Drastic growth effect explains
sympatric cannibalistic
polymorphism
Journal of Theoretical Biology 226:69-77 (2004)

Joe Yuichiro Wakano
Department of Biological Sciences
The University of Tokyo2006 March 101

Agenda{ Basic introduction to evolutionary gametheory{ Cannibalistic polymorphism

cost/benefit of cannibal morphz food web structurez{ Model 1z

without drastic growth effect{ Model 2z

with drastic growth effect{ Conclusion2

Basic introduction toEvolutionary 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 otheranimals.
If there are only two types of behavior (A & B), fitnessof each behavior depends on the frequency of A
(frequency-dependent selection).
Such a situation is studied by evolutionary gametheory.
3

Basic introduction toEvolutionary Game Theory{

Terminology
player = animal / individualz strategy = behavior (genetically determined)z payoff = fitnessz equilibriumz stabilityz{

Classic game theory
rational players with full knowledge of payoff matrix
z rational decision makingz{

Evolutionary game theory
blind players
z natural selectionz4

Basic introduction to
Evolutionary Game Theory
p1-p

HawkDoveHawk-1

2-p+2(1-p)

Dove

0

11-p

F ( Hawk ) = −3 p + 2
F ( Dove) = − p + 1
If p=1/2, both strategies are equally successful.
p=1/2 is equilibrium.
5Basic introduction to
Evolutionary Game Theory{ Stable Equilibrium

F ( Hawk ) = −3 p + 2
F ( Dove) = − p + 1p (freq. of Hawk)
01/2

16

Basic introduction toEvolutionary Game Theory{ Unstable Equilibrium

F (B )F ( A)p (freq. of A)
01/2

17

Cannibalistic polymorphism
Hynobius retardatus salamander larvae

cannibalmorph

typicalmorph

Photo by Y. Kohmatsu

8Cannibalistic polymorphism{ Sympatric coexistence of cannibalmorphs and typical morphsz Ambystoma salamanders
z Hynobius salamanders
z Scaphiopus toads

9Benefit of cannibal morph{ Advantage in cannibalism

gape limited

cannibal morph

typical morph

10

Cost of cannibal morph{ not very clearz Costs such as eating relatives,

pathogen infection or handling time are
the cost of cannibalism.

{ Larger mortality(due to decreased swimming speed)

11

Food webnatural enemy
predation

predation

cannibalism

cannibal morph

typical morph
regular food(small insects, planktons)

12Assumptions{ Two distinct strategies:

cannibal and typical morphs{ The strategy is geneticallydetermined{ experimental studies shows both genetic

basis and phenotypic plasticity{ Only cannibal morphs can performcannibalism

13Goal of the study{ Explain the cannibalistic

polymorphism as the stablecoexistence solution in an
evolutionary game model{ Determine the condition underwhich such a polymorphic solution
exists14

Model 1:
parameters
Freq. of cannibal morphpFreq. of typical morph

1− pIntensity of cannibalismγ

Freq. of cannibalismΓ = γp (1 − p )(Population size is normalized to unity.)15

Model 1:
Body size at the end of growth period
Typical morph
Cannibal morph

growth byregular foodconsumption

AΓkAp+B

size of victim
probability with which each cannibal
morph can eat a typical morphk : Energy conversion factor (<1)(the efficiency of digestion)
16Model 1:
Fitness(fitness) = (body size) X (survival rate)

Cannibal morph
Typical morph
Relative survival rateof cannibal morph⎛ Γ⎞
FC = ⎜⎜ kA + B ⎟⎟ ⋅ m⎝ p⎠
1− p − Γ
FT = A ⋅1− pm (< 1)(cost of cannibal morph is 1-m)
17

Model 1:
The result
FC = (− mkAγ ) p + (mB − mkAγ )FT = (− Aγ ) p + ApSince mk<1, even if the intersection exists,
it is always unstable equilibrium.

Coexistence will not evolve.
18

Drastic growth effect{ Larvae initially have similar body size.
{ A cannibalism promotes the cannibal tooutstanding size (a giant){ Once giants appear, they monopolize
most cannibalisms(Difference between typical/cannibal morphs becomes negligible.)

normals

typicalgiants

cannibal
19

Drastic growth effectFitness of a giant is naturally expected to bevery high.

The aim of taking cannibal morph
strategy might be to increase theprobability of becoming a giant by
developing a slightly larger organ(cf. head width).

20

Model 2:
Assumptions{ consider only normal individuals and level1 giants{ Fitness of a giant is constantA level 1 giant will proceed to level 2 by performing
successful cannibalism. For simplicity, however, we neglectsuch a chain. The most important point is how manycannibal morphs become giants.
In real salamanders, almost all occurrences of cannibalism
are monopolized by a few giants. However,rare occurrences among normal larvae, which produces a giant,are important.
21

Model 2
Phase I

Phase IIfitness

giant

gnormal

1

dead

0

cannibal
morph

p1-p
typicalmorph

22Model 2
m1-m

cannibalmorph

p

giant
normal1-ptypicalmorph

deadΓ
23

Model 2:
Analysis1
FC = [Γg + ( p − Γ)]⋅ mp
1− p − Γ
FT =1− p
FC = −2mγ ( g − 1) p + m{2γ ( g − 1) + 1}
FT = −2γp + 1
p* =m{2γ ( g − 1) + 1}− 12γ {m( g − 1) − 1}24Model 2:
Analysis
If

1m>
g −1

(stability condition)

1m>
2γ ( g − 1) + 1
m < 1− 2γ(existence conditionof an equilibrium)then, there is a globally stable interior equilibrium, that is,stable coexistence solution25Model 2:
Necessary condition for stability1m>g −1(stability condition)1
g > 1+mg>226

Evolutionarily stable frequency ofcannibal morphs

p1

10.75
0.5
0.25

0.9

unstable

00.8
10.7

stable

20.6

34g

m

0.5
560.4

27

Summary of the analysis{ Evolutionary outcome is one ofi) domination of typical morphii) domination of cannibal morphiii) stable coexistence{ Coexistence evolves when g is large

and m is intermediate.

{ For coexistence, g>2 is necessary28

Is g >2 realistic ?
Experimental data

Ambystoma tigrinum salamander’s case:No more than one cannibal (=giant) appeared per aquarium
containing 16 larvae (Pfennig & Collins 1993).

Fitness of a giant is much larger than twotimes fitness of non-giants.
This promotes the stable coexistence oftypical/cannibal morphs or polymorphism.
29

Conclusion{ Evolutionary game models of cannibalistic

polymorphism are proposed{ The condition of stable coexistence oftypical/cannibal morphs is studied{ Considering drastic growth effect, large

fitness of a giant (>2) promotes evolutionof cannibalistic polymorphism.

30Thank you for your attention.

31