International Conference on Mathematical and Theoretical Biology, Hawaii, July 16-19 2001

(Center for Ecological Research, Kyoto University)

Yukihiro Kohmatsu
Norio Yamamura(Department of Chemical System Engineering, University of Tokyo)

Joe Yuichiro Wakano

若野 友一郎1

Evolutionary dynamics of growth strategy
in game-theoretical situationin cannibalistic amphibians

at metamorphosis.

Larvae that cannibalize conspecifics has larger body size

Cannibalism is widely observed in amphibian larvae.

Empirical Facts

2

cannibal morph

cannibalize
typical morph
3

Cannibalistic polymorphism is known in some species.

Empirical Factstypical morph

4

Relative head sizes determine occurrence of cannibalism.

cannibal morph

Larger Head → Larger Mouth→ Advantageous in cannibalism

Empirical Factscannibal morph

natural enemy(larvae of dragon fly)typical morph
5

Cannibal morph is more often predated by natural enemy.

Empirical Factscannibal morph
6

Unbalanced body shape is thought tochange behavior or decreaseswimming speed.

Why ?

Cannibal morph is more often predated by natural enemy.

Empirical Facts

Evolutionary game model of optimal growth scheduleabsolute body shapesuffers from high predation pressure.

relative head size

benefits from cannibalism.

Cannibal morph ...

Empirical Facts : Summary

7

Head Size sModell

Body length

at given time t.

8head size and body length

the total energy to growth of

distribution function u(t) of

Growth strategy is aModelinvestment to head size

investment to body length

dsl= uEdt

dls = (1 − u ) E
dt

9total incoming energyd(sl ) = E
dtregular foodinteraction term
due to cannibalism

10

predation bynatural enemys
E = asl + C ( s, p ( s )) − B ( 2 )lbeing predated by cannibal or natural enemy.

For simplicity, we substitute loss of energy intake forModel2F = (s (T )l (T ) )Fitness is measured by volume at time T.

11B( x) = b( x − k ) 2 for computer simulationss2(equationholdswhens=kl)
B( 2 ) ≥ 0l

Unbalanced body shape brings higher intensity of predation.

Model

BOUNDARY CONDITIONS

220 ≤ u ≤1

Initial body shape is uniformly fixed.

s(0) = k{l (0)} , s(T ) = k{l (T )}at time 0 and T.

12

For successful metamorphosis, body shape must be balanced

Modelu0(t) = u*(t)Necessary condition for ESS is13

If a growth function u * (t ) is an ESS, the fitness of mutant strategy
with growth function u (t ) must be maximum at u (t ) = u * (t ) .

Optimal mutant strategy u0(t) against wild type strategy u*(t) is
derived by Pontrjagin’s theory.

Analytic Result2c 4s = kl + l6b

Balanced growthHigh predation pressure
Low cannibalistic interaction

When we define C’(0)=c and B( x) = b( x − k ) 2 ,14As mutant is rare, growth trajectory of a wild type individual (s*,l*)is independent of mutant strategy. We assume the interaction termof mutant as C ( s − s*) . Necessary condition for ESS iss2l C ' ( 0) = 3 B ' ( 2 )lAnalytic Result15

16

Analytic Results(0)Head sizel(0)Balanced
growthBody length

OptimalGrowth17

si2EaslCb(k)=+−− ii i2li s&Ei
 = uili si
 si
 l&Ei = (1 − ui )
 li
li si

i :genotype

18

We perfom computer simulation including growth dynamics andevolutionary dynamics.

Derived analytic solutions are only candidates of ESS.

Computer simulation 1

u(t)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

t

0.5

strategy

0.6

0.7

0.8

0.9

u0
u1
u2
u3
u4
u5
u6
u7
u8
u9
u10
u11
u12
u13
u14

su15
u14

u2

lu3

u1

trajectory

Genetically fixed 15 strategies

Computer simulation 119

genotype jδ : encouter ratep j : frequency ofγϕ (x)η20Head Size DifferenceΓ ≡ γδ corresponds to C’(0) in analytic model.


2ϕ ( x) ≡ η − 1
− 2γx


 1 + exp( η ) (−η < ϕ ( x) < η , ϕ (0) = 0, ϕ ' (0) = γ )j =1

C = δ ∑ p jϕ ( si − s j )15Interaction term C has an explicit form as

Computer simulation 1

growth dynamics

evolutionary dynamics

Random mutation occurs every 100 generations.

21

Genotype distribution in the next generation is determined so thatfrequency of each genotype is proportional to its fitness.

Fitness (squared volume at time T) is determined by this growthsimulation.

Growth of each genotype is simulated from t=0 to t=T.

Initial genotype distribution is uniform.

Computer simulation 1

1+

1+

1+

1+

1+

0.4

0.6

0.8

1.0

1
0

0.2

Ω

3+

3+

3+

3+

3+

3
1 .3

4+

4+

4+

4+

4+

4
2

5+

5+

5+

5+

4,5

5
2.8

6+

6+

6+

6+

4,5

6
3.5

7+7+

7+

6,7

3,4,5

7
4 .2

8+

8+

8

6,7

3,4,5

9+

9+

8

5,6,7

3,4,5

10+

10+

8,9

5,6,7

3,4,5

3,4,5

12
7 .3

3,4,5

13
7.7

11+

10,11

12,13

10,11

6,7,8,9

4,5,6,7

3,4,5

14
8

6,7,8,9

5,6,7

3,4,5

15
8 .3

13

Some ESS candidates are not always realized.
22

13,15

15

10,11 9,10,11,12 ,10,11,1

6,7,8,9 6,7,8,9 6,7,8,9

4,5,6,7 4,5,6,7 4,5,6,7

3,4,5

11
6.8

Genotypes with more than 10% time average frequency are shown.
i+ means correspoding genotype has >90% time average frequency.

2+2+2+2+2+2
0.6

corresponding strategy
Γ891055.6
6.2

Result (Ω ≡ δη Γ ≡ γδ )Computer simulation 1

u1generation

balanced

Computer simulation 1

strategy
unbalancedu1523

Why cyclic ?

similar to Taxon cycle

Endless replacement of the dominating strategies.

24

2. Fitness of the most unbalanced strategy is very low andbalanced strategy invades.

1. Slightly more unbalanced growth than population majority
is the most adaptive.

Computer simulation 1

Qualitative results remain the same.

Encounter rate = population density
Probability of cannibalism = function of difference of head sizes
Cannibalized / Predated individuals are killed;
population density monotonously decreases (actually observed)
Cannibal receives 50% of volume of the victim as energy intake;successful cannibal grows very suddenly (actually observed)
Individual who succeeded in cannibalism by chance has larger probabilityof the next successful cannibalism (actually observed)

25Individual based model which directly deals death events ofindividuals is also analyzed. The model can also represent
stochastic effect.

Computer simulation 2

0100200300400500600

7008009001000

Strategy#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15

Computer simulation 2

Generation

10
11001000

1000

26

0100200300

400500600

700

8009001000

Strategy#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15

Computer simulation 2Generation

1011001000

1000

27

Total Fitness

0100200300

Computer simulation 2500600Generation

400

700

800

9001000

11Dominating Strategy

28

29

Under cyclic evolution, the dominating strategy would becompletely different between populations whose environmental
factors are the same (but evolutionary phases differ).

This depends on environmental parameters.

convergence to the single strategy
orthe cyclic evolution

Evolution of growth strategy might be either

Conclusion