drastic growth effect explains sympatric …joewakano.sakura.ne.jp/research/presentation8.pdf1...
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Drastic growth effect explainssympatric cannibalistic polymorphism
Joe Yuichiro Wakano
Department of Biological SciencesThe University of Tokyo
2006 March 10
Journal of Theoretical Biology 226:69-77 (2004)
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Agenda
Basic introduction to evolutionary game theoryCannibalistic polymorphism
cost/benefit of cannibal morphfood web structure
Model 1without drastic growth effect
Model 2with drastic growth effect
Conclusion
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Basic introduction to Evolutionary Game Theory
The frequency of a gene that gives higher fitness to its carrier will increase (Darwinian theory of evolution).
If animal behavior is determined by a gene, the best behavior will evolve.
The best behavior often depends on behavior of other animals.
If there are only two types of behavior (A & B), fitness of each behavior depends on the frequency of A (frequency-dependent selection).
Such a situation is studied by evolutionary game theory.
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Basic introduction to Evolutionary Game Theory
Terminologyplayer = animal / individualstrategy = behavior (genetically determined)payoff = fitnessequilibriumstability
Classic game theoryrational players with full knowledge of payoff matrixrational decision making
Evolutionary game theoryblind playersnatural selection
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Basic introduction to Evolutionary Game Theory
10Dove
2-1Hawk
DoveHawk
p 1-p
-p+2(1-p)
1-p
1)(23)(
+−=+−=
pDoveFpHawkF
If p=1/2, both strategies are equally successful.
p=1/2 is equilibrium.
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Basic introduction to Evolutionary Game Theory
Stable Equilibrium
0 11/2
1)( +−= pDoveF
23)( +−= pHawkF
p (freq. of Hawk)
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Basic introduction to Evolutionary Game Theory
Unstable Equilibrium
0 11/2
)(BF
)(AF
p (freq. of A)
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Cannibalistic polymorphism
cannibalmorph
typicalmorph
Hynobius retardatus salamander larvae
Photo by Y. Kohmatsu
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Cannibalistic polymorphism
Sympatric coexistence of cannibal morphs and typical morphs
Ambystoma salamandersHynobius salamandersScaphiopus toads
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Benefit of cannibal morph
Advantage in cannibalism
gape limited
cannibal morph typical morph
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Cost of cannibal morph
not very clearCosts such as eating relatives, pathogen infection or handling time are the cost of cannibalism.
Larger mortality (due to decreased swimming speed)
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Food webnatural enemy
cannibal morph typical morph
predation predation
cannibalism
regular food(small insects, planktons)
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Assumptions
Two distinct strategies:cannibal and typical morphs
The strategy is genetically determined
experimental studies shows both genetic basis and phenotypic plasticity
Only cannibal morphs can perform cannibalism
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Goal of the study
Explain the cannibalistic polymorphism as the stable coexistence solution in an evolutionary game modelDetermine the condition under which such a polymorphic solution exists
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Model 1:parameters
)1( pp −=Γ γFreq. of cannibalism
Freq. of cannibal morph p
p−1
Intensity of cannibalism γ
Freq. of typical morph
(Population size is normalized to unity.)
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Model 1:Body size at the end of growth period
Typical morph A
Bp
kA +Γ
k : Energy conversion factor (<1)(the efficiency of digestion)
Cannibal morph
growth by regular food consumption
probability with which each cannibal morph can eat a typical morph
size of victim
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Model 1:Fitness
Cannibal morph
Typical morphp
pAF
mBp
kAF
T
C
−Γ−−
⋅=
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛+
Γ=
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Relative survival rate of cannibal morph(cost of cannibal morph is 1-m)
)1(<m
(fitness) = (body size) X (survival rate)
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Model 1:The result
p
ApAFT +−= )( γ
Coexistence will not evolve.
)()( γγ mkAmBpmkAFC −+−=
Since mk<1, even if the intersection exists, it is always unstable equilibrium.
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Drastic growth effect
Larvae initially have similar body size.A cannibalism promotes the cannibal to outstanding size (a giant)Once giants appear, they monopolize most cannibalisms
giants
typical cannibal
normals
(Difference between typical/cannibal morphs becomes negligible.)
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Drastic growth effect
The aim of taking cannibal morph strategy might be to increase the probability of becoming a giant by developing a slightly larger organ (cf. head width).
Fitness of a giant is naturally expected to be very high.
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Model 2:Assumptions
consider only normal individuals and level 1 giants
Fitness of a giant is constant
A level 1 giant will proceed to level 2 by performing successful cannibalism. For simplicity, however, we neglect such a chain. The most important point is how many cannibal morphs become giants.
In real salamanders, almost all occurrences of cannibalismare monopolized by a few giants. However, rare occurrences among normal larvae, which produces a giant, are important.
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Model 2
fitness
g
1
0
Phase I Phase II
typicalmorph
cannibalmorph
p
1-p
dead
giant
normal
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normal
giant
Γ
dead
Model 2
typicalmorph
p
1-p
m 1-m
cannibalmorph
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Model 2:Analysis
[ ]
ppF
mpgp
F
T
C
−Γ−−
=
⋅Γ−+Γ=
11
)(1
{ }12
1)1(2)1(2+−=
+−+−−=pF
gmpgmF
T
C
γγγ
{ }{ }1)1(2
11)1(2*−−−+−
=gm
gmpγγ
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Model 2:Analysis
If
11−
>g
m
1)1(21
+−>
gm
γ
γ21−<m
then, there is a globally stable interior equilibrium, that is,
stable coexistence solution
(stability condition)
(existence condition of an equilibrium)
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Model 2:Necessary condition for stability
11−
>g
m (stability condition)
mg 11+>
2>g
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Evolutionarily stable frequency of cannibal morphs
stable
unstable
1
2
3
4
5
6g 0.4
0.5
0.6
0.7
0.8
0.9
1
m
00.25
0.50.75
1
p
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Summary of the analysis
Evolutionary outcome is one ofi) domination of typical morph
ii) domination of cannibal morphiii) stable coexistence
Coexistence evolves when g is large and m is intermediate.
For coexistence, g>2 is necessary
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Is g >2 realistic ?Experimental data
Ambystoma tigrinum salamander’s case:
No more than one cannibal (=giant) appeared per aquariumcontaining 16 larvae (Pfennig & Collins 1993).
Fitness of a giant is much larger than two times fitness of non-giants.
This promotes the stable coexistence of typical/cannibal morphs or polymorphism.
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Conclusion
Evolutionary game models of cannibalistic polymorphism are proposedThe condition of stable coexistence of typical/cannibal morphs is studied
Considering drastic growth effect, large fitness of a giant (>2) promotes evolution of cannibalistic polymorphism.
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Thank you for your attention.