1

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)

2

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

3

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.

4

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

5

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.

6

Basic introduction to Evolutionary Game Theory

Stable Equilibrium

0 11/2

1)( +−= pDoveF

23)( +−= pHawkF

p (freq. of Hawk)

7

Basic introduction to Evolutionary Game Theory

Unstable Equilibrium

0 11/2

)(BF

)(AF

p (freq. of A)

8

Cannibalistic polymorphism

cannibalmorph

typicalmorph

Hynobius retardatus salamander larvae

Photo by Y. Kohmatsu

9

Cannibalistic polymorphism

Sympatric coexistence of cannibal morphs and typical morphs

Ambystoma salamandersHynobius salamandersScaphiopus toads

10

Benefit of cannibal morph

Advantage in cannibalism

gape limited

cannibal morph typical morph

11

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)

12

Food webnatural enemy

cannibal morph typical morph

predation predation

cannibalism

regular food(small insects, planktons)

13

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

14

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

15

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.)

16

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

17

Model 1:Fitness

Cannibal morph

Typical morphp

pAF

mBp

kAF

T

C

−Γ−−

⋅=

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+

Γ=

11

Relative survival rate of cannibal morph(cost of cannibal morph is 1-m)

)1(<m

(fitness) = (body size) X (survival rate)

18

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.

19

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.)

20

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.

21

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.

22

Model 2

fitness

g

1

0

Phase I Phase II

typicalmorph

cannibalmorph

p

1-p

dead

giant

normal

23

normal

giant

Γ

dead

Model 2

typicalmorph

p

1-p

m 1-m

cannibalmorph

24

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

25

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)

26

Model 2:Necessary condition for stability

11−

>g

m (stability condition)

mg 11+>

2>g

27

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

28

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

29

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.

30

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.

31

Thank you for your attention.