ARTICLE IN PRESS

Theoretical Population Biology 66 (2004) 249–258
www.elsevier.com/locate/ytpbi

Evolution of social learning: a mathematical analysis
Joe Yuichiro Wakanoa,, Kenichi Aokia, Marcus W. FeldmanbaDepartment of Biological Sciences, The University of Tokyo, Hongo 7-3-1, Bunkyoku, Tokyo 113-0033, Japanb
Department of Biological Sciences, Stanford University, Stanford, CA 94305, USA
Received 1 April 2004
Available online 11 September 2004

Abstract
Social learning is an important ability seen in a wide range of animals including humans. It has been argued that individual
learning, social learning, and innate determination of behavior are favored by natural selection when environmental changes occur
at short, intermediate, and long intervals, respectively. Only recently, however, has the hypothesis been examined by means ofmathematical models. In this paper, we construct a simple model in which each organism uses one of three genetically determinedstrategies — it is an individual learner, a social learner or an ‘‘innate’’ — and the three types of organisms are in direct competition
with each other. A reduced model, involving only the individual learners and innates, is effectively linear, and we show that bysolving the eigenvalue problem of this reduced system we arrive at a good approximation to the global dynamics of the full model.
We also study the effect of stochastic environmental changes and reversible mutations among the three strategies. Our results are
consistent with the predictions of previous studies. In addition, we identify a critical level of environmental constancy below which
only individual and social learners are present.
r 2004 Elsevier Inc. All rights reserved.
Keywords: Social learning; Individual learning; Innate behavior; Environmental change

1. Introduction
The behavior of a biological organism is a complex
process involving the expression of relevant information
possessed by that organism. For example, as we writethese words, we are presumably accessing learned
knowledge stored in our brains that we believe applies
to the problem we hope to solve. At the risk ofoversimpliﬁcation (Gould and Marler, 1987), it ispossible to classify behaviors by the three distinct ways
in which this information can be obtained (CavalliSforza and Feldman, 1983a; Boyd and Richerson, 1985;
Laland et al., 2000; Henrich and McElreath, 2003;
Alvard, 2003). This trichotomous classiﬁcation labels
behavior as being either ‘‘innate,’’ ‘‘socially learned,’’ or‘‘individually learned’’.

Corresponding author. Fax: +81-33818-7547.

E-mail address: joe@biol.s.u-tokyo.ac.jp (J.Y. Wakano).
0040-5809/$ - see front matter r 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.tpb.2004.06.005

A behavior is innate when it entails the direct
expression of information encoded in the genes, which
are inherited from the parents via the germ cells. Social
learning entails the transfer of information betweensocially interacting individuals, as a result of which the
behavior exhibited by a ‘‘model’’ is adopted by an‘‘observer’’ (Galef, 1988; Whiten and Ham, 1992; Heyes,1993). This rubric covers teaching, imitation (goaldirected copying of an action pattern), local enhancement (attention drawn to a particular object by the
behavior of another, leading to independent discovery ofthat behavior), and various other psychological processes. Finally, individual learning refers to learning that
occurs independently of any social inﬂuences. Examples
are trial-and-error and insight.
Social learning is of special interest to anthropologists
because it is the process that supports cultural inheritance (Cavalli-Sforza and Feldman, 1981; Durham,
1991). As noted above, social learning is a generic term
describing many different processes, and it is exhibited

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J.Y. Wakano et al. / Theoretical Population Biology 66 (2004) 249–258

by a wide variety of animals. Although much attention
has been paid to social learning in humans, we consider
here the evolution of social learning in general.
The utility of social learning (and cultural inheritance)
was ﬁrst studied quantitatively by Cavalli-Sforza andFeldman (1983a,b) and Boyd and Richerson (1985) (seealso Feldman and Cavalli-Sforza, 1976). The key pointis that social learners are equally receptive to both
adaptive and maladaptive behaviors. The ﬁtness ofsocial learners depends on the individuals from whomthey acquire information, and this complicates theevolutionary dynamics of social learning (Cavalli-Sforza
and Feldman, 1983a,b; Aoki and Feldman, 1987, 1989;
Aoki, 1990; Takahasi and Aoki, 1995).
Rogers (1988) proposed a simple model of competition between individual and social learners in a
temporally variable environment. In this model, theenvironment may change between generations. There isan optimal behavior appropriate to each environmental
state. Individual learners achieve this optimal behavior
on their own, but suffer a ﬁtness cost due to errors madein learning. Social learners ‘‘copy’’ an organism of theparental generation chosen at random (oblique transmission, Cavalli-Sforza and Feldman, 1981), at a direct
cost that is smaller, but run the risk of copying an
inappropriate behavior. Since only the individual
learners are able to accurately track the changing
environment — the social learners are parasitic on theindividual learners — this risk increases as the frequency
of individual learners decreases (or equivalently thefrequency of social learners increases). Hence, we
intuitively expect that the ﬁtnesses of individual andsocial learners may be equal at some intermediate
frequency, and that this should deﬁne a polymorphic
equilibrium. Feldman et al. (1996) rigorously reformulate the Rogers (1988) model for both periodically andrandomly changing environments. Their analysis conﬁrms the prediction (Rogers 1988) that social learners
will be eliminated if the environment changes too often.
Social learners can be maintained in the population if
the environment is sufﬁciently constant, and at higher
frequency the greater is this stability.
Most previous studies have compared only two
strategies. Recently, we extended the model of Feldman
et al. (1996) to propose a new model in which all threestrategies (innates, social learners and individual learners) compete at the same time (Aoki et al., submitted).
Numerical analysis of this model conﬁrms the consensus
view that individual learning, social learning (from theparental generation), and innate determination of
behavior are favored by natural selection when theinterval, measured in generations, between environmental changes is short, intermediate and long, respectively.
Moreover, we showed that there is a critical frequency
of environmental change. When the environmentchanges at a frequency higher than this critical level,

the individual and social learners coexist while theinnates go extinct. When this frequency is lower than thecritical level, innates dominate while both individual andsocial learners are eliminated. These transitions seen in
the numerical analysis are very sharp.
In this paper, we investigate further the model of Aoki
et al. (submitted). First, we argue that the behavior ofthis model can, to a good level of approximation, be
described in terms of the eigenvalue problem of the
reduced evolutionary dynamics in which social learners
are neglected. This assumption permits us to obtain ananalytic expression for the critical frequency of environmental change under both periodically changing andstochastically changing environments. Second, we extend the model to incorporate random reversible
mutations among the three strategies and show thatthe qualitative results remain similar in the extended
model, but that there are also some interesting
differences.

2. Models and results
2.1. The basic model
Assume a population of haploid asexual organisms. A
tri-allelic ‘‘strategy’’ locus determines whether anorganism is an innate, a social learner, or an individual
learner (abbreviated as G, SL and IL, respectively). A
social learner copies a random member of the parental
generation. Its behavior will be adaptive or maladaptive
depending on whether or not the information obtainedis appropriate to the current environmental state. An
individual learner collects information directly from theenvironment and its behavior is always adaptive. The
behavior of an innate is determined by the allele it
carries at the ‘‘innate information’’ locus and is adaptive
only if this allele ‘‘matches’’ the present environmental
state. The innate information locus is not expressed inindividual and social learners. Note that there may be avariety of adaptive behaviors, which by assumption are
equally adaptive (i.e. have the same ﬁtness), andsimilarly for maladaptive behaviors.
We posit a model of mutation and variation at the
innate information locus involving two classes of alleles.
The ‘‘resident’’ alleles produce behavior that is adaptivein the current environment state, whereas the ‘‘mutant’’
alleles produce maladaptive behavior. Each of the twoclasses of alleles comprises many selectively neutral
variants. A small subset amounting to a fraction r ofeach class possesses the special property of being
adaptive when the environment changes. These variants
can be regarded as ‘‘preadapted’’ alleles awaiting
environmental change.
Our basic premise with regard to the environment isthat when it changes it never reverts to an earlier stateARTICLE IN PRESS
J.Y. Wakano et al. / Theoretical Population Biology 66 (2004) 249–258(inﬁnite environmental state model, Feldman et al.,1996). Rogers (1988) assumed that the environment ﬂips
back and forth between two states. Comparative
analysis of the two environmental state and the inﬁnite
environmental state models yielded qualitatively similar
results (Feldman et al., 1996). The real world likely
comprises a ﬁnite but large number of possible statesamong which transitions can occur. If the environment
includes other evolving species, it is a reasonableassumption that the probability of any particular state
recurring will be small. The inﬁnite environmental statemodel is an idealization of this situation, which we
introduce for the mathematical simpliﬁcation it
achieves.
In the inﬁnite state model, none of the preexisting
behaviors can be adaptive after an environmental
change. Innate behavior in our model is the result of
an interaction between the allele carried by an organism
and the environment. Hence, the behavior of an innate
carrying a preadapted allele may differ in the pre- andpost-change generations, and consistent with our basic
premise, we assume that the behavior in the pre-change
generation is maladaptive in the post-change generation.
In particular, this implies that a social learner cannot
acquire correct behavior by copying an innate with apreadapted allele when the environment changes.
By assumption, individual learners always behave
appropriately for the given environmental state, butsocial learners and innates can get it either right orwrong. Hence, our model distinguishes ﬁve phenogenotypes (genotype–phenotype combinations, Feldman
and Cavalli-Sforza, 1984), which we abbreviate: GC(genetic correct, i.e., innate with resident allele), GW(genetic wrong, i.e., innate with mutant allele), SLC(social learner correct), SLW (social learner wrong), andIL (individual learner). The ﬁve phenogenotypes, theirfrequencies among reproductive adults, and theirﬁtnesses are summarized in Table 1. The ﬁtnesses are
relative viabilities and are assigned to the ﬁve phenogenotypes in the following way. There is a baselineﬁtness of 1 for adaptive behavior, and maladaptive
behavior causes the ﬁtness to be reduced by s. Social
learners bear a direct cost of developing and maintaining a nervous system supportive of learning, which is

Table 1
The ﬁve phenogenotypes, their ﬁtnesses and frequencies among
reproductive adults
Phenogenotype

Fitness

Frequency

GC (genetic correct)
GW (genetic wrong)
SLC (social learner correct)
SLW (social learner wrong)
IL (individual learner)

1
1s
1d
1sd
1c

u
v
x
y
z

251

translated into a ﬁtness loss d. Individual learners suffera similar direct cost but in addition they are adversely
affected by mistakes made before they mature; the total
penalty for them is c.
In our basic deterministic model, we ignore mutation
at the strategy locus and assume that the environmentchanges with a ﬁxed period l. The life cycle events areasexual reproduction, learning, and natural selection, in
this order. Then, the recursions relating the frequenciesof the ﬁve phenogenotypes in the offspring generation,
indicated by primes, to those in the parental generation
can be written as follows.
First, when the environment remains constant between generations the recursions are:W 1 u0 ¼ u
W 1 v0 ¼ ð1  sÞv;W 1 x0 ¼ ð1  dÞðx þ yÞðu þ x þ zÞ;
W 1 y0 ¼ ð1  s  dÞðx þ yÞðv þ yÞ;
W 1 z0 ¼ ð1  cÞz:ð1ÞSecond, when the environment changes between generations they are:W 2 u0 ¼ rðu þ vÞ;
W 2 v0 ¼ ð1  sÞð1  rÞðu þ vÞ;
W 2 x0 ¼ 0;
W 2 y0 ¼ ð1  s  dÞðx þ yÞ;
W 2 z0 ¼ ð1  cÞz;ð2Þ

where we used the assumption that the constant fraction
r of all alleles at the information locus produces
adaptive behavior in the post-change generation. In
both recursions, W i is a normalization factor for thefrequencies. The environment changes every l generations, so one set of iterations entails l  1 applications of(1) followed by one application of (2).
2.2. The reduced model and analysis
Aoki et al. (submitted) studied the basic model
numerically when docos: They found that the frequencies of the ﬁve phenogenotypes converge to aperiodic solution with period l: Moreover, they showedthat the solution as a function of l shows a sharptransition at a critical value of l; which we denote l  : Inother words, the global behavior of the basic model isclassiﬁed into two modes. When l4l  ; they found thatG dominates; on the other hand when lpl  ; G die out
and SL and IL coexist. The transition between the twomodes is very sudden and discrete. These results wereobtained numerically. In this section we explore theanalytic framework and derivation of l  :The social learners are parasitic on the individual
learners because only individual learners are able toaccurately track the changing environment. Hence, if a

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252

population of innates is stable against invasion byindividual learners, it should be able to resist invasion bya mixture of individual and social learners. Therefore,
consider the competition between innates and individual
learners. Let transformation f be
0 01 0
10 1
u
1
0
0
u
B 0C BCB C
0 A @ v A:
@v A ¼ @0 1  s
0
0
0
1c
z
z
As u, v and z are frequencies, a normalization
operator Z follows f: The state after t generations istherefore given by
0 1
0 1
u0
ut
B C
C
tB
@ vt A ¼ ðZ  fÞ @ v0 A:
zt

z0

0

1

the state after nl generations becomes
0 1
0 1
u0
unl

B C 
C
l1 n B@ vnl A ¼ ðZ  fE ÞðZ  fÞ
@ v0 A:
znl

ð3Þ

z0

In this equation, normalization applies every generation. However, as f is linear, one normalization sufﬁces(see Appendix A). Hence,
0 1
0 1
0 1
u0
u0
unl


B C 
C 
C
l1 n Bl1 n B@ vnl A ¼ ðZfE ÞZf@ v0 A ¼ ZfZEf@ v 0 A;
znl

z0

z0

where the latter equality holds since E and Z commute,i.e. EZ ¼ ZE: As f has an inverse matrix and ZfZ ¼
ðZfÞ2 f1 ¼ Zf2 f1 ¼ Zf; we obtain
0 1
0 1
0 1
u0
u0
unl



B C 
C
C
l1 n Bl1 n B@ vnl A ¼ ZfEf
@ v0 A ¼ Z fEf@ v0 A ð4Þ
znl

z0

z0

because fEfl1 is linear. Although Z is not linear, theglobal behavior of the system is completely determinedby the linear transformation Efl :u0 ¼ rðu þ ð1  sÞl vÞ;v0 ¼ ð1  rÞðu þ ð1  sÞl vÞ;
z0 ¼ ð1  cÞl z:
For simpler notation, we deﬁne
A ¼ ð1  sÞl ;C ¼ ð1  cÞl :

z0

0

0

C

z

and the eigenvalues
l1 ¼ r þ ð1  rÞA;l2 ¼ 0;l3 ¼ Care all real and non-negative. The corresponding
eigenvectors are
0
1 0
1 0 1
r
A
0
B
C B
C B C
@ 1  r A; @ 1 A; @ 0 A;
0Environmental changes occur every l generations. Using
the matrix
0
1
r
r
0
B
C
E ¼ @ 1  r 1  r 0 A;
0The transformation Efl can then be written as
0 01 0
10 1
r
rA
0
u
u
B 0C B
CB C@ v A ¼ @ 1  r ð1  rÞA 0 A@ v A

01

respectively.
As n ! 1; the system is dominated by the eigenvector associated with the eigenvalue having the largest
absolute value. With continuing iteration of Efl thesystem asymptotically approaches either
0
1
0 1
r
0
B
C
B C
k1 ln1 @ 1  r A or k2 C n @ 0 A
01
where k1 and k2 are constants depending on the initialstate. It is clear that when we include normalization (i.e.,
Eq. (4) the system also approaches either of these twomodes. In the ﬁrst mode (l1 4l3 ), the state P1,
0 1 0
1
u
r
B C B
C
@vA ¼ @1 rAz

0

is the unique globally stable equilibrium. Convergenceto P1 means that innates outcompete individual
learners. The frequency of GC is small (r) at P1 because
P1 is the state just after the environment changes. In thesecond mode (l1 ol3 ), the state P2,0 1 0 1u
0
B C B C@vA ¼ @0Az
1
is the unique globally stable equilibrium. Convergenceto P2 means that individual learners outcompete
innates.
The transition between the two modes occurs at l ¼
l  ; which is determined by l1 ¼ l3 or
r þ ð1  rÞA ¼ C:

ð5ÞSolving (5) for l gives the analytic expression of l : Asl ! 1; A ! 0 and C ! 0: Thus, for sufﬁciently large l,
l1 4l3 and P1 is globally stable. Assume s4cr; thenl1 ol3 when l ¼ 1: Both eigenvalues are monotonicallydecreasing functions of l, and l1 converges to r faster

ARTICLE IN PRESS
J.Y. Wakano et al. / Theoretical Population Biology 66 (2004) 249–258

than l3 converges to zero. This means that there exists a
unique solution of (5) in an interval ½1; 1Þ (Fig. 1). In
other words, there exists a unique integer l  such that P1
is always approached when l4l  and that P2 is always
approached when lpl  :Convergence to P1 or P2 in the reduced model implies
convergence to the periodic solution with period l in thefull model (Eqs. (1)–(2)). Here we discuss whether thedifferences between the reduced and full models are
negligible or not. With regard to the ﬁrst mode, stabilityof P1 (ﬁxation of G) in the reduced model shows that IL
cannot invade by itself in the full model. Intuitively, we
expect that presence of SL at a low frequency should not
facilitate invasion by IL, since SL are parasitic on IL,
but we cannot rule out this possibility. On the other
hand, it can be rigorously shown that SL cannot invade
the population of G in the full model, whether or not IL
are also present (see Appendix B).
For the second mode, stability of P2 (ﬁxation of IL)
does not necessarily entail that IL will be ﬁxed, as SL
may coexist with IL in the full model. Nevertheless, theimportant point is that G cannot invade. In fact, theﬁtness of IL is always 1  c regardless of the frequenciesof the other phenogenotypes, and at the equilibrium
periodic solution the geometric mean ﬁtness of SL
should equal 1  c: (This condition determines theequilibrium ratio of SLC to SLW.) Hence, it will beequally difﬁcult for G to invade a polymorphism of IL
and SL as for it to invade a monomorphism of IL. WhenP2 is stable, the geometric mean ﬁtness of G is less than1  c; whence G cannot invade.
In conclusion, in the full model we expect the innates
to dominate in the ﬁrst mode (l4l  ) and to go extinct inthe second mode (lpl  ). With Aoki et al.’s parameters(s ¼ 0:1; c ¼ 0:02 and r ¼ 106 ), (5) gives l  ¼ 683;
which is exactly the same as in the numerical result(Fig. 2). A comparison between the numerical results of

253

Fig. 2. Frequencies of individual learning, social learning, and innate
behavior, averaged over one period at equilibrium, are plotted against
the period length (l) on a log scale for the periodic environmental
change model. The curves are labeled by the frequencies: z for
individual learners, x þ y for social learners, and u þ v for the innates.
Parameters are s ¼ 0:1; c ¼ 0:02 and r ¼ 106 :Table 2Comparison of numerical and analytical results
Parameters

Results

s

c

d

r

l

l  analytic

0.1

0.02

0.01
0.1

106

683
683
269
165
683
797

683
683
269
165
683
797

0.05
0.08
0.2
107

l  and l  analytic are derived by the numerical iteration of (1)–(2) andfrom (5), respectively. Parameters are s ¼ 0:1; c ¼ 0:02; d ¼ 0:01; r ¼
106 unless otherwise speciﬁed. l  analytic is monotone decreasing in
the parameters r and c, monotone increasing in the parameter s(proofs not shown), and independent of the parameter d. Rows 1 and 5of the table predict the same apparent value of l  analytic although s
has been varied, because ð1  sÞl 5ð1  cÞl holds for the given
parameters.

the full model and the analytical results of the reduced
model is summarized in Table 2. Our analysis gives theexact l* value in all cases tested.

2.3. A stochastic model and analysis

Fig. 1. Eigenvalues as a function of period. Parameters are s ¼ 0:1;
c ¼ 0:02 and r ¼ 106 :Aoki et al. (submitted) also dealt with the case whereenvironments change with probability p at each generation. In this case also, they observed two modes,
although the transition was continuous. Here we try to
clarify whether a threshold p* exists and to derive it
analytically.

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254

We study the reduced stochastic model in which thetransformation with stasis is Zf while the transformation with change is ZEf: Let us consider the asymptotic
behavior of the ratio of the frequency of individual
learners relative to the frequency of innates, i.e. z=ðu þvÞ: In the deterministic model, if the environment does
not change for l  1 generations and then changes (lgenerations in total), this ratio changes by the factor ofgðlÞ ¼C:r þ ð1  rÞAThis is because the component in the second eigenvector
corresponding to the eigenvalue l2 (¼ 0) disappears.
Fortunately, the ﬁrst and third eigenvectors do notdepend on the period, and the IL/G ratio is also
multiplied by gðlÞ in the stochastic model.
The probability that the interval between an environmental change and the next one is l isPðlÞ ¼ ð1  pÞl1 p:
After sufﬁciently many generations, we expect thatthe initial IL/G ratio is multiplied by1YgðlÞPðlÞ :Fig. 3. Average frequencies of individual learning, social learning, andinnate behavior in the stochastic environmental change model. The
values at the initial 50,000 generations are discarded to eliminate theeffect of the initial values and the average of the subsequent 50,000
generations are used. This process is repeated for 100 different random
number seeds and the average result is plotted. The horizontal axis, 1/
p, is a mean interval of environmental changes. The curves are labeledby the frequencies: z for individual learners, x þ y for social learners,
and u þ v for the innates. Parameters are s ¼ 0:1; c ¼ 0:02 and r ¼
106 :

l¼1

Therefore, if
1XPðlÞ ln gðlÞo0

ð6Þ

l¼1

then the population of innates is stable against invasion
by individual learners (and most likely social learners).
By this analysis, we expect that there exists a thresholdp and the system essentially has two modes. The
inequality gives an analytic estimate of p : With Aoki etal.’s parameters (s ¼ 0:1; c ¼ 0:02; d ¼ 0:01 andr ¼ 106 ), (6) gives p ¼ 0:0016 while the transition
occurs around p ¼ 0:0015 in our numerical calculationof the full stochastic model (Fig. 3).
2.4. Reversible mutations at the strategy locus
Given that innate behavior was the primordial state,neither individual nor social learning could have arisen
without mutation. Hence, introducing reversible mutation among the three strategies enhances the realism of
our model. When an individual learner (IL) or a social
learner (SLC or SLW) mutates to an innate (G),the allele it carries at the innate information locus will
be expressed. Hence, it is necessary to keep trackof the class of allele, resident or mutant, occurringat the innate information locus in individual andsocial learners as well as in the innates. We do
this by distinguishing the frequencies of non-innateorganisms with resident and mutant alleles by subscripts
r and m, respectively. Thus, for example, the frequency

of a SLC with a resident allele is denoted by xr : There
are a total of eight variables, seven of which are
independent.
Recall that resident alleles produce adaptive behavior
whereas mutant alleles produce maladaptive behavior in
the current environmental state. The class, resident or
mutant, is deﬁned relative to the environmental state.
When the environment changes, each organism inherits
the parental allele but the class of the allele changes. We
assume that a small fraction (r) of all alleles at the
innate information locus becomes resident alleles in the
post-change generation, regardless of the status in the
pre-change generation. In other words, the class of anallele at the innate information locus is reset when theenvironment changes: it becomes a resident allele withprobability r and a mutant allele with probability 1  r:
Let the mutation rate at the strategy locus be m. We
assume reversible unbiased mutation among the threestrategies, G, SL and IL, which occurs before asexual
reproduction. The frequencies of genotypes after mutation are denoted byu ¼ uð1  mÞ þ ðxr þ yr Þm=2 þ zr m=2;v ¼ vð1  mÞ þ ðxm þ ym Þm=2 þ zm m=2;
xnr þ ynr ¼ um=2 þ ðxr þ yr Þð1  mÞ þ zr m=2;
xnm þ ynm ¼ vm=2 þ ðxm þ ym Þð1  mÞ þ zm m=2;
znr ¼ um=2 þ ðxr þ yr Þm=2 þ zr ð1  mÞ;
znm ¼ vm=2 þ ðxm þ ym Þm=2 þ zm ð1  mÞ:

ð7Þ

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The recursions for environmental stasis are0n

W 1u ¼ u ;W 1 v0 ¼ ð1  sÞvn ;W 1 x0r ¼ ð1  dÞðxnr þ ynr Þðu þ xr þ xm þ zr þ zm Þ;
W 1 x0m ¼ ð1  dÞðxnm þ ynm Þðu þ xr þ xm þ zr þ zm Þ;
W 1 y0r ¼ ð1  s  dÞðxnr þ ynr Þðv þ yr þ ym Þ;
W 1 y0m ¼ ð1  s  dÞðxnm þ ynm Þðv þ yr þ ym Þ;
W 1 z0r ¼ ð1  cÞznr ;W 1 z0m ¼ ð1  cÞznm :ð8ÞThe recursions for environmental change are
W 2 u0 ¼ rðun þ vn Þ;
W 2 x0r ¼ 0;
W 2 x0m ¼ 0;
W 2 y0r ¼ ð1  s  dÞðxnr þ ynr þ xnm þ ynm Þr;
W 2 y0m ¼ ð1  s  dÞðxnr þ ynr þ xnm þ ynm Þð1  rÞ;
W 2 z0r ¼ ð1  cÞðznr þ znm Þr;
W 2 z0m ¼ ð1  cÞðznr þ znm Þð1  rÞ:

ð9Þ

The model is rather complicated, but the recursions
for mutation and the recursions for innates andindividual learners are still linear. Therefore, we canperform an analysis similar to the previous one. First,
we consider periodically changing environments. Consider the reduced dynamical system in ðu; v; zr ; zm ÞT :Using matrices
0
1
1m
0
m
0
B 0
1m
0
m C
B
C
M¼B
C;
@ m
0
1m
0 A
m

0

1
0
B0 1 s
B
x¼B
@0
0
0
0

0

1m
1

0
0

0
0

1c
0

0
1c

r

0

0

1

1r
0

0
r

0
r

C
C
C;
A

0

1r

1r

C
C
C;
A

and
0

r

B1  r
B
F ¼B
@ 0
0

transformation is linear, but it is too complicated to deal
with analytically. We numerically examined the largest
eigenvalue and the corresponding eigenvector ofxFMðxMÞl1 by the power method (assuming parametervalues s ¼ 0:1; c ¼ 0:02; r ¼ 106 and m ¼ 106 ). For
most l values, the dominant eigenvector changes
gradually as l is varied. However, it changes very
sharply from (109, 0.003, 106, 0.996) to (106, 0.979,
108, 0.02) as l changes from 683 to 684. Such atransition occurs only once and we estimate l  ¼ 683:
Associated with this transition, the average frequencies
of innates and individual learners over l generations,derived froml11X
ðxM Þi h;l i¼0

W 2 v0 ¼ ð1  sÞð1  rÞðun þ vn Þ;0255

the state after nl generations is given by the following
transformation:

n 
n
ðZxFMÞðZxMÞl1 ¼ ðZxFMÞZðxMÞl1

n

n
¼ ZxFMðxMÞl1 ¼ Z xFMðxMÞl1 ;where we used MZ ¼ ZM; F Z ¼ ZF and ZxZ ¼ Zx: Thus,xFMðxMÞl1 determines the behavior of the system. The

where h is a dominant eigenvector, change from(104, 0.999) to (0.354, 0.645). These results mean thatthe overall behavior remains similar when we
introduce reversible mutations at the strategy locus.
On the other hand, a major difference predictedby the reduced model is that individual learners
will coexist with innates at signiﬁcant frequency( m) when l4l  : A possible reason for this polymorphism of individual learners and innates is that therate of increase (or multiplication factor) of theindividual learners is no longer a constant (i.e., 1  c)divided by the mean ﬁtness, but depends on thefrequencies of the other phenogenotypes (Eq. (7)).
This frequency dependence, although weak, may maintain the individual learners in a polymorphic state evenwhen l4l  ; particularly in the neighborhood of thecritical period l  where selection favoring the innates isalso weak.
The results of the numerical iterations of the fullmodel (Eqs. (7)–(9)) are shown in Fig. 4. The thresholdoccurs at l ¼ 684; which is in good agreement with thepredicted value of l  ¼ 683: Individual learners andinnates dominate when l4684 (the ﬁrst mode), while
individual and social learners dominate when lp684(the second mode). The frequency of individual learners
in the ﬁrst mode is non-negligibly large as is predicted by
the reduced model. Due to the mutations at the strategy
locus, social learners also exist in the ﬁrst mode but theirfrequency is of the order of mutation rate, m; andsimilarly for the frequency of innates in the second
mode. Thus, at the evolutionary equilibrium, interactions between social learners and innates are rare.
Lastly, we study the model including both stochastically changing environments and mutations at thestrategy locus. The numerical analysis shows that theoverall tendency is similar to the previous models(Fig. 5). The only difference is that the overlapping
region is larger.

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J.Y. Wakano et al. / Theoretical Population Biology 66 (2004) 249–258

Fig. 4. The result of the model with mutations at the strategy locus
and periodic environmental changes. The graph is drawn in the same
way as Fig. 2. Parameters are s ¼ 0:1; c ¼ 0:02; r ¼ 106 and m ¼
106 :

Fig. 5. The result of the model with mutations at the strategy locus
and stochastic environmental changes. The graph is drawn in the same
way as Fig. 3. Parameters are s ¼ 0:1; c ¼ 0:02; r ¼ 106 and m ¼
106 :

3. Discussion
Building on the work of Boyd and Richerson (1985,
1988), Rogers (1988), and Feldman et al. (1996), Aoki etal. (submitted) showed that individual learning, social
learning, and innate determination of behavior arefavored by natural selection when environmental
changes occur at short, intermediate, and long intervals,
respectively. In this paper, we analyzed and extended
Aoki et al.’s models in order to give more strong
evidences supporting the hypothesis. First, we carriedout a mathematical analysis of the model. Analyzing the
reduced linear dynamics, the evolution of learning
ability is described in terms of the eigenvalue problem.

The analysis not only supports the hypothesis but also
explains the sharp transition of two modes and gives ananalytic expression for the critical frequency of environmental change. Second, we introduced more biologicallyrealistic assumptions including reversible mutation at
the strategy locus, which was ignored in the basic model.
The general model also supports the hypothesis.
In the basic model, we have shown that only innates
dominate and both social and individual learners go to
extinction when the period of environmental change islong (l4l  ). The introduction of reversible mutations
results in the coexistence of strategies. The frequency ofeach strategy is maintained above a certain levelsupported by the continuous ﬂow of mutations. As themutation rate is low, we would not expect mutation tohave a major effect on these frequencies. However, ouranalysis shows that the equilibrium frequency of
individual learners is signiﬁcantly larger than that
expected by the mutation rate alone. This is not only
mathematically interesting but also reminds us that
simpliﬁcations (such as neglecting rare mutations) do
not always produce the expected outcome. In conclusion, when we assume a low mutation rate at thestrategy locus, innates and individual learners coexist
when the period of environmental change is long.
When the environment changes periodically, social
learners and innates do not coexist at signiﬁcantfrequencies. Therefore, only those behaviors that canbe ‘‘invented’’ by the individual learning process can besocially transmitted. The ‘‘content of culture’’ is limitedby the capacity for individual learning as much as by thecapacity for social learning. This argument was originally made by Aoki et al. (submitted), and the present
study shows that it is valid even when we introduce
mutations at the strategy locus. On the other hand,when the environment changes stochastically, thetransition of two modes is more gradual. Moreover,
the overlapping region is larger when we introduce
mutations among the strategies. These results suggestthat the content of culture basically comes from what
individual learners invented, but it may also comefrom innate behaviors when the environment changes
randomly.
There remains the task of testing the predictions
against data on the phylogenetic distribution of social
learning. This would be a formidable undertaking, andwe do not attempt it here. One problem lies with ourmodeling approach, where we have followed Rogers(1988) in assuming that behavior is obligate. Any real
behavior cannot easily be classiﬁed as innate, sociallylearned, or individually learned, and many behaviors
will actually involve all three processes. Hence, thepredictions of our models serve a heuristic purpose but
cannot readily be applied to data. Boyd and Richerson(1985, 1988) have introduced more realistic models thatrecognize the facultative nature of behavior. Unfortu-

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257

nately, the price of this realism has been lack ofamenability to detailed mathematical treatment (butsee Feldman et al., 1996).

when x; y; z  1: Substituting the explicit expression for
W 1 ; we obtain

Acknowledgments

where we neglected the higher-order terms of x; y and z:
This equation means x þ y decreases while the environment remains constant. When the environment changes,

We thank the anonymous referees for helpful comments. Research supported in part by NIH Grant
GM28016.

A normalization operator is deﬁned asx
Z : x7!
;
kxk

xi

i¼1

is a scalar. Here we proveðZzÞn  x ¼ Zzn  x

ðA:1Þ

for any positive integer n and any linear transformationz: Note thatcx
¼Zx
Z  ðcxÞ ¼kcxk
andz  ðcxÞ ¼ cz  x
for any non-zero scalar c. The proof uses the principle ofmathematical induction. When n ¼ 1; Eq. (A.1) is true
by deﬁnition. If it is true when n ¼ m; then it is also true
when n ¼ m þ 1 because
mþ1

ðZzÞ

zm x
 x ¼ Zz  m ¼ Z z x

1
zmþ1  xzm x!

¼ Zzmþ1  x

This completes the proof, of Eq. (A.1).

Appendix B
Here, we prove that social learners cannot invade apopulation consisting mostly of innates but in which
individual learners may also be present at a low
frequency. Recursions (1) can be rewritten as
W 1 u0 ¼ u;
W 1 v0 ¼ ð1  sÞv;W 1 x0 ¼ ð1  dÞðx þ yÞu;W 1 y0 ¼ ð1  s  dÞðx þ yÞv;W 1 z0 ¼ ð1  cÞz;W 2 ðu0 þ v0 Þ ¼ ð1  s þ srÞðu þ vÞ
W 2 ðx0 þ y0 Þ ¼ ð1  s  dÞðx þ yÞ

Referenceswhere x ¼ ðx1 ; x2 ; :::; xN Þ is an arbitrary vector andkxk 

ð1  dÞu þ ð1  s  dÞv
1  d  sv
ðx þ yÞ ðx þ yÞ;u þ ð1  sÞv
1  svand thus x þ y decreases. This completes the proof.
Notice that the result is independent of parametervalues.

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