Evolutionary Ecology Research, 2002, 4: 719–736

Evolutionary dynamics of frequency-dependent growthstrategy in cannibalistic amphibians
J.Y. Wakano,* Y. Kohmatsu and N. Yamamura
Centre for Ecological Research, Kyoto University, Kamitanakami Hiranocho, Otsu,
Shiga 520-2113, Japan

ABSTRACT
Cannibalistic amphibian larval morphs, which have greater head widths than typical morphs
and therefore have larger mouths, provide the basis for the construction of a simple modelof cannibalism that describes the growth dynamics of body shape. As the relative head sizes ofinteracting individuals determine the occurrence of cannibalism, the situation is frequencydependent. Because natural enemies prey more often upon individuals with a more unbalanced
body shape, an optimal growth schedule should exist. The necessary conditions for an evolutionarily stable strategy (ESS) are derived analytically; these conditions are checked by evolutionary simulation. When the probability of cannibalism is low, an ESS exists. In such cases,
the body shape is more balanced (less adaptive to cannibalism) when predation pressure ishigher and population density is lower. When the probability of cannibalism is high, there is no
ESS. For such cases, a computer simulation of the evolutionary dynamics revealed that thedominant growth strategy changes cyclically. The development of a more detailed model ofindividual-based population dynamics showed that the qualitative results of the simple model
held for the individual-based model. Accompanied by cyclic evolution, the number of surviving
individuals at metamorphosis oscillated. The frequency-dependent models suggested that theevolutionary dynamics of cannibalism change dramatically depending upon environmental
conditions.
Keywords: cannibalism, cannibalistic morph, evolutionary cycling, evolutionary dynamics,
frequency dependence, polymorphism.

INTRODUCTION
Animals adapt not only to abiotic environments, but also to social interactions. Intraspecificpredation, or cannibalism, is one of the most important factors among a variety of social
interactions, being commonly observed in many animal taxa (for a review, see Polis, 1981).
It may be a strong factor in natural selection, because being cannibalized means instantdeath of the individual.
* Address all correspondence to Joe Yuichiro Wakano, Department of Chemical System Engineering, The
University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan. e-mail: joe@chemsys.t.u.-tokyo.ac.jp
Consult the copyright statement on the inside front cover for non-commercial copying policies.
© 2002 J.Y. Wakano

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In amphibians, cannibalism is common during the period from hatching until metamorphosis. An unbalanced growth strategy (cannibalistic morph), which is characterized bya greater head width (and thus larger mouth) relative to body length, has been recorded in
Ambystoma salamanders (Rose and Armentrout, 1976), Hynobius salamanders (Wakahara,
1995) and Scaphiopus toads (Pfennig, 1990). As an individual can eat conspecifics with asmaller head width (Kusano et al., 1985; Ohdachi, 1994), the cannibalistic morph is considered to be an adaptive growth strategy (Fig. 1). However, if all individuals becomecannibalistic morphs, they cannot cannibalize other individuals. Thus, the situation isgame-theoretical or frequency-dependent.
Several experiments have shown that the magnitude of imbalance in body shape ofthe cannibalistic morph was greater when the population density was higher (Nishihara,
1996; Hoffman and Pfennig, 1999). Even when the density was very high, some individuals
retained the typical morph features, resulting in a polymorphic population (Pfennig and
Collins, 1993; Wakahara, 1995). Generally, the frequency of cannibalism increases asfood resources decrease (Polis, 1981), as has been clearly demonstrated in amphibian larvae(Crump, 1992; Wakahara, 1995). However, the induction of the cannibalistic morph is
not related to food availability (Collins and Cheek, 1983; Lannoo and Bachmann, 1984).
The benefit of an unbalanced growth strategy is clear, while individuals do not develop
cannibalistic morphs without stimulus from other individuals (Hoffman and Pfennig,1999). Factors such as pathogen infections (Pfennig et al., 1991, 1998; Kiesecker andBlaustein, 1999) and the eating of relatives (Pfennig and Collins, 1993) are considered to becosts of executing cannibalism. However, cannibalistic morph individuals are less likely
to be cannibalized by conspecifics and the strategy in which an individual develops
as a cannibalistic morph but does not perform cannibalism is adaptive when the cost of
executing cannibalism is greater than the benefits that are accrued. Therefore, the inductionof typical morphs cannot be explained without regard to the cost of having a cannibalistic
morph. As an example of such cost, a recent study has shown that an imbalance of body
shape (greater head width) causes a change in behaviour or a decline in swimming speed,which results in increased risk of predation by natural enemies such as dragonfly
larvae (Y. Kohmatsu et al., in prep.). The existence of benefits and costs implies that the
optimal growth strategy depends upon environmental factors such as population density or
predation pressure.
Previous theoretical studies on cannibalism have focused mainly upon population

Fig. 1. Schematic representation of amphibian larvae. Larger headed individual (cannibalistic
morph) cannibalizes smaller headed individual (typical morph). Cannibalism occurs even when theirbody sizes are almost the same.

Optimal growth strategy in cannibalism

721

dynamics and the dynamics of size distribution (e.g. Cushing, 1992; Fagan and Odell, 1996;
Henson, 1997). Recently, Claessen et al. (2000) ran an integrated simulation of population
dynamics, to which size-dependent resource competition was introduced together with sizedependent cannibalism, resulting in the dynamics showing complex behaviour. However,
the evolutionary dynamics of cannibalistic characters remains unexplored.
In this study, we develop theoretical models to analyse the evolution of cannibalistic
morphs. Two distinct phenotypes, typical morph and cannibalistic morph, have been notedin the study of morphological characteristics in some amphibian species (Pfennig, 1989; fora review, see Crump, 1992). However, relative head width does not always show a bimodal
distribution (Nishihara, 1996), which suggests that intermediate morphs exist between the
typical and cannibalistic morphs. For this reason, a continuous set of strategies for body
shape is adopted in the model. The model has an explicit representation of body shape
growth dynamics because growth strategy (allocation of incoming energy to head widthand body length at each stage of growth) plays an important role in cannibalism among
amphibian larvae. We assume the growth strategy to be genetically determined andunchanging during growth (see the Discussion for empirical evidence supporting thisassumption). Head width dependent cannibalism and body shape (ratio of body length andhead width) dependent mortality from natural enemies are introduced to population
dynamics.
The evolutionary dynamics of the growth strategy are analysed to determine whether
polymorphism could be attained and whether the appearance of the cannibalistic morph
depends on food availability. First, a dynamic model of the growth process including body
shape is developed and the optimal growth strategy is sought. Candidates for an evolutionarily stable strategy (ESS) are derived analytically and whether they are actual ESSs is tested
by computer simulation. The evolutionary dynamics in the absence of an ESS is also examined. Thereafter, an individual-based model is developed, a computer simulation demonstrating the evolutionary dynamics of the growth strategies, together with the population
dynamics.
The distinct feature of this study is that two-dimensional traits are considered. When only
a one-dimensional trait (size, in most studies) is considered, there is only one social relationship, larger or smaller. In the case of two-dimensional traits, two independent relationships exist, which is exactly the case with amphibian larvae – that is, wider or narrowerhead width (trait regarding cannibalism) and balanced or unbalanced body shape (trait
regarding predation). Growth of a trait benefits growth of the other trait by increasingenergy income. Thus, maximization of fitness is the problem of optimal growth trajectory in
two-dimensional trait space.
MODEL AND RESULTS
Optimal growth modelTwo types of predation, cannibalism and predation by natural enemies, are considered.
Although individuals are killed if cannibalized or caught by a natural enemy in reality, an
integrated model of individual and population growth is too complicated to be dealt withanalytically and in the first step we assume that a decrease in mean growth rate is substituted
for death. Later, an individual-based model, including the death process, is developed (seeTable 1 for primary assumptions of models in this paper).

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Table 1. Primary assumptions of three models

Strategy set
Individual death

Stochastic effect on growth
Energy transfer from victim tocannibal

Analytic
model

Evolutionary
simulation

Individual-basedmodel

Continuous
Replaced by
reduction of
growthNone
Indirectu1, u2, . . . , u15
Replaced by
reduction of
growthNone
Indirectu1, u2, . . . , u15
Directly included

Included
Direct (50%)

For simplicity of analysis, the body shape of amphibian larvae is taken as a column
defined as {(x, y, z) ∈ R3 : x2 + y2 ≤ r2, 0 ≤ z ≤ l }. The cross-sectional area s is regarded ashead size and the length l as body length. Body shape is considered to be balanced whens = kl 2, where k is a constant, because the shape remains unchanged with growth under thisrelationship. The rate of growth of the volume sl is determined by energy intake per unit
time E (we omit conversion factors for the sake of clarity):d(sl) = Edt(1)where E is composed of three terms:
E = asl + C(s, p(ŝ)) − B

s

冢l 冣
2

(2)

The first term is energy intake resulting from consumption of regular food, being proportional to the volume. The coefficient a is a constant, representing food availability. The
second term, C(s, p(ŝ)), is energy intake or loss following cannibalism. Each individual caneat other individuals with smaller heads but may be eaten by individuals with larger heads.
Thus C(s, p(ŝ)) is a function of head size distribution in the population, p(ŝ), as well as
2
its own head size, s. The third term, B(s/l ), is the cost of imbalance, representing risk of
predation from natural enemies. At balanced growth, there is no cost – that is, B(k) = 0.
Because cost increases as the magnitude of imbalance increases, we set B(s/l 2) = (b (s/l2) − k)2,
where b represents intensity of predation pressure. This term could be interpreted as cost of
imbalance in terms of lowered foraging efficiency.
The growth of volume sl is written asddsdl(sl) = l + s
dt
dt
dt

(3)

where the first and second terms indicate increase in head size and body length, respectively.
The distribution function of the total energy to increase in head size at a given time is a
growth strategy, u(t):
l

ds
= uE
dt

(4)

Optimal growth strategy in cannibalism

723

and
s

dl
dt

= (1 − u)E(5)

We assume that both adult survival rate and adult fecundity are proportional to body size
sl and that both constants of proportionality are one. Fitness F is defined as their product atmetamorphosis time T:
F = (s(T)l(T))2(6)

Because a balanced body shape is required for metamorphosis (Y. Kohmatsu, in prep.), theterminal condition is s(T) = kl(T)2. Initial values, s(0) and l(0), are given such that they arealso balanced: s(0) = kl(0)2.
Necessary conditions for an evolutionarily stable strategy
The strategy u*(t) is defined as an ESS when any rare mutant sub-population has lower
fitness in a monomorphic u*(t) population. Thus, if the growth function u*(t) is an ESS,
the fitness of a mutant strategy with growth function u(t) must be at a maximum at
u(t) = u*(t). An optimal growth schedule u0(t) against a wild-type schedule u*(t) can bederived using Pontryagin’s theory (see Appendix 1). The necessary condition for an ESS
is u0(t) = u*(t). In other words, we can analytically derive a candidate of ESSs. When
we consider only two strategies, the cannibalistic interaction term of a mutant, C(s, p(s)),
can be written as C(s − s*), a function of difference in head sizes between wild-type andmutant, which is used in ESS analysis. If an ESS exists, the growth trajectory of the ESS
must satisfys = kl +
2

c 4
l
6b

(7)

as long as 0 < u(t) < 1, where c = C⬘(0), which represents the intensity of gain resulting fromcannibalism when an individual interacts with other individuals of similar size. Note thatthe result (7) is independent of the actual form of function C(s − s*). As s = kl 2 represents abalanced shape, the second term represents the extra investment in a greater head size. Since
the initial and terminal conditions are s = kl 2, evolutionarily stable growth should be as
shown in Fig. 2: only s increases first, then both s and l increase according to (7), and finally
only l increases (see Appendix 1 for proof). The second term says that the organism can put
energy into growth of both head size and length only if its shape is already sufficiently
biased towards head size. The magnitude of the bias is determined only by intensity ofcannibalism, c, and intensity of predation pressure, b. As expected in (7), high c leads
to unbalanced growth and high b leads to balanced growth. This is independent of foodavailability, a, or the balanced body shape, k.
Existence of ESS and results of evolutionary simulation
Generally, an ESS or the best single strategy does not always exist. In the previous section,
we showed that if a monomorphic population cannot be invaded by a rare mutant, then it
must have the form of Fig. 2. But we have not shown the converse, that a Fig. 2 strategy can

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Fig. 2. An example of analytically derived optimal growth (solid line) and balanced growth (dottedline).

be a monomorphic ESS. Namely, we have not derived sufficient conditions for the existenceof an ESS. In this section, evolutionary simulation is performed to explore the question.
In the simulation, each growth strategy is assumed to be genetically fixed. Fifteen
strategies {u1(t), u2(t), . . . , u15(t)} are chosen from candidates of ESSs, corresponding todifferent values of C⬘(0) (see Fig. 3), and introduced into the simulation. Thus, energy
intake and growth of an individual with genotype i are given as:2s Ei = asi li + C − b 2i − k
li


E
 ṡi
 = ui
s
l
i si
 i
 l̇
 i = (1 − ui) E
 li
li si

冢

冣

(8)

where the interaction term, C, is given as
15

C=δ冱p θ(s − s )
j

i

j

(9)

j=1

with


2
θ(x) ≡ η 
− 1


−2γx

 1 + exp
η



冢

冣

(− η < θ(x) < η, θ(0) = 0, θ⬘(0) = γ)(10)Optimal growth strategy in cannibalism

725

Fig. 3. Candidates for evolutionarily stable strategies. The growth trajectory of an individual withstrategy ui (t) when the population is monomorphic is shown. Note that actual realization of the
trajectory is varied due to interactions among strategies. u1(t) corresponds to balanced growth andu15(t) to the most extreme case of unbalanced growth. ui (t) is an analytically derived candidate ofan evolutionarily stable strategy when Γ = 0.0, 0.6, 1.3, 2.0, 2.8, 3.5, 4.2, 5.0, 5.6, 6.2, 6.8, 7.3, 7.7, 8.0,
8.3, respectively.

where pi is the frequency of genotype i. For positive x, θ(x) represents the benefit per
encounter in which the opponent’s head size is smaller by x. For negative x, θ(x) is negative,
which represents cost per encounter in which the opponent’s head size is larger by | x |. The
parameter δ represents the encounter rate, η is the maximum benefit/cost per interactionand γ represents benefit/cost per encounter when the head size difference is very small. Hereγδ corresponds to C⬘(0), as shown in Appendix 1, and thus γδ is an important parameter.
By defining Ω ≡ δη and Γ ≡ γδ, the equation for C is rewritten as


2



Γ
C(si ) = Ωpj  1 + exp − 2 (si − sj ) 
j=1
Ω


15

冱

冦

冧

(11)

The initial genotype distribution is taken to be uniform. The dynamics of genotype
frequency is determined by the following rule (roulette selection). Genotype i is chosen withprobability
pi fi

冱p f

j j

j

where fi is fitness of genotype i:fi = (si (T)li (T))

2

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Wakano et al.

Selection is continued until the population of the next generation reaches N, so that the
population size is always N. Smaller N represents greater genetic drift. For the computer
simulation, N is set at 1000 and random mutations are introduced. At the beginning ofeach generation, each individual’s genotype mutates into a genotype randomly chosen from{u1(t), u2(t), . . . , u15(t)} at probability ε per individual per generation; ε is set to 10 − 5 in thesimulation.
The results are summarized in Table 2, each strategy of average frequency greater than
10% being indicated. As a result of the simulation, when one strategy dominates, it is ananalytically derived ESS candidate (equation 7) in most cases (Table 2). One example of theevolutionary dynamics for this case is shown in Fig. 4a. However, the simulation revealed
that an analytically derived ESS candidate does not always dominate. In such cases, thestrategy frequency distribution averaged over generations exhibited polymorphism. In
particular, in most cases, the dynamics of strategy frequency is not stable, changing
cyclically (Fig. 4b). This strategy distribution is unimodal at each instant in time. Suchcyclic evolution is observed when Γ is large and Ω is small (upper right in Table 2). In suchcases, the magnitude of imbalance is determined mainly by Ω. As Ω becomes larger, selectedstrategies become more unbalanced.
Why is cyclic evolution found in such a parameter space? A balanced growth strategy
benefits from a low risk of predation but suffers from cannibalization. Conversely, an
unbalanced growth strategy benefits from cannibalization but suffers from a high risk of
predation. An ESS candidate (equation 7) is at equilibrium where an increase in benefit
from cannibalization matches an increase in cost from predation. However, the mostbalanced strategy (strategy 1) might invade the ESS candidate if the benefit from a low riskof predation overcomes the cost of being cannibalized by the latter. The magnitude of
imbalance of an ESS candidate is determined by Γ. When Γ is larger, an ESS candidate ismore unbalanced and suffers from a higher risk of predation. The maximum of benefit/costfrom cannibalization is represented by Ω. When Ω is smaller, strategy 1 suffers less fromcannibalization. Therefore, when Γ is large and Ω is small, an ESS candidate is not theactual ESS because it is invaded by strategy 1. When strategy 1 invades and becomesdominant, strategy 2 has greater fitness as it can cannibalize strategy 1 without muchdifference in intensity of predation. Following the invasion of strategy 2, which becomesdominant, strategy 3 does likewise, and so on. The displacement of the dominating strategy
continues until strategy 1 invades again.
INDIVIDUAL-BASED MODEL
In the previous model, death is replaced by a decrease in mean growth rate, which is an
artificial assumption. An individual-based model that deals directly with mortality due tocannibalism and predation is developed. When cannibalism occurs, a victim is killed andits fitness becomes zero. Consequently, the cannibal grows very quickly. The more the latter
succeeds in cannibalism, the easier the next act of cannibalism becomes. It has been
observed that individuals that have cannibalized are much larger among larvae that arecollected from the same population and hatched at the same time (Wakahara, 1995;
Kohmatsu et al., 2001). Such a phenomenon cannot be represented without developing
an individual-based model. The evolutionary simulation of an individual-based model isperformed (for details, see Appendix 2).
Both a monomorphic solution and a cyclic solution are found (Fig. 5). The cyclic solution

1
0
1+
1+
1+
1+
1+

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

3
1.3
3+
3+
3+
3+
3+

4
2
4+
4+
4+
4+
4+

5
2.8
4, 5
5+
5+
5+
5+

6
7
8
3.5
4.2
5
4, 5 3, 4, 5 3, 4, 5
6+
6, 7
6, 7
6+
7+
8
6+
7+
8+
6+
7+
8+

9
5.6
3, 4, 5
5, 6, 7
8
9+
9+

10
11
12
13
14
15
6.2
6.8
7.3
7.7
8
8.3
10
3, 4, 5 3, 4, 5
3, 4, 53, 4, 53, 4, 53, 4, 53, 4, 5
5, 6, 7 4, 5, 6, 7 4, 5, 6, 7 4, 5, 6, 7 4, 5, 6, 7
5, 6, 7
5, 6, 78, 9 6, 7, 8, 9 6, 7, 8, 9 6, 7, 8, 9 6, 7, 8, 9
6, 7, 8, 9
6, 7, 8, 9
10 +
10, 11
10, 11
10, 11 9, 10, 11, 12 9, 10, 11, 12 8, 9, 10, 11, 12, 13
10 +
11 +
12, 13
13
13, 1515
15 +Corresponding strategy
Note that Ω = δη and Γ = δγ, where δ represents population density or encounter rate, η is maximum benefit/cost per interaction and γ represents benefit/cost per
encounter when head size difference is very small. Each ESS candidate is determined only by the value of Γ. Corresponding strategy types (1, 2, 3, . . . , 15) are shown
above each Γ value. Average frequencies of strategies between 1000th and 10,000th generation are calculated for each parameter set (Γ, Ω). All strategies of averagefrequency greater than 10% are indicated; i + indicates average frequency of strategy i greater than 90%.

0.2
0.4
0.6
0.8
1.0



Table 2. Results of evolutionary simulation under some parameter sets (Γ, Ω)728

Wakano et al.

Fig. 4. Examples of the results of evolutionary simulation. The higher strategy frequency is represented by the darker grey. Although the original data contain time-series of frequencies of only 15
strategies, continual visualization over the strategy axis is applied to make it easier to grasp the
dynamics. Γ = 6.8, which corresponds to strategy 11 (u11). (a) When Ω = 1.0, strategy 11 dominates. (b) When Ω = 0.6, the dominating strategy gradually changes from strategy 4 to strategy 9, thensuddenly reverts to strategy 4. The time period for the cycle is constant.

is qualitatively similar to the one in the previous model, but the period of the evolutionary
cycle is not constant. The other types of dynamics (such as chaos, etc.) are not found.
The result implies that the behaviour of an individual-based model is essentially the same asthat of the previous model. The total fitness of the population at the cessation of larval
growth (time T), determined by the number of surviving individuals and average body size
at metamorphosis, are also surveyed (Fig. 6). The total fitness varies depending on theOptimal growth strategy in cannibalism

729

Fig. 5. Examples of the results of individual-based simulations, with γ = 1000 and ρ = 1. (a) δ = 5000,
strategy 9 dominant. (b) δ = 200, irregular cycle of dominating strategy. The amplitude and time
period for the cycle are not constant.

position, or phase, of the evolutionary cycle. In the phase where more balanced growthstrategies dominate, the total fitness is high.

DISCUSSIONAs represented by equation (7), analytically derived optimal growth is more unbalanced
when the intensity of cannibalism (c) is high or the intensity of predation pressure (b) is low,

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Wakano et al.

Fig. 6. Dynamics of total fitness at metamorphosis (thin line) and dominating strategy (thick line),with δ = 200, γ = 1000, ρ = 1. Total fitness is greater when the number of the dominating strategy issmaller (more balanced).

while it is independent of food availability. When food availability is low, the relative
importance of cannibalism is greater. The frequency of cannibalism actually increases insuch circumstances (Crump, 1992; Wakahara, 1995). However, some empirical studies haveshown that the appearance of cannibalistic morphs is independent of food availability(Collins and Cheek, 1983; Lannoo and Bachmann, 1984), which is consistent with thepresent results that optimal growth is independent of food availability.
As a result of simulation, unbalanced strategies dominate when Γ or Ω is increased (seeTable 2). Note that Ω ≡ δη (product of population density and maximum benefit/cost perinteraction) and Γ ≡ δγ (product of population density and benefit/cost per interaction withvery small head size difference). The value of γ could be measured by the reciprocal of theminimum difference in head size for cannibalism to occur. Thus, the prediction can be madethat amphibian larvae acquire an unbalanced shape when maximum benefit/cost per interaction is large, population density is high or the difference in head size necessary forcannibalism is small. The relationship between induction of a cannibalistic morph
and population density has been reported in several studies (Pfennig and Collins, 1993;
Wakahara, 1995; Nishihara, 1996; Hoffman and Pfennig, 1999), while the effect of theother two factors should be studied in the future.
Convergence to an ESS candidate occurs with small Γ and large Ω, although we
performed no analysis for convergence stability (Christiansen, 1991). The result suggeststhat an ESS candidate has convergence stability as well as evolutionary stability in corresponding parameter regions. Moreover, in the case of cyclic evolution, we cannot exclude
the possibility that an ESS candidate is actually an ESS but not convergently stable. Due
to limitation of numerical calculation, we cannot rule out the possibility of a stable
polymorphism with just the right mix of strategies.
Cyclic evolution occurs with large Γ and small Ω. Whether evolutionary dynamics is
cyclic or convergent is determined for the most part by Γ/Ω, as seen from Table 2. Since Γ/ΩOptimal growth strategy in cannibalism731

is equivalent to γ/η, the reciprocal of the ratio of head size difference necessary for cannibalism to maximum benefit/cost per interaction plays a very important role in the evolutionary
time-scale dynamics of the system.
This cyclic evolution could be considered as one realization of Van Valen’s (1973) Red
Queen hypothesis. As we dealt with the actual biological phenomenon, the model is rather
complicated and mathematically sophisticated analysis of interesting characters such as aperiod of cycle is almost impossible. However, using a more abstract model to derive
analytical results (e.g. Dieckmann et al., 1995) is another powerful tool. Evolution of traits
related to cannibalism has received little attention and thus we expect various interesting
behaviours of this system to be discovered in the future.
Differences in the frequencies of cannibalistic morphs among populations have been
widely observed (Pfennig, 1990; Nishihara, 1996), the possible main cause being differencesin environmental factors such as population density or predator density. In the present
model, when population density is higher, the optimal growth becomes more unbalanced,
which has been observed previously (Nishihara, 1996; Hoffman and Pfennig, 1999). The
model predicts that a decrease in predator density also unbalances the optimal growth. Thisremains to be confirmed by field and experimental studies. Cannibalistic polymorphism
among populations might also be explained by phase differences during cyclic evolution.
Cyclic evolution predicts that completely different strategies may be observed between
populations under similar environmental conditions. The time period for an evolutionary
cycle is dozens of generations, even though a high mutation rate is set in computer simulations. Consequently, direct verification of this hypothesis for amphibian larvae is extremely
difficult. If a large difference in the frequency of cannibalistic morphs among populations
under similar environmental conditions is observed, it may be indirect evidence in supportof the hypothesis.
The cost of having a cannibalistic morph is also introduced to the model. The cost of
executing cannibalism is not the same as the cost of having a cannibalistic morph. For
example, when all individuals in a population have cannibalistic morphs, cannibalism does
not occur. In such a case, the cost of having a cannibalistic morph is applicable, whereas
the cost of executing cannibalism is not. However, when polymorphism occurs, thecannibalistic morph has a greater probability of eating conspecifics. Thus, the cost of
executing cannibalism might affect the former more strongly. An apparent cost of executing
cannibalism is vulnerability to predators during handling time of eating conspecifics. Whena cannibal has a victim that has a similar body size to itself, it might be easily caught by anatural enemy. Eating relatives is another cost of executing cannibalism, although somekin-recognition ability has been reported in amphibian larvae (Pfennig and Collins, 1993).
An increase in mortality due to pathogen infection has also been suggested as a cost of
executing cannibalism (Pfennig et al., 1991, 1998; Sadler and Elgar, 1994; Kiesecker andBlaustein, 1999). The introduction of such costs into the present model appears to be anecessary next step.
Some studies have shown that growth patterns are environmentally controlled, which is
known as phenotypic plasticity (Collins and Cheek, 1983; Pfennig and Collins, 1993),
although another study has demonstrated differences in allozyme frequency between
cannibalistic and non-cannibalistic morphs (Pierce et al., 1981). The factors responsible forthe induction of a cannibalistic morph are not clear (Crump, 1992). The assumption thatgrowth strategies are genetically fixed in evolutionary simulation does not necessarily hold,since plasticity is known in amphibian larvae. Although an analysis based on plasticity was

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Wakano et al.

not undertaken here, a likely result might be as follows. If an ESS exists, all individuals withplasticity will choose it. If an ESS does not exist (cyclic evolution), their strategy choice will
be polymorphic, for the following reason. Assume all individuals take the same strategy X.
As strategy X is not an ESS, there exists strategy Y, which increases fitness. Accordingly,
some individuals will choose strategy Y. This leads to polymorphism. The present study
suggests that polymorphism can be expected, even if the growth strategy choice is affectedby plasticity.
ACKNOWLEDGEMENTS
We thank S. Yachi for valuable comments and suggestions. We also thank P. Taylor for his carefulreading and many helpful comments. We used GCC on Linux for computer simulations and thank allof the developers and programmers who support Linux, the GNU project and other useful freeware.
This work was supported in part by a grant-in-aid for scientific research from the Japan Ministry of
Education, Science, Sports and Culture (Creative Basic Research Program: DIVER).

REFERENCES
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APPENDIX 1
If a growth function u*(t) is an ESS, the fitness of a mutant strategy with growth function u(t) mustbe at a maximum at u(t) = u*(t), when the wild-type grows as (s*, l*) with growth function u*(t).
As a mutant is very rare, it interacts with only wild-type individuals. The interaction term for amutant C(s) is a function of s and s*, and we assume C(s) is a function of s − s*. Defining new
variables G ≡ E/ls, x ≡ ln s and y ≡ ln l, equations (2), (4) and (5) in the text are rewritten as
G=a+

1
sl

s

冦C(s − s*) − B 冢l 冣冧
2(A1)ẋ = uG(A2)ẏ = (l − u)G(A3)

and the terminal condition s = kl 2 as
x = ln k · 2y

(A4)Maximization of (sl )2 is equivalent to maximization of x + y. Hamiltonian and adjoint variables inthis system are written as
H = λxuG + λy (1 − u)G(A5)

冦−−λλ ==∂∂HH==l ∂s H∂ H

(A6)

x

y

We rewrite adjoint equations as

x

y

s

l

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G
冦−−λλ =={λ{λuu++λλ(l(l− −u)}lu)}s∂
∂G
x

x

y

y

x

s

y

(A7)

l

or

 λ̇ = −{λ u + λ (l − u)}s − 1 (C − B) + 1 C⬘(s − s*) − 1 B⬘ s
x
y
 x
ls2
ls
l2
l2


 λ̇ = − {λ u + λ (l − u)}l − 1 (C − B) + 1 2 sB⬘ s
x
y
 y
sl 2
ls l 3
l2


冤

冦

冤

冦

冢 冣冧冥

冢 冣冧冥

(A8)

An optimal control variable u is chosen so that Hamiltonian H is maximized. If optimal u is neither
0 nor 1 for a given time period, λx = λy must hold and thus λ̇x = λ̇y holds:

1s
C ⬘(s − s*) − 2B⬘ 2
l
l
l

1

冦

冢 冣冧

2
s
= 3B⬘ 2
l
l

冢冣

(A9)

or

s

冢l 冣

l 2C⬘(s − s*) − 3B⬘

2

(A10)

The above equation represents the optimal growth trajectory for a mutant when a wild type grows as(s*, l*). If u* is an evolutionarily stable strategy, u = u*, s = s* and l = l* must hold. Therefore,

s
l2

冢冣

l 2C⬘(0) = 3B⬘(A11)

Note that this is a necessary condition. Also note that this equation depends on only C⬘(0), the actual
form of C(x) being independent of this condition. Defining B(x) = b(x − k)2 and c = C⬘(0) results in the
trajectory of optimal growth beings = kl 2 +

c
6b

l

4

(A12)

Applying (A12) to equations (4) and (5) in the text, u(t) is given as a function of l(t)2k + 4βl
2k + 1 + 5βl 2

(A13)

c
6b

(A14)

2

u=
where

β≡

The interaction term C(s) is zero when all individuals in the population take the same strategy.
Therefore,Optimal growth strategy in cannibalism
2
1 s
E= al − b 2 − k
s
s l

冢

冣

735
(A15)

Inserting s = kl 2 + βl 4 into (A15) and equation (5) in the text, we finally obtain (after some calculation)2bβ l
1 + βl 3dl=a−ldt 2k + 1 + 5βl 2
k + βl 2

冢

冣

(A16)

It is difficult to solve this equation and explicit solutions of s(t), l(t) and u(t) cannot be derived.
Accordingly, in computer simulations, numerically calculated values of u(t) are used.
A growth trajectory of an ESS is derived where 0 < u(t) < 1. The optimal trajectory consists of thecurve derived above, line segments where u(t) = 0 and line segments where u(t) = 1. We here prove that
optimal growth is as shown in Fig. 2. The case c ≡ C ⬘(0) > 0 is considered owing to greater head sizes
benefiting more from cannibalism. Two variables are defined:≡ λx − λy
φ≡

d
dt
s

冢l − k冣 − l c

= 6b

2

2

Note that φ is a function only of s and l. Optimal control u is chosen so that u(t) = 0 where (t) < 0
and u(t) = 1 where (t) > 0. The body shape space (l, s) is divided according to the sign of φ; thepositive φ space, zero φ curve and negative φ space (Fig. A1). Both the starting and finishing points
are in the negative φ space. As λx and λy are continuous functions, = 0 must hold where the value of
u changes. If u changes from 0 to 1, changes from negative to positive and then d /dt = φ > 0.
Therefore, u does not change from 0 to 1 in negative φ space. By the same consideration, u does not

Fig. A1. Body shape space is divided into three parts; positive φ space, zero φ curve and negative φspace. Optimal growth is proven as in Fig. 2b using this figure and Appendix 1.

736

Wakano et al.

change from 1 to 0 in positive φ space. The trajectory, which meets the above conditions, is restricted
to that shown in Fig. 2, where only s increases at first (u = 0), then both s and l increase (0 < u < 1) and,
finally, only l increases (u = 0).

APPENDIX 2
Details of the individual-based model are as follows. At the beginning of growth and population
dynamics, N individuals are alive. Each individual has its own strategy u, head size s, body length l andalive/dead flag f. Energy intake of each individual consists of the regular food consumption term only,
which is proportional to the volume. The effect of cannibalism and predation by a natural enemy isdealt with in the following way. Two randomly chosen individuals encounter one another at a constantprobability per unit time δ. When one of the chosen individuals is dead, the model proceeds to thenext encounter – that is, the actual encounter rate is dependent upon population density (the numberof alive individuals). At each encounter, the probability that the larger headed individual eats thesmaller headed individual isc(x) =

2
1 + exp(− 2γx)−1(0 ≤ x, 0 ≤ c(x) < 1)
x = | s1 − s2 |With large γ, only a slight difference in head size is sufficient for cannibalism to occur. Whencannibalism occurs, the victim is marked as ‘dead’ and 50% of the volume of the victim is immediately
added to the cannibal’s energy intake. The probability that predation by a natural enemy occurs perunit time per each individual is
p=

s

2

冢 l − 1冣 · ρ
2where ρ is a constant representing predator density. This probability corresponds to B in the previous
model. The individual that becomes the prey is also marked as ‘dead’; the population, therefore,
decreases monotonously with time. At the end of growth and population dynamics, the fitness of eachindividual is calculated (volume squared). Averaging fitness of individuals adopting the same strategy
produces the fitness of each strategy.
The strategy set consists of 15 strategies (u1, u2, . . . , u15). Initial genotype distribution is taken
as uniform. The dynamics of genotype frequency is determined by the roulette selection rule withpopulation size N = 1000 in the same way as the previous simulation. The random mutation rate ε isset at 10−5. See text for details of the roulette selection and the mutation.