The Prisoner’s Dilemma Applied to the Interaction of Black Flies and Their Residents Maria Byrne...
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The Prisoner’s Dilemma Applied to the Interaction of Black
Flies and Their Residents
Maria Byrne – Math & StatsJohn McCreadie – Biology
University of South Alabama
MAA Local Meeting University of West FloridaFriday, November 18th
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.)
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)
• Payoff Matrix
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
SilentA: 1 month jail
B: 1 month jail
A: free
B: 1 year jail
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From a Global Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
SilentA: 1 month jail
B: 1 month jail
A: free
B: 1 year jail
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From a Global Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
Silent2 months jail
“fair”
A: free
B: 1 year jail
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From a Global Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
Silent2 months jail
“fair”
1 year jail
“ not symmetric”
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From a Global Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
Silent2 months jail
“fair”
1 year jail
“ not symmetric”
Defect1 year jail
“ not symmetric”
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From a Global Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
Silent2 months jail
“fair”
1 year jail
“ not symmetric”
Defect1 year jail
“ not symmetric”
6 months jail
“fair”
Prisoner A
Pris
oner
B
• Payoff Matrix – From a Global Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
Silent2 months jail
“fair”
1 year jail
“ not symmetric”
Defect1 year jail
“ not symmetric”
6 months jail
“fair”
Prisoner A
Pris
oner
B
• Payoff Matrix – From Prisoner’s Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
SilentA: 1 month jail
B: 1 month jail
A: free
B: 1 year jail
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From Prisoner’s Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
SilentA: 1 month jail
B: 1 month jail
A: free
B: 1 year jail
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From Prisoner’s Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
SilentA: 1 month jail
B: 1 month jail
A: free
B: 1 year jail
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From Prisoner’s Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
SilentA: 1 month jail
B: 1 month jail
A: free
B: 1 year jail
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
• Payoff Matrix – From Prisoner’s Perspective
Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)
Silent Defects
SilentA: 1 month jail
B: 1 month jail
A: free
B: 1 year jail
DefectA: 1 year jail
B: free
A: 3 months jail
B: 3 months jail
Prisoner A
Pris
oner
B
Either way – should defect!
• 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)
Prisoner’s Dilemma Applied toInteraction 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.
Prisoner’s Dilemma Applied toInteraction 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.
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)
Prisoner’s Dilemma Applied toInteraction 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.
Is the relationship parasitic or mutualistic?H
ost S
urvi
val (
%)
20
40
60
80
100
Commensalistic: Fungi have no effect on host survival.
McCreadie et al, 2005
FED
Is the relationship parasitic or mutualistic?H
ost S
urvi
val (
%)
20
40
60
80
100
Commensalistic: Fungi have no effect on host survival.
Hos
t Sur
viva
l (%
)
20
40
60
80
100
FED STARVED
Mutualistic: Fungi improve survival rate of flies in starvation conditions
McCreadie et al, 2005
Is the relationship parasitic or mutualistic?H
ost S
urvi
val (
%)
20
40
60
80
100
Commensalistic: Fungi have no effect on host survival.
Hos
t Sur
viva
l (%
)
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
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
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 lx and the expected number of offspring at that age mx
(Brommer 2000, Roff 2008)
xxmlxF )(
Black Fly Fitness
• lx is the survival rate– Trichomycetes increase this in starvation conditions.
• mx 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.
xxmlxF )(
• Let f0 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
bE
bE
ERa
RfRF 0
Trichomycetes Benefit on FitnessVia Survival Term
Trichomycetes Benefit on FitnessVia Survival Term
• Let f0 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 EWith N Microbes
bE
bE
ENkRa
NkRfNRF
1
10,
Trichomycetes Cost on FitnessVia Fertility Term
• Let f0 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
d
d
Nkc
Nkf
2
20 1
Net Result on Fitness
• Net effect on fitness (F(RE,N) compared to f0) depends upon amount of available resource E and number of trichomycetes N.
tfertilitybenefitsurvivalfNRF E cos, 0
d
d
bE
bE
ENkc
Nk
NkRa
NkRfNRF
2
2
1
10 1,
Net Result on Fitness
d
d
bE
bE
ENkc
Nk
NkRa
NkRfNRF
2
2
1
10 1,
In practice, commensalism would be a band because some minimal difference is needed before beneficial or parasitic effects would be detectable.
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.
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.
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.
Fly and Trichomycetes “Options”
• Trichomycetes– Parasitic– Non-Parasitic
• Larval black flies– Tolerant– SemiTolerant (eject fungi during molt)– Intolerant
Fly and Trichomycetes “Options”
• Payoff Matrix
Resistant
To FungiTolerant Until Molt
Tolerant
Of FungiDoesn’t Hijack
Ovaries
Fly: No benefit.
T: No available host, extinction.
Fly: benefits in starvation conditions
Fly: benefits in starvation conditions
Hijacks
Ovaries
Fly: No benefit.
T: No available host, extinction.
T: benefits with extra
reproduction
T: benefits with extra
reproduction
Black Fly
Tric
hom
ycet
es
Fly and Trichomycetes “Options”
• Payoff Matrix
Resistant
To FungiTolerant Until Molt
Tolerant
Of FungiDoesn’t Hijack
OvariesFly Defects Cooperative
Hijacks
OvariesFungi Defects
Black Fly
Tric
hom
ycet
es
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)
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.
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.
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.
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.
Preliminary Model
• Payoff Matrix
Resistant
To FungiTolerant Until Molt
Tolerant
Of FungiDoesn’t Hijack
Ovaries
Fly: No benefit.
T: No available host, extinction.
Fly: benefits in starvation conditions
Fly: benefits in starvation conditions
Black Fly
Tric
hom
ycet
es
Preliminary Model Results
Reproductive Number R0
Tolerant -- 0.96SemiTolerant -- 0.78
Intolerant – 0.05
Preliminary Model Results
Reproductive Number R0
Tolerant -- 0.96SemiTolerant -- 0.78
Intolerant – 0.05
Number Of Gut TrichomycetesTolerant -- 48
SemiTolerant -- 26Intolerant – 0
Preliminary Model Results
Reproductive Number R0
Tolerant -- 0.96SemiTolerant -- 0.78
Intolerant – 0.05
Number Of Gut TrichomycetesTolerant -- 48
SemiTolerant -- 26Intolerant – 0
Prob of MaturityTolerant – 0.016
SemiTolerant – 0.007Intolerant – 0.000
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.
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)
• 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
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.