their gut. –The fungi (Harpella Zygomycota, “trichomycetes”) require black fly larvae for reproduction, so for them the interaction is beneficial. –If the fungi benefit the black fly, the relationship is mutualistic; if the fungi harm the black fly the relationship is parasitic.

18. Larval Black Fly Host

19. Prisoner’s Dilemma Applied to Interaction of Black Flies and their Resident Fungi How to characterize their relationship? –Black flies spend larval stage in moving water, where they may encounter fungi that take residence in their gut. –The fungi (Harpella Zygomycota, “trichomycetes”) require black fly larvae for reproduction, so for them the interaction is beneficial. –If the fungi benefit the black fly, the relationship is mutualistic; if the fungi harm the black fly the relationship is parasitic.

20. Trichomycetes group of cosmopolitan filamentous fungi obligate endosymbiotes in the guts of arthropods 300 species world wide 35 species in black flies trichospores (water column)

21. Prisoner’s Dilemma Applied to Interaction of Black Flies and their Resident Fungi How to characterize their relationship? –Black flies spend larval stage in moving water, where they may encounter fungi that take residence in their gut. –The fungi (Harpella Zygomycota, “trichomycetes”) require black fly larvae for reproduction, so for them the interaction is beneficial. –If the fungi benefit the black fly, the relationship is mutualistic; if the fungi harm the black fly the relationship is parasitic.

22. Spectrum of Host-Resident Interactions Parasitic Mutualistic 

23. Spectrum of Host-Resident Interactions Parasitic Mutualistic Commensalistic 

24. Is the relationship parasitic or mutualistic? Host Survival (%) 20 40 60 80 100 Commensalistic: Fungi have no effect on host survival. McCreadie et al, 2005 FED

25. Is the relationship parasitic or mutualistic? Host Survival (%) 20 40 60 80 100 Commensalistic: Fungi have no effect on host survival. Host Survival (%) 20 40 60 80 100 FED STARVED Mutualistic: Fungi improve survival rate of flies in starvation conditions McCreadie et al, 2005

26. Is the relationship parasitic or mutualistic? Host Survival (%) 20 40 60 80 100 Commensalistic: Fungi have no effect on host survival. Host Survival (%) 20 40 60 80 100 FED STARVED Mutualistic: Fungi improve survival rate of flies in starvation conditions Parasitic: Some species invade larval germ tissue and ‘hijack’ the ovaries of the female adult fly. McCreadie et al, 2005

27. Conceptual Framework The trichomycete-simuliid relationship changes with environmental factors. A model is developed to explore movement of the relationship along the P-C-M axis depending upon the number of fungi and host food supply. M P C

28. Developing a Model of Cost-Benefit of Fungi on Simuliid Fitness Fitness: A measure of the reproductive success of an individual allele, organism or species, depending on the context. Formal definition: the fitness F at age x is sum of the products of the relative rate of survival to a certain age l x and the expected number of offspring at that age m x (Brommer 2000, Roff 2008)

29. Black Fly Fitness l x is the survival rate –Trichomycetes increase this in starvation conditions. m x is the reproductive rate –Trichomycetes decrease this. There is a fitness trade-off, where trichomycetes exert a benefit for one term and a cost for the other.

30. Let f 0 be the mean adult fitness (reproductive number) of an individual fly when resource E is not limiting, in the absence of parasitism. Model of Limiting Resource E Trichomycetes Benefit on Fitness Via Survival Term

31. Let f 0 be the mean adult fitness (reproductive number) of an individual fly when resource E is not limiting, in the absence of parasitism. Model of Limiting Resource E With N Microbes  

32. Trichomycetes Cost on Fitness Via Fertility Term Let f 0 be the mean adult fitness (reproductive number) of an individual fly when resource E is not limiting, in the absence of parasitism. Model of Fertility Cost of N Microbes

33. Net Result on Fitness Net effect on fitness (F(R E,N) compared to f 0 ) depends upon amount of available resource E and number of trichomycetes N.

34. Net Result on Fitness

36. In practice, commensalism would be a band because some minimal difference is needed before beneficial or parasitic effects would be detectable.

37. Summary So Far We have developed a cost-benefit fitness model that shows quantitatively how a host-resident relationship can vary from parasitic to mutualistic depending upon environmental factors.

38. Back to the Prisoner’s Dilemma Combine our cost-benefit model for fitness with an evolutionary model where different fly and trichomycetes types compete for survival. Will the flies and trichomycetes defect or cooperate over time? Species ‘choose’ options by increasing their frequency in the dynamic, time- evolved population model according to the immediate fitness of that option.

39. Fly and Trichomycetes “Options” Trichomycetes do not have to be parasitic. –Only some species invade larval germ tissue of the fly. Larval black flies eject trichomycetes when they molt. –Will consider the hypothetical cases that some species of larvae may retain trichomycetes and some species may be resistant to residence.

40. Fly and Trichomycetes “Options” Trichomycetes –Parasitic –Non-Parasitic Larval black flies –Tolerant –SemiTolerant (eject fungi during molt) –Intolerant

41. Fly and Trichomycetes “Options” Payoff Matrix Resistant To Fungi Tolerant Until Molt Tolerant Of Fungi Doesn’t Hijack Ovaries Fly: No benefit. T: No available host, extinction. Fly: benefits in starvation conditions Hijacks Ovaries Fly: No benefit. T: No available host, extinction. T: benefits with extra reproduction Black Fly Trichomycetes

42. Fly and Trichomycetes “Options” Payoff Matrix Resistant To Fungi Tolerant Until Molt Tolerant Of Fungi Doesn’t Hijack Ovaries Fly DefectsCooperative Hijacks Ovaries Fungi Defects Black Fly Trichomycetes

43. Stochastic Evolutionary Model Initialization Parameters: –A specific number of each fly and trichomycetes species to ‘seed’ the simulation. –A fixed fly resource level (low, high) which determined the daily probability of a fly finding food (in the absence of microbes)

44. Stochastic Evolutionary Model All flies are initially 0 days old and have the same growth rate if food is found. When flies reach growth stage 1 they molt, at growth stage 2 they begin laying eggs. All trichomycetes are initially free- swimming. Gut microbes divide and produce spores (free-swimming trichomycetes) at a constant rate.

45. Stochastic Evolutionary Model Evolution. At each time step: –Free swimming trichomycetes have a probability of encountering a fly (mass action) and occupy that fly if that fly is tolerant. –Flies have a probability of encountering food (mass action). The presence of microbes in their gut increases the probability of finding food. Encountering food results in growth. –Resident fungi, if parasitic, have a probability of invading the fly germ cells.

46. Stochastic Evolutionary Model All fly and trichomycetes states are stored as values in a matrix. Growth Stages –At growth stage 1, flies molt and possibly eject the resident trichomycetes (become free- swimming). –At growth stage 2, flies begin laying eggs or trichomycetes.

47. Preliminary Model Start with three fly species –Intolerant to fungi –Semi-tolerant (eject fungi during molt) –Tolerant to fungi Consider only non-parasitic trichomycetes. Predict that the most tolerant fly species will be most fit.

48. Preliminary Model Payoff Matrix Resistant To Fungi Tolerant Until Molt Tolerant Of Fungi Doesn’t Hijack Ovaries Fly: No benefit. T: No available host, extinction. Fly: benefits in starvation conditions Black Fly Trichomycetes

49. Preliminary Model Results

50. Reproductive Number R 0 Tolerant -- 0.96 SemiTolerant -- 0.78 Intolerant – 0.05

51. Preliminary Model Results Reproductive Number R 0 Tolerant -- 0.96 SemiTolerant -- 0.78 Intolerant – 0.05 Number Of Gut Trichomycetes Tolerant -- 48 SemiTolerant -- 26 Intolerant – 0

52. Preliminary Model Results Reproductive Number R 0 Tolerant -- 0.96 SemiTolerant -- 0.78 Intolerant – 0.05 Number Of Gut Trichomycetes Tolerant -- 48 SemiTolerant -- 26 Intolerant – 0 Prob of Maturity Tolerant – 0.016 SemiTolerant – 0.007 Intolerant – 0.000

53. Future Directions Determine the most fit fly/trichomycetes species in the case of parasitic trichomycetes phenotypes. Will explore the possibility of stable “cheaters” – small populations of flies and trichomycetes that benefit from the cooperative behavior of most species. Determine the most fit fly/trichomycetes species in the case of a stochastically or spatially varying resource environment and a constant flux of small numbers of each phenotype.

54. Myxobacteria ‘Cheaters’ During myxobacteria fruiting body formation, most myxobacteria cooperate to form the fruiting body. Most cells will become structural, some will become spores. Some bacterial cells won’t contribute to the fruiting body structure but always become spores. (Velicer, Kroos, Lenski, 2000)

55. Principle of competitive exclusion: no two species can occupy the same niche. Gause, 1934 “No pure strategy is evolutionarily stable in the repeated Prisoner's Dilemma game “ Boyd, R. and Lorberbaum, J. P. 1987. A Couple Ecological Principles

56. References McCreadie, J. W., C. E. BEARD, and P. H. Adler. 2005. Context- dependent symbiosis between black flies (Diptera: Simuliidae) and trichomycete fungi (Harpellales: Legeriomycetaceae). Oikos 108:362-370. McCreadie JW, Adler PH, Larson R. 2010. Variation in larval fitness of a black fly species over heterogeneous habitats. Aquatic Insects. In press. Axelrod, R. and Hamilton, W. D. 1981. The evolution of cooperation. - Science 211: 1390-1396. Brommer J. E. 2000 The evolution of fitness in life-history theory. Biol. Rev. (Camb.) 75, 377–404. Roff D. A. 2008 Defining fitness in evolutionary models. J. Genet. 87, 339– 348 Feldmann, M. W. and Thomas, A. C. 1987. Behaviour dependent contexts for repeated plays of the prisoner's dilemma. II. Dynamical aspects of the evolution of cooperation. - J. theor. Biol. 128: 297-315. Velicer, G.J., Kroos, L. and R.E. Lenski. Developmental cheating in the social bacterium Myxococcus xanthus. Nature 404:598-601. Boyd, R. and Lorberbaum, J. P. 1987. No pure strategy is evolutionary stable in the repeated prisoner's dilemma game. Nature 327: 58-59.

57. Thank You

58. Thank You to University of West Florida Thank You