The Prisoner’s Dilemma Applied to the Interaction of Black Flies and Their Residents Maria Byrne – Math & Stats John McCreadie – Biology University of

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.


• Payoff Matrix


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


Silent Defects


B: 1 month jail

A: free

B: 1 year jail


B: free

A: 3 months jail

B: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail

“fair”

A: free

B: 1 year jail


B: free

A: 3 months jail

B: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail

“fair”

1 year jail

“ not symmetric”


B: free

A: 3 months jail

B: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail

“fair”

1 year jail


Defect1 year jail


A: 3 months jail

B: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail

“fair”

1 year jail


Defect1 year jail


6 months jail

“fair”

Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail

“fair”

1 year jail


Defect1 year jail


6 months jail

“fair”

Prisoner A

Pris

oner

B

• Payoff Matrix – From Prisoner’s Perspective


Silent Defects


B: 1 month jail

A: free

B: 1 year jail


B: free

A: 3 months jail

B: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects


B: 1 month jail

A: free

B: 1 year jail


B: free

A: 3 months jail

B: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects


B: 1 month jail

A: free

B: 1 year jail


B: free

A: 3 months jail

B: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects


B: 1 month jail

A: free

B: 1 year jail


B: free

A: 3 months jail

B: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects


B: 1 month jail

A: free

B: 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 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.

Larval Black Fly

Host






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)






Spectrum of Host-Resident Interactions

Paras

itic

Mut

ualis

tic

Spectrum of Host-Resident Interactions

Paras

itic

Mut

ualis

tic

Comm

ensa

listic

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


ost S

urvi

val (

%)

20

40

60

80

100


Hos

t Sur

viva

l (%

)

20

40

60

80

100

FED STARVED

Mutualistic: Fungi improve survival rate of flies in starvation conditions



ost S

urvi

val (

%)

20

40

60

80

100


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.


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


• Model of Limiting Resource EWith N Microbes

bE

bE

ENkRa

NkRfNRF

1

10,

Trichomycetes Cost on FitnessVia Fertility Term


• 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,


tfertilitybenefitsurvivalfNRF E cos, 0


d

d

bE

bE

ENkc

Nk

NkRa

NkRfNRF

2

2

1

10 1,


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.


• Trichomycetes– Parasitic– Non-Parasitic

• Larval black flies– Tolerant– SemiTolerant (eject fungi during molt)– Intolerant


• 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


Hijacks

Ovaries

Fly: No benefit.


T: benefits with extra

reproduction

T: benefits with extra

reproduction

Black Fly

Tric

hom

ycet

es


• Payoff Matrix

Resistant


Tolerant


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)


• 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.


• 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.


• 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


Tolerant


Ovaries

Fly: No benefit.




Black Fly

Tric

hom

ycet

es

Preliminary Model Results


Reproductive Number R0

Tolerant -- 0.96SemiTolerant -- 0.78

Intolerant – 0.05




Intolerant – 0.05

Number Of Gut TrichomycetesTolerant -- 48

SemiTolerant -- 26Intolerant – 0




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.

Thank You

Thank You to University of West Florida

Thank You

Documents

The Prisoner’s Dilemma Applied to the Interaction of Black Flies and Their Residents Maria Byrne – Math & Stats John McCreadie – Biology University of