ARTICLE IN PRESS

Journal of Theoretical Biology 226 (2004) 69–77

Drastic growth effect may explain sympatriccannibalistic polymorphism
Joe Yuichiro Wakano*Department of Chemical System Engineering, University of Tokyo-Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan
Received 4 October 2002; received in revised form 1 July 2003; accepted 15 August 2003

Abstract
Cannibalistic polyphenism is observed in many ﬁshes and amphibians. In the case of amphibian larvae, cannibal morph andtypical morph coexist. Beneﬁts and costs of the cannibal morph have been studied empirically but the mechanism of themaintenance of polymorphism is not well known. Here, we construct a game model of typical and cannibal morph strategies toobtain the condition of stable coexistence. Generally, once an individual succeeds in cannibalism, it grows very quickly, whichfacilitates the next cannibalism. In a model without this ‘drastic growth effect’, stable coexistence cannot occur. To represent drastic
growth effect, it is assumed that cannibal/typical morph stage is followed by giant/normal stage. A cannibal morph that performs
cannibalism in the ﬁrst stage can become a ‘giant’ in the next stage. This model allows stable coexistence of cannibal and typicalmorphs. The condition for coexistence is that payoff of a giant is two times larger than normal individuals. As long as direct
consumption of victim’s body is considered as reward for successful cannibalism, coexistence cannot be explained. When the reward
is considered as social standing of being outstanding size in a population, sympatric cannibalistic polymorphism is possible, without
regard to the initial size variation or resource shortage.
r 2003 Elsevier Ltd. All rights reserved.
Keywords: Cannibalism; Cannibalistic polyphenism; Cannibalistic polymorphism; Drastic growth effect

1. Introduction
Intra-speciﬁc predation, or cannibalism, is a commonly observed ecological interaction in many animal
taxa (for review, Polis, 1981). Cannibalistic polyphenism
is known in many ﬁshes and amphibians (Polis, 1981;
Crump, 1992). Cannibalistic polyphenism consists of
two cases. First, all members of the population arepotentially cannibal, although small differences insuccess of cannibalism ultimately result in an explicit
polymorphism, for example, in body size. Second,cannibalistic and typical individuals coexist as alternative strategies or tactics with different morphology.
There are few experimental studies which clearly
distinguish the two cases. If only size variation isobserved, it might be the result of cannibalism, i.e. theformer case. On the other hand, cannibal morph, which*Corresponding author. Present address: Department of Biological
Sciences, University of Tokyo Hongo 7-3-1, Bunkyo-ku, Tokyo 1138656, Japan. Tel.: +81-3-5841-4492; fax: +81-3-3818-7547.
E-mail address: joe@biol.s.u-tokyo.ac.jp (J.Y. Wakano).
0022-5193/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2003.08.005

is characterized by a greater head width (and thus larger
mouth) relative to body length, is known in Ambystoma
salamanders (Rose and Armentrout, 1976), Hynobius
salamanders (Wakahara, 1995) and Scaphiopus toads(Pfennig, 1990). As cannibalism is gape-limited, cannibal morph has an advantage in performing and avoiding
cannibalism. The cannibal morph appears in Hynobius
salamanders in very early developmental stage when
they do not perform cannibalism (Nishihara-Takahashi,1999). In addition, cannibal morph usually has specialized oral structures (Crump, 1992). These facts suggestthat cannibalistic polymorphism in amphibian is of thesecond kind. Such kind of cannibalistic polymorphism
may exist in other species in which individual with thedisproportionately large organ related to cannibalism isobserved.
It is known that cannibal morphology is a plastic
phenotypic response to environmental conditions(Nishihara, 1996; Hoffman and Pfennig, 1999; Nishihara-Takahashi, 1999; Kohmatsu et al., 2001). Thesestudies might imply that tactics (cannibal or typical) are
different phenotypes of a single genotype. On the otherARTICLE IN PRESS70

J.Y. Wakano / Journal of Theoretical Biology 226 (2004) 69–77

hand, there is a study demonstrating differences inallozyme frequency between cannibal and non-cannibal
morphs (Pierce et al., 1981). The study implies that eachmorph has a corresponding genotype (or genotypes).
These empirical studies suggest two opposite hypotheses
and the reality probably lies in between.
In the present study, we study the case in which eachtactic results from each strategy and the condition forthe stable coexistence of two pure strategies is mathematically analysed. Evolution of cannibalistic polymorphism is discussed later. The model might beapplied to the different situation where a single mixed
strategy dominates the population. This point will also
be discussed later.
In this paper, we study the simple case with only two
strategies, typical and cannibal morphs. For example,
Ambystoma salamanders meet the condition (Pfennig,1989). Generalized ‘‘cannibal morph’’ is deﬁned as a
strategy which specializes in cannibalism and enjoys
larger probability of cannibalism and smaller probability of being cannibalized but suffers larger natural
mortality.
Even when population density is very high, many
individuals retain the typical morph features, resulting
in a polymorphic population (Pfennig and Collins, 1993;
Wakahara, 1995). To understand the phenomenon, theprevious studies focused on the cost of cannibalism(Pfennig et al., 1998; Pfennig, 2000; Kohmatsu et al.,2001). Eating relatives (Pfennig and Collins, 1993;
Wakahara, 1995) and pathogen infection (Pfennig
et al., 1991, 1998) are considered as such cost. Thesestudies have answered the question why all individuals
do not always develop cannibal morph. As cost andbeneﬁt of cannibal morph depend on the environment,
cannibal morph is not always adaptive. The nextquestion is how two morphs can coexist under the sameenvironmental conditions. In order to establish stablecoexistence, frequency dependence of the payoff mustsatisfy certain criteria such as in a Hawk–Dove game
(Maynard-Smith, 1982). Some studies (Pfennig, 1992;
Maret and Collins, 1997) explained cannibalistic polymorphism based on the experiments, suggesting that theresources for each morph are limited and independent,
i.e. cannibal morphs are escapee from intense competition among typical morphs. However, cannibalism hasvery strong and complex impact on population and thus
cannibalistic polymorphism might be explained without
regard to resource shortage. The present study explores
such possibility.
Most previous theoretical studies on cannibalism havefocused upon sized-population dynamics (Van den
Bosch et al., 1988; Henson, 1997; Cushing, 1992; Fagan
and Odell, 1996; Claessen et al., 2000; Wakano et al.,2002). In these studies, the occurrence of cannibalism is
not caused by the difference of strategies but size
dependent. There are few theoretical studies where

alternative strategies regarding cannibalism are deﬁned
and the condition for the polymorphism is obtained (butsee, Dercole and Rinaldi, 2002).
Here, we propose a simple game model to explain themechanism of the maintenance of sympatric cannibalistic polymorphism. First, the simplest game model oftypical and cannibal morph strategies is analysed. The
model appears unable to account for coexistence. The
reason why coexistence is not reproduced can be
explained in a general way (independent of the functionform). ‘‘Drastic growth effect of cannibalism’’ isintroduced as a possible mechanism of coexistence.
The extended model is proposed and the mathematicalanalysis shows that it can reproduce stable coexistence.
Further analysis and numerical calculation conﬁrms
that the result remains qualitatively the same whenassumptions of analysis are relaxed.

2. Models and results
2.1. The simplest model
The strategies in the game are cannibal morph andtypical morph, which are ﬁxed from birth. For
simplicity, all individuals start growing at the same timewith the same size. The initial population is deﬁned as
one without loss of generality. Let p and 1  p be theinitial frequency of cannibal morph and typical morph,
respectively. For simplicity, we assume that there is only
one opportunity for cannibalism, which occurs at theend of the growth period and that at every encountercannibal morph preys on typical morph. The frequency
of encounters between cannibal and typical morphs is
G ¼ min ½gpð1  pÞ; 1  p(
gpð1  pÞ ðgpo1Þ;¼
1pðgpX1Þ;ð1Þ

where g represents population density. Minimum function is required to represent the situation in which all
typical morphs are cannibalized. G typical morphs outof 1  p become victims of cannibalism. Energy gain of
typical morph consists of only regular food consumption, denoted by A: As described above, in order to
exclude the effect of resource shortage, regular foodconsumption is assumed to be constant. Energy gain ofcannibal morph is
kA

G
þ B:
p

ð2Þ

The ﬁrst term is energy gain from cannibalism. kA
corresponds to energy transfer per victim, which is
proportional to the energy gain of typical morph.
0oko1 is a conversion factor. Gp represents the averagefrequency of victims per cannibal morph. The second

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71

term B comes from regular food consumption. Payoff ofeach strategy is deﬁned as the product of the energy gain
during growth and the survival rate:
FT ¼ A
(
¼

1pG
1p
A  Agp0

ðgpo1Þ;ðgpX1Þ;

ð3aÞ




G
FC ¼ kA þ B m
p
8mkAg
þ mB  mkAgp ðgpo1Þ;<¼
1p: mB þ mkA
ðgpX1Þ:p

ð3bÞ

The model is summarized in Table 1. Survival rate oftypical morph in the absence of cannibalism is assumed
to be one without loss of generality. Survival rate ofcannibal morph is denoted by mo1; representing cost ofcannibal morph strategy.
Hereafter, we consider the payoff as the ﬁtness of eachstrategy. For simplicity, we assume no generation
overlap. Then, the frequency of cannibal morph strategy
in the next generation, p0 ; is written as
p0 ¼

pFC
pFC þ ð1  pÞFTð4Þand the evolutionary dynamics of two strategies can beanalysed. Both FT and FC are monotone decreasing
linear functions of the frequency p when gpo1: If theinclination of FC is steeper, FT  FC is decreasing
function of p; which means stable coexistence. However,as 0omo1 and 0oko1; Ag > mkAg always holds andthus the relationship will be as illustrated in Fig. 1. For
arbitrary parameter values A; B; g; m; k; there are only
three possibilities: (1) typical morph is always successfulðmoðA  AgÞ=BÞ; (2) cannibal morph is always successful ðm > A=ðkAg þ BÞÞ and (3) bistable (otherwise). In
other words, if rare cannibal morphs are more successful
than resident typical morphs, cannibal morphs arealways successful regardless of frequency. Thus, thestable coexistence of two pure strategies is not expected.
This model cannot explain cannibalistic polymorphism
as a stable equilibrium of the game.

Table 1
Energy income and survival rate of each strategy when gpo1
Energy income
Cannibal morph

Typical morph

kA

G
þB
p

A

Payoff is measured by a product of them.

Survival rate
m

1pG
1p

Fig. 1. Payoff of cannibal morph (solid line) and typical morph(dashed line) drawn against cannibal morph frequency. A bistable caseis shown. The position or existence of the intersection may vary, but
the latter is always steeper.

2.2. Where does the difficulty come from?
The difﬁculty comes from the difference in sensitivity
to frequency, i.e. the coefﬁcient of p in the payoff
function. Regardless of the details of the model, both
payoffs are decreasing function of p . For FT ; increase incannibal morph means increase in frequency of being
cannibalized, resulting in decrease in survival rate oftypical morph. For FC ; the frequency of potential
victims per cannibal morph is proportional to 1  p;i.e. decreasing function of p . The other factors, such as
regular food consumption, energy transfer rate, additional mortality of cannibal morph, etc., are independent of frequency of cannibal morph.
In this simplest model, the sensitivity to frequency isalways larger for the typical morph. The reason is asfollows. Frequency-dependent term in the payoff function is determined by the frequency of cannibalistic
interactions. At each cannibalistic interaction, a typicalmorph is killed; however, not all energy lost by typicalmorph population is transferred to cannibal morph
population. This is due to conversion factor ko1 andadditional mortality 1  m > 0: Therefore, payoff ofcannibal morph is always less sensitive to cannibalistic
interaction. In conclusion, as long as the payoff of eachstrategy is measured by total energy income, stablecoexistence cannot occur. Although the model is very
simple, it grasps the essence. Generally, as long as only
energy balance is considered, at each cannibalism, theamount of energy lost by typical morph population isalways larger than that transferred to cannibal morph
population. It is because energy conversion factor cannever exceed one, regardless of formulation. Therefore,
in order to explain the coexistence, the biologicallyreasonable mechanism should be studied in which thesensitivity of cannibal morph strategy becomes larger.

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J.Y. Wakano / Journal of Theoretical Biology 226 (2004) 69–77

2.3. Drastic growth effect of cannibalism
Most cannibalistic amphibian populations consist of
many normal-sized individuals and a few very largesized individuals, or giants (Pfennig and Collins, 1993;
Wakahara, 1995). Giants are also observed in ﬁsh(LeCren, 1992; Claessen et al., 2000). Giants are thosethat succeeded in cannibalism in the early stage of thegrowth period. Supposing that all individuals have
similar size at the beginning of growth and that acannibal makes use of most of the victim’s body, a giant
will be almost two times as large as normal. Head widthdifference between giants and normals is much greaterthan difference between typical morphs and cannibal
morphs of a similar size (usually less than 20%). Oncean individual becomes a giant, the next cannibalism
becomes much easier. Subsequent cannibalism most
likely occurs between a giant and a normal. As a result,once a giant appears in population, advantage of acannibal morph strategy becomes negligible comparedwith overwhelming size of a giant. Thus, the aim oftaking cannibal morph strategy is thought to be
increasing the probability of becoming a giant by
developing a slightly larger head width. In this paper,
the advantage gained by becoming a giant is called the‘drastic growth effect’.
Does considering drastic growth effect change theform of frequency dependence? If a cannibal morph
individual succeeds in cannibalism to become a giant, it
receives great reward, while it pays only cost if it cannot
become a giant. When the frequency of cannibal morph
is larger, the probability of becoming giant is presumably smaller. Therefore, considering drastic growtheffect, the sensitivity of cannibal morph’s payoff to thefrequency might be increased. In the next section, thepossibility of stable coexistence is studied in a modelwith drastic growth effect.
2.4. Extended model with drastic growth effect ofcannibalism
Here, we consider an extended model of population
with giants (Fig. 2). Growth period is split into twophases: no giant phase and giant normal phase. In theﬁrst phase, the population consists of typical morphs
and cannibal morphs both of which have similar bodysize. The existence of giants is neglected. Individuals
who performed cannibalism become giants in the second
phase. Individuals who did not perform cannibalism but
survive the ﬁrst phase become normals in the second
phase. As cannibalism is monopolized by giants, thetype of morph of a normal individual in the ﬁrst phase
does not change the ﬁnal payoff. The initial population
size is taken as one without loss of generality. As aframework, we consider the general case in which bothmorphs potentially perform cannibalism in the ﬁrst

Fig. 2. Schematic illustration of the model with drastic growth effect.

phase. It is natural to assume that the likelihood ofcannibalism is determined by head width difference. So,
it is assumed that the probability of cannibalism among
the same morphs is the same. Let c as probability ofcannibalism among the same morph and c0 as probability of cannibal morph eating typical morph. Notethat the initial difference in head width precludes thepossibility of typical morph eating cannibal morph. Let
CXY be the frequency of cannibalism where a morph X
cannibalizes a morph Y : In order to calculate CXY ; anencounter rate must be deﬁned, which is assumed to be
included in c and c0 : The frequencies of cannibalism aredenoted by
CCC ¼ cp2 ;ð5aÞ

CCT ¼ 2c0 pð1  pÞ;

ð5bÞ

CTC ¼ 0;

ð5cÞ

CTT ¼ cð1  pÞ2 ;ð5dÞ

respectively. Here, we assume that cannibalism in theﬁrst phase does not occur very often and can neglect thepossibility of extinction of typical morph. In the ﬁrst
phase, no individual cannibalizes more than twice. Thus,CXY represents the frequency of cannibals and victims.
The frequency of cannibal morphs which succeed incannibalism, i.e. become giants is
GC ¼ CCC þ CCT :ð6Þ

The frequency of typical morphs which become giants is
GT ¼ CTT :ð7Þ

The frequency of cannibal morphs which are cannibalized is
DC ¼ CCC :ð8Þ

The frequency of typical morphs which are cannibalizedisDT ¼ CCT þ CTT :ð9Þ

Payoff of a normal individual is taken as one without
loss of generality. Payoff of a giant is g > 1: As in the

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73

previous model, natural survival rates of typical andcannibal morphs in the ﬁrst phase are assumed to be 1
and m; respectively. Finally, payoff functions arecalculated as follows:1
FC ¼ ½GC g þ ðp  GC  DC Þmp¼ m½ðcg  2c0 g  2c þ 2c0 Þp þ 2c0 ðg  1Þ þ 1;1½GT g þ ð1  p  GT  DT Þ
1p¼ ðcg þ 2c  2c0 Þp þ cg þ 1  2c:

ð10aÞ

FT ¼

ð10bÞ

2.5. Analysis of a simple caseFirst, we deal with the special case of c ¼ 0; i.e.
cannibalism in the ﬁrst phase occurs only between
cannibal morph and typical morph. We also assume
ﬁtness of a giant g is constant. Then, payoff functionsare
FC ¼ 2mc0 ðg  1Þp þ mf2c0 ðg  1Þ þ 1g;

ð11aÞ

FT ¼ 2c0 p þ 1;

ð11bÞ

which are linear functions of p with the coefﬁcients now
depending on g: Difference in payoffs is
FC  FT ¼  2c0 fmðg  1Þ  1gp
þ mf2c0 ðg  1Þ þ 1g  1:
Thus, equilibrium p is
mf2c0 ðg  1Þ þ 1g  1:
2c0 fmðg  1Þ  1g
For any 0op o1; p is stable if and only if
p ¼

m>

1
:
g1

ð12Þ

ð13Þ

ð14Þ

When this stability condition holds, p > 0 holds if andonly if
1
m> 0
:
ð15Þ
2c ðg  1Þ þ 1
Similarly, when the stability condition holds, p o1
holds if and only if
mo1  2c0 :ð16Þ

These three conditions (Eqs. (14)–(18)) and corresponding p value are illustrated in Fig. 3. All three curvesintersect at ðg; mÞ ¼ ðð2  2c0 Þ=ð1  2c0 Þ; 1  2c0 Þ: Thep ¼ 0 curve is monotonically decreasing for g > 1:
Therefore, the necessary condition for a stable coexistence solution to exist isg>

2  2c0:
1  2c0

ð17Þ

Fig. 3. Analytic result of the case c ¼ 0 and c0 ¼ 0:1: (a) Conditions
for stability (bold line), p ¼ 1 (solid line) and p ¼ 0 (dashed line) are
shown in ðg; mÞ space. Parameter region where a stable equilibrium
0op o1 is realized is shown by shaded area. (b) Corresponding p isshown in three-dimensional graph. Only 0op o1 surface is drawn.
Five contour lines drawn in the bottom of the graph correspond to
p ¼ 0; 0:25; 0:5; 0:75 and 1.0, respectively. Notched line corresponds toa stability condition where p -7N: Beyond the stability condition,p becomes unstable equilibrium, which means bistability.

2.6. Relaxing the assumptions
In this section, we relax the two assumptions.
(A) constant ﬁtness for giants and (B) c ¼ 0: In
order to keep the model tractable, we deal withthe special case of c; c0 51 and apply approximation. In other words, only few cannibalisms occur inthe ﬁrst phase. Let gðqÞ and nðqÞ be payoffs of a giant
and a normal, whereq¼¼

mGC þ GTmðp  DC Þ þ ð1  p  DT Þ
mcp2 þ 2mc0 pð1  pÞ þ cð1  pÞ2
ð18Þ
mðp  cp2 Þ þ f1  p  2c0 pð1  pÞ  cð1  pÞ2 g

is frequency of a giant in the second phase. The
denominator represents the initial population of thesecond phase while the numerator represents thefrequency of giants. Due to c; c0 51 assumption, q51
holds. Applying the arguments leading to Eq. (3), theARTICLE IN PRESS
J.Y. Wakano / Journal of Theoretical Biology 226 (2004) 69–77

74##
actual forms of gðqÞand nðqÞ
becomes# ¼ kAg þ B  kAgq;
gðqÞ

ð19aÞ

# ¼ A  Agq;
nðqÞ

ð19bÞ

where we have used gqo1: Additional mortality of agiant is assumed to be zero. The relative ﬁtness of a giant
to a normal iskAg þ B  kAgq
g¼
A  AgqBB
E kg þ  kgq þ kg þ
gq:
ð20Þ
A
A
Notice that gðqÞ can be written as a linear function of theinﬁnitesimal variable q: Putting these results into
Eq. (4.6) we get
FC ¼ m½ðcg  2c0 g  2c þ 2c0 Þp þ 2c0 ðg  1Þ þ 1;ð21aÞ

FT ¼ ðcg þ 2c  2c0 Þp þ cg þ 1  2c:

ð21bÞ

In these equations, g always appears in a product with cor c0 ; which are small by assumption. Therefore, to theﬁrst order we can neglect the dependence of g on q:
Putting
B
ð22Þ
gEkg þ
A
into Eq. (21) yields
FC  FT ¼ ½mfcg  2c0 g  2c þ 2c0 g  fcg þ 2c  2c0 gp
þ mf2c0 ðg  1Þ þ 1g  ð1 þ cg  2cÞ:Thus, equilibrium p is
p ¼

mf2c0 ðg  1Þ þ 1g  ð1 þ cg  2cÞ
:mfcg  2c0 g  2c þ 2c0 g þ fcg þ 2c  2c0 g

and thus, when following conditions
ð23Þ

For any 0op o1; p is stable if and only ifm>ð2a  1Þ þ ðg  1Þ;
ð2a  1Þðg  1Þ þ 1

ð24Þ

where a c0 =c > 1: When this stability condition holds,p > 0 holds if and only if
1 þ ðg  2Þc
E1 þ ðg þ 2a  2  2agÞc:ð25Þ
m>2acðg  1Þ þ 1
Similarly when the stability condition holds, p o1
holds if and only if
1  2c0mo
E1  ðg þ 2a  2Þc
1 þ cðg  2Þ

¼ 2ða  1Þðg  2Þ

2ða  1Þðg  2Þ > 0

and1 þ cðg  2Þ > 0

hold, there is region in the ðc; mÞ parameter space where
0op o1: The area of this region increases with a or g:
If g > 2; these conditions are always met as a > 1:
Moreover, as the right hand of Eq. (24) is a decreasing
function of g; there always exists 0omo1 which meetsthe stability condition if g > 2: Therefore, the condition
for a stable coexistence isg > 2:

ð27Þ

Note that Eq. (17) converges to Eq. (27) when the ﬁrstorder approximation for c0 is applied.

ð26Þ

when 1 þ cðg  2Þ > 0: When 1 þ cðg  2Þo0; the sign ofthe inequality is reversed. These three conditions(Eqs. (24)–(26)) are illustrated in Fig. 4. p ¼ 0 line
and p ¼ 1 line intersect at ðc; mÞ ¼ ð0; 1Þ: The differencein the inclination of the two lines in ðc; mÞ space is ðg þ 2a  2Þ  ðg þ 2a  2  2agÞFig. 4. Analytic result of the case c; c0 51: (a) Conditions for stability(bold line), p ¼ 1 (solid line) and p ¼ 0 (dashed line) are shown in
ðc; mÞ space. Parameter region where a stable equilibrium 0op o1 isrealized is shown by shaded area. g ¼ 6; a ¼ 2: (b) Analytic result of
the case c ¼ 0:01 and c0 ¼ 0:02: Equilibrium p ; contour lines and astability condition are shown as in Fig. 3b.

2.7. Numerical calculation
Without applying any approximation, Eqs. (18)–(21)are directly numerically calculated by computer. Whenc; c0 51 condition is relaxed, payoff function becomes acomplicated nonlinear function of p: A stable equilibrium is searched numerically (Fig. 5). The search wasdone for an appropriate region in ðg; mÞ space and thestable coexistence region is achieved (Fig. 6). Validity of

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analysis in the previous sections is conﬁrmed. All
coexistence solutions are found at g > 2: Moreover, thegeneral tendency remains the same that coexistence
more likely occurs when c0 =c or g is larger.

3. Discussion
The stable coexistence of cannibal and typical morphs
can be explained by introducing drastic growth effect of

Fig. 5. Difference in payoff of cannibal and typical morphs, FC  FT ;is shown against frequency of cannibal morph, p: In this case (c ¼ 0:1;
c0 ¼ 0:2; k ¼ 0:5; g ¼ 4; A ¼ B ¼ 1; m ¼ 0:45), both unstable (opencircle) and stable (ﬁlled circle) equilibriums exist. The p value of stableequilibrium is numerically obtained.

75

cannibalism. The intuitive reason for coexistence is asfollows. To grasp the mechanism easily, assume that the
typical morph does not perform cannibalism, g (payoffof a giant) is a constant and the frequency ofcannibalism is very small in the ﬁrst stage. In addition,
assume that payoff is linear function of p (frequency
of cannibal morph). Survival rate and hence payoffof typical morph decreases as p increases with the
coefﬁcient X ; which corresponds to loss of life. Payoff ofa cannibal morph is also decreasing function of p; whose
coefﬁcient is Y : Y represents beneﬁt received by acannibal after cannibalism. The important point is that
Y is not the amount of energy (or meat or nutrition,etc.) but the social standing of being a giant. As long as
Y is considered as energy, the conversion rate cannot
exceed one and thus X > Y always holds. Coexistence
cannot occur when X > Y as we have argued above.
Cannibalism in the early stage gives very large social
proﬁt to a cannibal, which is exactly what Y means.
Then, Y can be larger than X : So, in a sense, the drastic
growth effect model is a k > 1 version of the simplest
model. Drastic growth effect explains how k can exceed
one. This is the essential mechanism of coexistence of
cannibal and typical morph strategies in our model.
At polymorphic equilibrium, cost and beneﬁt ofcannibal morph should be balanced, which has been
discussed a lot. For example, population density(encounter rate g) or the ability to cannibalize (c0 or a)increases equilibrium frequency of cannibal morph. The
present study focuses upon stability of polymorphism

Fig. 6. Numerically derived stable coexistence solution is shown as in Fig. 3b. Energy transfer coefﬁcient is ﬁxed at k ¼ 0:5: Energy gain from regularfood consumption A ¼ B ¼ 1: g is varied to change payoff of a giant. The shown interval gA½0; 10 roughly corresponds to gA½1; 6: (a) c ¼ 0:01;
c0 ¼ 0:02 (corresponding to Fig. 4a), (b) c ¼ 0:01; c0 ¼ 0:1 and (c) c ¼ 0:05; c0 ¼ 0:1:

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J.Y. Wakano / Journal of Theoretical Biology 226 (2004) 69–77

and we conclude that cannibalistic polymorphism
occurs when ﬁtness of a giant is larger than two. Ifenergy conversion rate is 0.5, performing two cannibalisms doubles a cannibal’s size. In real salamander larvae
population, there are only a few cannibals. In Ambystoma tigrinum, no more than one cannibal appeared per
aquarium containing 16 larvae (Pfennig and Collins,
1993). In Hynobius retardatus, there were 102 typicalmorphs and 17 cannibal morphs (Wakahara, 1995).
From such fact, one can guess that a few (around 10%)cannibals monopolize cannibalism and one can easily
imagine that ﬁtness of these despots is much larger thanthe others.
Cost of cannibal morph in our study is assumed to be
additional mortality only. It is important to distinguishthe cost of performing cannibalism from the cost of
having cannibal morph. Cannibal morph is less likely to
be attacked by conspeciﬁcs and thus the strategy inwhich an individual develops as a cannibal morph but
does not perform cannibalism, is adaptive when the costof performing cannibalism is greater than the beneﬁts.
Therefore, the induction of typical morph cannot be
explained without regard to the cost of having cannibal
morph (Wakano et al., 2002). Eating relatives orpathogen infection are the cost of performing cannibalism and should not be considered as the cost of cannibal
morph in our model. It is reported that the imbalanced
body shape of cannibal morph causes a change inbehavior or decline in swimming speed, which results inincreased risk of predation by natural enemies (Kohmatsu et al., in preparation). It is also known thatcannibal morph is slow to metamorphose under drying
condition (Rose and Armentrout, 1976; Kohmatsu et al.,in preparation). On the other hand, cannibal morph fed
with only regular food grows as quick as typical morph(Kohmatsu et al., 2001). Thus, in our model, additionalmortality is considered as the cost of cannibal morph.
In the model, growth period is assumed to consist of
two phases. In the ﬁrst phase, cannibalisms mostly occur
between cannibal morphs and typical morphs and theexistence of giants are neglected. Therefore, we assumed
that cannibalism in the ﬁrst phase does not occur very
often. In the second phase, most cannibalisms are
performed by giants and cannibalism among normal
individuals are neglected. The last assumption can bejustiﬁed by the following argument. For normal
individuals, potential victims of cannibalism are as large
as cannibals, which means long handling time and thus
the vulnerability to predators. Under the existence of agiant, such vulnerability is too costly and cannibalisms
among normals might be suppressed. It is also assumed
that growth period consists of two discrete phases andthat there is only one opportunity of cannibalism at theend of growth period. These are introduced just formathematical simplicity and do not seem very realistic
because cannibalisms in reality are performed by many

individuals in continuous time space. However, dealing
continuous time directly makes the model very complicated, so we decided to keep the model as simple aspossible to achieve the result analytically. As a candidateof the full model, we could assume a ‘normal’population and a ‘giant’ population and deﬁne theprobability of the transition in continuous time space.
At each cannibalism, a victim is removed from a‘normal’ population and a cannibal is moved from
‘normal’ into a ‘giant’ population. However, theprobability of cannibalism is dependent on the frequencies of cannibal morph, typical morph and a giant.
Thus, this process is not Markov chain, which makes
analysis of the system very difﬁcult. Computer simulation might be able to be performed, which is left for
future works.
Conditions for coexistence of two morphs is independent of regular food abundance. The result is achieved
under the assumption of constant regular food consumption rate. Relaxing this assumption may cause theescape effect from intense competition for regular foodand may increase the possibility of the coexistence(Pfennig, 1992; Maret and Collins, 1997).
We studied the game model of two pure strategies andthe condition for the stable coexistence is shown. In
order to explain the observed cannibalistic polymorphism by evolutionary scenario, two points should be
discussed. The ﬁrst point is the relationship between the
payoff and ﬁtness. It is natural to assume that the bodysize at the end of larval period has positive correlation
with the adult survival rate and the number of offspring.
Cannibalism on adult stages is generally rare inamphibians (Crump, 1992), suggesting that it is theadult size rather than the type of morph in the larval
period that affects ﬁtness directly. To the author’s
knowledge, there is no evidence of the dependence ofreproductive success on the morph itself in ﬁsh. The
second point is the relationship between tactics andstrategies. In evolutionary biology, strategy must havegenetic basis. Phenotypic trait without genetic basis iscalled tactic in general. However, as is discussed in the
introduction, genetic basis of the morph determinationis unclear. The study showing difference in allozyme
frequency does not necessarily indicate an allelicdimorphism on a locus (or loci) concerning the morph
development. For example, different migration rate
among morphs or assortative mating can also result in
the difference in allozyme frequency. On the other hand,the observed phenotypic plasticity of morph determination does not necessarily deny the existence of genetic
basis of morph determination. Further experimental
study would be necessary for the point.
Under the assumptions of the model, the present
study implies that the coexistence of two strategies canbe established as an evolutionarily stable state. As
genetic background of cannibalism is unclear, weARTICLE IN PRESS
J.Y. Wakano / Journal of Theoretical Biology 226 (2004) 69–77

assumed two pure strategies. However, evolution of
single mixed strategy can also be analysed with few
alterations to the present model and the result will notchange. Moreover, all analytic result not only showsthat coexistence is evolutionarily stable but also showsthat the convergence to the stable state is guaranteed
because the payoff is a linear function of frequency.
Considering drastic growth effect, sympatric cannibalistic polymorphism can be explained as an evolutionary
consequence.

Acknowledgements
I thank Y. Kohmatsu for carefully reading of theearly draft. I also thank K. Aoki for many valuablesuggestions and improving English. I am also grateful toanonymous referees for valuable comments.

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