J. theor. Biol. (1998) 193, 153–165
Article No. jt980697

Dynamic Model of Optimal Age Polyethism in Social Insects under Stable andFluctuating Environments
J. Y. W,* K. N†  N. Y**Center for Ecological Research, Kyoto University, and the †Department of Zoology,
Faculty of Science, Kyoto University, Kyoto, 606-8502 Japan(Received on 29 July 1997, Accepted in revised form on 20 February 1998)

Social insect workers from many species perform division of labor based on age (temporal division oflabor). The age polyethism hypothesis postulates that the kind of labor is determined by age,
independent of the environmental condition. Previous theoretical studies on this subject have used only
static models which do not describe time developmental factors such as population growth or age
structure dynamics. However, age structure dynamics is an essential factor for temporal division oflabor. In this paper, we develop an age-structured population model with division of labor between tasks
inside the nest and a foraging task outside the nest and analyse the adaptive form of age polyethismby computer simulation. We show that in a constant environment age polyethism is adaptive when
specialization for a single job is efficient or when the mortality during foraging task is higher than thatduring inside task. And computer simulations suggest that when the excess foraging mortality exists
and life expectancy of workers decreases monotonously with age, an age polyethism where workers
drastically switch from inside to outside labor at a certain age is adaptive, which is consistent with fieldobservations. In a fluctuating environment, the balance of labor can not stay optimal because age
structure fluctuates due to temporal variation of newly produced workers and because the necessarybalance itself may fluctuate. Computer simulations reveal that when environmental fluctuation affects
the efficiencies of both inside and outside labors, a ‘‘soft age polyethism’’ form (in which each worker
of an age class performs both labors and the ratio of the labors gradually changes with age) becomes
more adaptive than any ‘‘hard age polyethism’’ form (in which all workers of an age class perform thesame single labor and the kind of labor abruptly changes at a certain age class) as the magnitude offluctuation increases. However, when environmental fluctuation affects only outside labor, hard age
polyethism forms tend to stay adaptive even as the fluctuation increases.
7 1998 Academic Press

Introduction
Division of labor among workers in social insects is
an important characteristic for their success because
it appears to enhance the efficiency of production ofworkers (Oster & Wilson, 1978). Relationships
between tasks and age of individual workers havebeen found in almost all ant species studied (forreview, see Hölldobler & Wilson, 1990), as well as inhoneybees (Seeley, 1982), wasps (Dew & Michener,
1981) and termites (Badertscher et al., 1983). Thisphenomenon is referred to as ‘‘temporal division of
0022–5193/98/130153 + 14 $30.00/0

labor’’ (TDL), and involves a universal tendency foryoung workers to stay inside their nest to care forbroods and reproductives, while old workers gooutside the nest to forage.
One assumption for the mechanism of attaining
TDL is that a worker’s task is determined by its age,
independent of external environments. This explanation is referred to as the ‘‘age polyethism
hypothesis’’ (AP hypothesis). Wilson (1968) andOster & Wilson (1978) developed a static modelfor optimal allocation of workers into different
7 1998 Academic Press154

. .    .

morphological castes and stated that task distribution
under AP hypothesis can be explained by classifying
individuals of the same age into functional castes (agecastes).
Division of labor is believed to be adaptive because
workers specialized in some tasks might be more
efficient than less specialized workers (Hölldobler &Wilson, 1990). However, the specialization effect can
not explain why younger workers always stay inside.
Jeanne (1986) used a model of AP to show
theoretically that the life expectancy of a worker islonger when it engages in inner tasks before
dangerous foraging labor than when there is nodivision of labor. He suggested that high mortality
during foraging can explain the tendency of TDL
because the longer life expectancy of workers may
result in a colony that retains a larger work-capacity
and hence higher productivity. We refer to thismechanism as the foraging mortality effect.
However, all of the theoretical studies mentioned
above are static in the sense that they do not describe
time developmental factors such as the population
growth or the dynamics of worker age structure which
may have a great influence on the optimality of AP.
In any model of AP, one should first formulate alabor allocation program that decides the amount ofeach labor allocated for each age. The allocation oflabor in a whole colony should then be derived fromthe dynamics of age structure with the labor
allocation program. Our first purpose is to make sucha dynamic model based on the AP hypothesis. Ourmodel is formulated as an age structured populationof workers with both specialization and foraging
mortality effects, and can determine the optimal labor
allocation program that maximizes worker production. Modeling AP, we permit various forms oflabor allocation programs, which we classify into‘‘soft AP’’ forms where each worker of an age classperforms both labors in a certain ratio and ‘‘hard AP’’forms where all workers of an age class perform thesame single labor. We apply this dynamic model toexamine whether the existence of foraging mortality
favors the AP form of ‘‘younger-inside and olderoutside’’.
Standard ant colonies have reproductive cycles
in which the colony continues worker production
until colony size reaches a certain level, at which thecolony switches to reproduction. The colony switches
to worker production again when colony size
decreases below a certain level (Hölldobler & Wilson,
1990). In social insects which reproduce by fission(such as honeybees and some ants), colony growth
by worker production directly contributes toreproduction. Thus, the fitness of a social insect

colony can be estimated by its worker production
rate [for further discussion, see Wilson, (1971)].
Therefore we evaluate fitness of a colony by its
growth rate.
AP has previously been considered to be disadvantageous under fluctuating environments such as
variable weather or food supply (Calabi & Traniello,1989). On the other hand, a phenomenon thatworkers change behaviors corresponding to artificially changed environments, known as behavioral
flexibility (ants: Gordon, 1989; bees: Cartar, 1992),
has been observed. Behavioral flexibility implies thatworkers do not show a rigid AP. Recently, many
theoretical studies have been done under theassumption that behavioral flexibility is an adaptive
response which arises from individual interactions(Tofts & Franks, 1992; Franks & Tofts, 1994; Bourke& Franks, 1995; for a review see Gordon, 1996).
However, few studies have actually demonstrated thatbehavioral flexibility has meaningful fitness consequences for the colony as a whole (Calabi & Nonacs,
1994) and there is even a report that behavioral
flexibility does not necessarily compensate theimbalance of labor allocation nor enhance the colony
productivity (Nakata, 1996). Furthermore, because of
the lack of theoretical studies of AP (Gordon, 1996),
it is not clear how and why AP is disadvantageous
in a fluctuating environment. Therefore, we here
concentrate on the study of AP in order to clarify
to what extent AP be adaptive in fluctuating
environments.
There are two effects of fluctuating environments
on AP. One is that the necessary labor allocation fora colony changes when the environment changes since
the ratio of the efficiencies of different labors varies.
The other is that age structure fluctuates with time
because the number of newly produced workers
fluctuates. The final purpose which we give the mostemphasis is to evaluate how these two effects influctuating environments influence the adaptive AP
form, including whether it is a hard AP form or a soft
AP form.

Model
We make the following assumptions in developing
the model:(1) worker behaviors are determined only according to worker age (age polyethism hypothesis).
Environment or age structure of a colony do notaffect their behaviors;(2) all workers change their behaviors in the same
way according to their age;     
(3) there are two kinds of labor, outside work
(foraging) and inside work. The labor allocation
program is defined as the frequency of inside work at
each age;(4) a colony population consists of a brood classand N age classes of workers. All broods successfully
grow up and move to the first age class of workers
after a given period of time. Thus the unit of time isthe time which an individual takes to grow up froman egg to a worker. All workers in the final age classdie at the next time step.
Under the above assumptions, the model isformulated as follows. In the following expressions,the subscript t denotes time (t = 0, 1, 2, . . .) andsubscript n denotes age class (n = 1, 2 , . . . , N).
Colony states
at time t are represented by age:
structure, At (each component Ant means the numberof workers of each age class n) and the number ofbroods, Bt (i.e. the total number of eggs, larvae andpupae).
Labor allocation program is denoted by labor
allocation proportion (LAP) xn (0 E xn E 1) which
specifies the proportion of inside labor at each age
class. Every worker has a certain basic capacity forwork and an additional capacity proportional to thedegree of specialization denoted by (12 − xn )2, which is
maximum when xn = 0 or xn = 1 and minimum whenxn = 12 . Thus, the total capacity per individual is
wn = b +

0

1

2

1
− xn ·4a2(bEwn E b + a)

where a is the magnitude of specialization effect andb is the base work-capacity per individual. Thus, thetotal inside work-capacity of the colony is
It = s Ant ·wn ·xnn155

rate rn ) and the probability of surviving foraging
1 − (1 − xn )·g:
pn = rn × 41 − (1 − xn )·g5.
We further assume that the number of broods
produced is limited by the smallest work-capacity.
The number of broods at time t + 1 is calculated fromwork-capacities at time t as
Bt + 1 = min(cl It , cF Ft )where cl It is the number of broods which the colony
can care for and cF Ft is the number of broods which
can be fed. Thus, ct and cF represent the potential
number of broods which can be produced per unit ofinside and foraging work-capacities, respectively. We
consider three different sets of assumptions regarding
brood production.
 1: I = 1, F = 
This model represents the case when there is noenvironmental fluctuation.
Models 2 and 3 concern about environmental
fluctuation. Let an unpredictable fluctuating environment be represented by a stochastic variable Rt . Rt
might correspond to temperature or food resource
abundance. Rt can take one of the following values
with equal probability:
1
1
1,, . . . , , 1, 2, . . . , s − 1, ss s−1
2
where s is a natural number representing themagnitude of fluctuation. In this stochastic process
the geometric mean of Rt has a constant value of 1,
independent of the value of s. Time-scale offluctuation is the same as unit time of this model
which is measured by worker’s developmental time.

and total foraging work-capacity of the colony is 2: I = Rt , F = Rt
Ft = s Ant ·wn ·(1−xn )nSome workers are killed during foraging. Foraging
mortality g is defined by the probability per unit time
period of being killed during foraging when a worker
forages for all of its time in an age class. The survivalrate, pn , is the product of the probability per unit time
period of avoiding natural death (internal survival

This model represents a situation, such as
fluctuating temperatures, in which environmental
fluctuation affects the efficiencies of both inside andforaging work-capacities equally. Insects are poikilothermal animals so that activity of both inside andoutside labor responds to fluctuating temperature.
The success of brood production may also be
influenced by humidity, disease or parasite density.
These factors may affect inside and outside labor. .    .

156

equally. Thus, this model assumption is relevant toeffects such as those imposed by fluctuations inweather, epidemic disease, or parasites.
 3: cI = 1, cF = cRThis model represents a case in which environmental fluctuation affects only the efficiency of foraging.
In nature, such fluctuations may be related to changes
in food resource density.
In all of these models, age structure changes withtime asF A1 JG A2 GG A3 G
G*G
fANjt+1
F0
Gp1
=G0
G*
f0

0
0
p2
*
0

0
0
0
*
0

...
...
...
...
pN−1

0J F A1 J FBtJ
0G G A2 G G 0 G
0G G A3 G +G 0 G
0G G * G G * G
0j fAN j t f 0 j

Our model is nonlinear and so fitness cannot be
defined so easily as in linear models. Especially, stable
age structure is never reached under fluctuating
environments in Models 2 and 3. Nevertheless,preliminary simulations showed that after enoughtime had passed colony population growth isapproximately exponential. In other words, whenN

Nt = Bt + s Ant ,ft =n=1log(Nt /N0 )tconverges when t approaches infinity. Moreover, theconvergent value of ft is independent of initial age
structure or actual realization of temporal sequence of
R. These results are proved by ergodic theorem in
linear stochastic matrix population models and theconvergent value is considered as the reasonablemeasure of expected growth rate in stochastic
environment (Caswell, 1989). Referring to the resultof our computer simulations, we consider this result
still holds in our model and define colony fitness as
the convergent value of ft .

Computer Simulation and Results
Due to nonlinear factors (especially minimum
function in determining the number of produced
broods), no good analytic method of approximation

could be found. Therefore, to estimate optimal labor
allocation program, we used a computer simulation inwhich we assigned xn 11 discrete values from 0 to 1
at steps of 0.1 and calculated fitnesses for all possible
LAPs. Preliminary simulation showed that fitness isnearly independent of initial age structure, therefore
we adopted an initial age structure for all simulations
defined asi−1

B = V, A1 = V, Ai = V t rj

(i e 2)j=1

where V is chosen to make the initial population sizeunity. This is a stable age structure when g = 0 andthe growth rate is unity, which is selected because it
causes ft to converge rather quickly. Different
realization of the temporal sequence of the randomvariable R is used for different values of a and g while
the same realization is used to compare the fitnessesof different LAPs for the same value of a and g.
In all simulations, the number of age classes N, thebase work-capacity per individual b and the ratio of
inside work efficiency to outside work efficiency c are
set at constant values (N = 6, b = 0.6 and c = 1.5)
while specialization effect a, foraging mortality g,environmental fluctuation s and internal survival rate
rn are varied for each simulation.
   1  
In order to examine the age polyethism (AP) understable conditions, all LAPs (116 cases) were calculated
each for several parameter sets using Model 1. The
optimal LAP is a set of labor allocation proportions
4xn 5 which reaches a maximum fitness in 1000 time
steps ( f1000 ). We adopted 1000 as the number of time
steps because ft converges before 1000 steps in almost
all cases. Results are summarized in Table 1 andFig. 1(a, b).
Division of labor among age classes is adaptive as
long as either specialization effect or foraging
mortality exists (a q 0 or g q 0). In the case ofmonotonously decreasing internal survival rate
[Table 1(a)], the existence of foraging mortality
increased the tendency for young workers to beallocated for inside labor and old workers for outsidelabor with or without specialization effects. On theother hand, when a q 0 and g = 0, AP is adaptive but
the tendency of ‘‘younger-inside and older-outside’’
does not appear. We also calculated the optimal LAP
in a case when internal survival rate is constant[Table 1(b)] and in a case when worker life expectancy
in each age does not decrease monotonously with age[Table 1(c)]. In all three cases, workers in age classesof shorter life expectancy are allocated more for     

157

T 1
The adaptive labor allocation proportion of inside work (LAP) and its fitness in stableenvironments (Model 1) for different specialization effect a and foraging mortality gin cases of (a) monotonous decreasing internal survival rate r, (b) constant r and (c)low survival rate at age class 1. Life expectancy at each age, based on r, is shown
(a)
1

2

4r5
Life expectancy
a
g

1.0
4.43

0.9
3.43

0
0.2
0.4
0.6
0
0.2
0
0.2
0
0.2

0.6
0.0
1.0
0.0
1.0
1.0
1.0
1.0
1.0
1.0

Age class
3
4

5

6

0.8
0.7
2.7
2.12
Adaptive LAR

0.6
1.6

0.0
1

0.6
1.0
0.0
1.0
1.0
1.0
1.0
0.9
0.8
0.8

0.6
1.0
1.0
1.0
0.2
0.0
0.0
0.0
0.0
0.0

0.6
1.0
0.0
1.0
0.0
0.0
0.0
0.0
0.1
0.0

0.6
0.0
1.0
1.0
0.0
0.0
0.0
0.0
0.2
0.0

0.6
0.7
0.1
0.0
0.0
0.0
0.2
0.0
0.0
0.0

1

2

Age class
3
4

5

6

4r5
Life expectancy
a
g

0.85
4.15

0.85
3.71

0.85
0.85
3.19
2.57
Adaptive LAR

0.85
1.85

0.0
1

0
0
0

1.0
1.0
1.0

0.9
1.0
0.8

0.4
0.0
0.0

0.0
0.0
0.0

0.0
0.0
0.0

0.0
0.1
0.0

1

2

Age class
3
4

5

6

4r5
Life expectancy
a
g

0.6
4

1.0
5

1.0
1.0
4
3
Adaptive LAR

1.0
2

0.0
1

0
0
0

0.8
0.6
0.5

1.0
1.0
1.0

0.0
0.0
0.0

0
0
0
0
0.2
0.2
0.4
0.4
0.6
0.6

(b)

0.2
0.4
0.6

(c)

0.2
0.4
0.6

1.0
1.0
1.0

foraging. In adaptive AP, both inside work-capacities
and foraging work-capacities (and colony size as well)increase exponentially after a short initial period anda balance of these labors are obtained, i.e. cl It = cF Ft[Fig. 1(a)]. In contrast, in non-adaptive AP, a balanceof labors is never reached [Fig. 1(b)].
   2  3
In simulations with a fluctuating environment, weconcentrated on the case of a monotonously
decreasing internal survival rate where we set
4ri 5 = 41.0 0.9 0.8 0.7 0.65. Based on the results for
Model 1, we made the following assumption to reduce
the amount of calculation: in adaptive AP wheng q 0, young workers are allocated more for insidelabor and old workers are allocated more for outside

0.1
0.5
0.8

Fitness0.12200.20080.2671
0.32280.10010.17710.07380.14550.03860.1063Fitness
0.0761
0.04270.0010Fitness
0.0
0.0
0.0

0.06340.0323−0.0022labor: we searched the adaptive LAP among LAPs
which decreases monotonously with age.
Various forms of adaptive LAPs were obtained
from these simulations. We classified them into three
categories: ‘‘hard age polyethism’’, ‘‘soft age
polyethism’’ and ‘‘non-age polyethism’’ (see Fig. 2). In
hard-AP, every worker labors inside when young, thenswitches to outside labor at a certain age and continues
to forage till death. In soft-AP, each worker in an age
class labors both inside and outside at a certain
frequency. The frequency changes gradually with age.
We assumed that all workers behave in the same way
so that the same age class never contains both workers
specialized in inside labor and workers specialized inoutside labor. In non-AP, LAP is constant with age,which means no temporal division of labor.

. .    .

158

100

10

(a)

(b)

10

1

1

0.1

0.1
10

0

20

0

30
100

100

10

1

1

20

30

10

20

30

20

30

(d)

(c)

10

10

0.1

0.1
10

0
100

20

0

30
100

(e)

10

10

1

1

0.1

(f)

0.1
0

10

20

30

0

Time

10
Time

Figure 1—(Caption opposite).

     

159

F. 1. Population growth and balance of labors in Model 1 for (a) an adaptive AP where 4xn 5 = 41, 0.9, 0, 0, 0, 05 and (b) non-adaptive
AP where 4xn 5 = 41, 1, 1, 0, 0, 05; in Model 2 for (c) an adaptive soft-AP where 4xn 5 = 40.8, 0.8, 0.7, 0, 0, 05 and (d) a hard-AP where
4xn 5 = 41, 1, 0, 0, 0, 05 which is adaptive when s = 0; and in Model 3 for (e) an adaptive hard-AP where 4xn 5 = 41, 1, 0, 0, 0, 05 and (f)a soft-AP where 4xn 5 = 40.8, 0.8, 0.7, 0, 0, 05 which is adaptive in Model 2 for the same parameter set (see Fig. 3). We set a = 0.2, g = 0.4
and 4ri 5 = 41, 0.9, 0.8, 0.7, 0.65 in all cases and s = 2 for Models 2 and 3. The line with squares represents Nt (population size including
brood), the bold line represents cl It (the number of broods which could be cared for) and the thin line represents cF Ft (the number of broods
which could be fed). For values of cl and cF , see the text. The minimum of cl It and cF Ft is the reproductive success, that is the numberof broods actually produced. In Model 1, equality is reached (cl It = cF Ft ) in (a) the adaptive AP while not in (b) non-adaptive AP. In
Model 2, the balance of labors is nearly reached (cl It 3 cF Ft ) in (c) the adaptive AP while it is weaker in (d) non-adaptive AP. In Model3, the balance of labors is hardly reached both in (e) the adaptive and (f) non-adaptive APs.

In this study, the age of workers was artificiallydivided into discrete age classes but in nature the age
distribution is continuous. When hard-AP in whichlabor allocation switches at a certain age is adaptive,
it is unlikely that the switching age lies exactly
between two of our arbitrary age classes. Consideringsuch a case, a LAP is referred to as a hard-AP if it
involves workers performing only inside labor or only
outside labor in all age classes except for one age class[Fig. 2(b)].
   2
The optimal LAP was calculated for several
parameter sets in Model 2 in which environmental
fluctuation affects both inside and foraging workcapacities equally. Results are shown in [Table 2(a)].
The range of fitnesses are from −a which means
extinction to 0.3 depending on strategies. The

difference in fitnesses of the best and the second best
AP varies from 0.001 to 0.01. The adaptive form oflabor allocation varies depending on parameters.
Hard-AP is adaptive when a or g is large while
non-AP is adaptive when s is large. In other words,
specialization effect and foraging mortality favor
hard-AP, while environmental fluctuation favors
non-AP. Soft-AP appears in intermediate situations
between the regions where hard-AP and non-AP are
adaptive. For example, when a = 0.1 and g = 0.4, theadaptive forms are hard AP for s = 1, soft AP fors = 2, and non AP for s = 3,4. To see how much thefitness of the hard-AP adaptive at s = 1 decreases ass increases in comparison to that of the APs adaptiveat each s, we plotted the fitnesses of those APs as afunction of s [Fig. 3(a)]. The fitness of the former AP
decreased drastically compared with that of the latterAPs as s increased. Moreover, to see why soft-AP isT 2
The adaptive forms of labor allocation program under fluctuating environment that affect theefficiencies of inside and outside labors equally (Model 2) and under fluctuating environment that
affect only the efficiency of outside labor (Model 3)(a) Model 2
Smalla (Specialization effect)
Weak
3
4
Strong

0
0.1
0.2
0.3
0.4

4
Strong

0
0.1
0.2
0.3
0.4

:

Large

0.1

0.2

0.3

0.4

0.5

0.6

HNNN
HNNN
HHSN
HHHS
HHHH

HNNN
HSNN
HHSN
HHHS
HHHH

HNNN
HSNN
HHSN
HHHS
HHHS

SSNN
HSNN
HHSS
HHHS
HHHS

HSNN
HSSN
HHSS
HHHS
SHHS

9

g (Foraging mortality)

:

Small

Weak
3

g (Foraging mortality)

SNNN
HNNN
HHNN
HHHS
HHHH

(b) Model 3a (Specialization effect)

9

Large

0.1

0.2

0.3

0.4

0.5

0.6

SSSS
HHHH
HHHH
HHHH
HHHH

HSSS
HHHH
HHHH
HHHH
HHHH

HHSS
HHHH
HHHH
HHHH
HHHH

HHSS
HHHH
HHHH
HHHH
HHHH

SHSS
HHHH
HHSH
HHHH
HHHH

HHSS
HHHH
HHHH
HHHH
SHHH

H,S,N represents hard-AP, soft-AP, non-AP, respectively (see also Fig. 2). Four letters correspond to four levelsof fluctuation amplitude (s = 1, 2, 3, 4). For example, HSNN means that the adaptive AP is hard for s = 1, softfor s = 2, and non-AP for s = 3 and 4.

. .    .

160

(a)
1.0
0.8
0.6
0.4
0.2
0.0
1

2

3
4
Age class5

6

Proportion of inside labor

Proportion of inside labor

Hard–AP

(b)
1.0
0.8
0.6
0.4
0.2
0.0
1

2

3
4
Age class

5

6

2

3
4
Age class

5

6

(c)
1.0
0.8
0.6
0.4
0.2
0.0
1

2

3
4
Age class

5

6

2

3
4
Age class5

6

Proportion of inside labor

Proportion of inside labor

Soft–AP

(d)
1.0
0.8
0.6
0.4
0.2
0.0
1

Proportion of inside labor

Non–AP

(e)
1.0
0.8
0.6
0.4
0.2
0.0
1

F. 2. Examples of hard-AP (a, b), soft-AP (c, d) and non-AP (e). See the text for definition of hard-AP, soft-AP and non-AP.

adaptive under fluctuating environment, population
growth of adaptive soft-AP and the hard-AP adaptiveat s = 1 are shown for s = 2 in Fig. 1(c, d). The figure
shows that adaptive soft-AP balances inside andoutside labors better than non-adaptive hard-AP.
   3
The same simulations performed for Model 2 werealso conducted for Model 3 in which environmental
fluctuation affects only the foraging work-capacity.
The range of fitnesses are from −a which means
extinction to 0.25 depending on strategies. The
difference in fitnesses of the best and the second best
AP varies from 0.001 to 0.01. These simulations showthat the region where hard-AP is favored is muchlarger than for Model 2 [Table 2(b)]. In contrast to

Model 2, non-AP is never optimal. For example,
when a = 0.1 and g = 0.4, all the adaptive forms are
hard-AP for s = 1, 2, 3 and 4. We plotted the fitnessof the AP adaptive at s = 1 against resource
fluctuation s with the fitness of AP adaptive ateach s as we did for Model 2 [Fig. 3(b)]. The fitnessof the adaptive AP is higher than that of the former
AP adaptive at s = 1 but the difference is not so large
as in Model 2. In this case, the frequency of inside
labor

0 1
s xn
n

in the adaptive AP increases as s increases. We
confirmed this tendency for other parameter sets.

     

161(a) Model 2(1)

Fitmess

0.08

0.00

–0.08
1

2
3
Magnitude of fluctuation

4(2)

Ratio of inside labor=1=2= 3 and 4

1.0
0.8
0.6
0.4
0.2
0.0
1

2

3
4
5
Age class

6

1

2

3
4
5
Age class

6

1

2

3
4
5
Age class

6(b) Model 3(1)

Fitmess

0.08

0.00

–0.08
3
2
Magnitude of fluctuation1

Proportion of inside labor(2)= 1 and 2

4

=3

=4

1.0
0.8
0.6
0.4
0.2
0.0
1

2

3
4
5
Age class

6

1

2

3
4
5
Age class

6

1

2

3
4
5
Age class6F. 3. Fitnesses of a hard-AP, which is adaptive when s = 1, (R) and the AP adaptive for each s value (R) in (a) the fluctuating
environments that affect the efficiencies of inside and outside labors equally (Model 2) and (b) the fluctuating environments that affectthe efficiency of only outside labor (Model 3). The difference between fitness of each AP and fitness of non-AP is shown relative to theamplitude of fluctuation s (1). The actual form of adaptive AP is also shown (2). a is set at 0.1 and g at 0.4.

. .    .

162

In order to see why soft-AP is not adaptive inthis case, population growth of the hard-AP
adaptive at s = 2 and the soft-AP, which is adaptive
in Model 2 for the same parameter set, are shown in
Fig. 1(e, f). In both APs, a balance of labors is notreached and the degrees of imbalance are similar toeach other.

Discussion
The results of simulations for Model 1 show thatdivision of labor among ages can be adaptive wheneither specialization effect or foraging mortality effectexists but that the simultaneous existence of both may
not be necessary for the evolution of AP (Table 1).
The results of simulations also suggest that life
expectancy is a better index than age itself todetermine whether a worker is allocated for
dangerous labor or not. The universal tendency intemporal division of labor (TDL) is that young
workers labor inside and old workers forage outside.
This phenomenon has been suggested to be adaptive
by an indirect consideration that this tendency
increases worker life expectancy (Jeanne, 1986). Usinga dynamic model with worker age structure, ouranalysis revealed that this universal tendency of TDL
is adaptive when: (1) some extra mortality during
foraging exists and (2) life expectancy of younger
workers is longer, even if specialization for a single
type of work in an age class does not increase theefficiency of labor. On the other hand, increased labor
efficiency due to specialization, without additional
foraging mortality, does not make the TDL tendency
adaptive. This suggests that foraging mortality effectis more important in explaining observed TDL thaneffects of specialization on labor.
When there is no environmental fluctuation,
adaptive age polyethism (AP) keeps a colony in the
optimal balance of labors [Fig. 1(a)]. This is because
a labor allocation program induces a corresponding
constant labor allocation in the whole colony
independent of time, as the colony reaches a stable
age structure and grows exponentially (Fig. 1). This
result shows that the optimal caste ratio discussed by
Wilson (1968) in his static model can be obtained
from our dynamic age structure model when there isno environmental fluctuation.
Another main purpose of our analysis was to study
AP under fluctuating environments. In Model 2,
where environmental fluctuation affects inside andoutside work-capacities equally, non-AP is selected
when the magnitude of fluctuation is large, provided
that specialization effect is not so high [Table 2(a)].

There was also a general tendency for adaptive AP tochange from hard-AP to non-AP through soft-AP as
the magnitude of fluctuation increases. This result
suggests that AP is no longer adaptive whenenvironmental fluctuation exceeds a critical value. On
the other hand, in Model 3 where environmental
fluctuation affects only the efficiency of foraging,soft-AP is rarely selected and non-AP is never selected[Table 2(b)]. This result suggests that AP is always
adaptive however high the level of fluctuation.
Moreover, the adaptive AP allocates more workers
for inside labor, contrasting to the tendency in Model
2 where AP changes from hard to soft, as the level offluctuation increases [Fig. 3(b)].
What causes differences between the results ofModels 2 and 3? There are two factors enhancing
worker productivity: one is the balance of inside vs.
outside labor and the other is the size of the total
work-capacity (the sum of inside and outside
work-capacities). Specialization and foraging mortality effects favor drastic task switching (hard-AP),
which leads to increased total work-capacity understable conditions. Therefore, hard-AP will be more
adaptive than soft-AP if the level of fluctuation is low.
In Model 2, the ratio of the efficiencies between insideand outside labors stays constant even though themagnitude of the efficiencies fluctuate. In a hard-AP,
the actual balance of labors varies due to fluctuations
in age structure, which means a decrease inproductivity. Therefore, soft-AP will become more
adaptive as the magnitude of fluctuation increases
because soft-AP can ameliorate the actual imbalance
resulting from fluctuations in age structure. In otherwords, balance of labor becomes a more dominantfactor than total work-capacity. Finally, whenenvironmental fluctuation exceeds a certain level,
non-AP is adaptive in which the balance of labors canstay optimal independent of the magnitude offluctuation. In Model 3, the ratio of efficiencies ofboth labors itself fluctuates, and thus no LAP canmaintain the actual balance of labors at an optimallevel [Fig. 1(e, f)]. This element does not exist in
Model 2 but is an important factor in Model 3.
Because of this element, soft-AP can not ameliorate
the actual imbalance resulting from fluctuations inage structure. Therefore, adaptive AP will be selected
so that total work-capacity is maximized. In otherwords, total work-capacity becomes a more dominantfactor than balance of labor. This is why hard-AP wasselected in almost all regions of parameter space,including regions with highly fluctuating environmental conditions.
The fluctuation of age structure is caused by thefluctuation of the total reproductive success over time     
because the individuals in different age classes are theresults of reproductive successes at different time
points. This total fluctuation of reproductive successover time is stronger in Model 2 than in Model 3[Fig. 1(c, e)] because both tasks are fluctuating inModel 2 but only one task is fluctuating in Model 3.
This fact also supports that hard-AP tends to beadaptive in Model 3.
One important finding of our study is thatenvironmental fluctuation is not always effective in
disfavoring the age-dependent division of labor. It
depends on the pattern of fluctuating environment. In
nature, fluctuations may involve environmental
factors of both types simulated in Models 2 and 3.
Our results indicate that adaptive AP differs
depending on which of the two types of factors isstronger. This suggests that we must consider effectsof fluctuating environmental factors on different
labors as well as fluctuation amplitude, foraging
mortality and specialization effect when we study theform of temporal division of labor in social insects.
The unit of time in our model is the time which anindividual takes in order to grow up from an egg toa worker. Available data indicate that this time isabout 1–6 month(s) in ants (Hölldobler & Wilson,
1990, p. 169, Table 3–3), for example 51 days on
average in Diacamma sp. (Nakata, 1996). We
assumed six worker age classes so that each worker
can potentially survive for about 6 months to several
years. Such potential life spans of workers seemrealistic for most ant species (Hölldobler & Wilson,
1990), for example about 300 days in Diacamma sp.
(Tsuji et al., 1996). The ratio of developmental time163

and potential life span in other social insects may
differ from six depending on species but the actual
value of the ratio is not essential to our analysis. The
time scale of fluctuation of our model is also based on
this unit of time since independent random variables
are given in the interval of the time unit. However,slower fluctuation may be realized by putting positive
correlation between adjoining R values, Rt and Rt + 1 .
In this case, age structure will be more stabilized thanin the present model because fluctuation of total
work-force is slower. Thus slower fluctuation may
prefer a hard AP form rather than a soft AP formby the similar reason as discussed on the result ofModel 2.
We have modeled AP with two labors for
simplicity. However, division of labor consisting ofmany kinds of labors is known, especially in ants andhoneybees. In most cases, inside labor consists ofdivision of labor among 5–20 kinds of tasks (ants:Hölldobler & Wilson, 1990; honeybees: Seeley, 1982).
What AP is adaptive for a division of labor within
inside labors in a fluctuating environment? We may
compare this situation to Model 2 with a small a value
when we assume that mortality differs slightly
depending on the kinds of inside labors andenvironmental fluctuation affects the efficiencies of all
kinds of inside labors equally. In Model 2, the regionin which soft-AP or non-AP is adaptive is larger thanin Model 3 (Table 2). Assuming that environmental
fluctuation affects the efficiency of inside labors andoutside labor differently, division of labor among
inside labors is comparable to Model 2 and divisionof labor among inside and outside labors is

Frequency

Inside C

OutsideInside B

Inside A

AgeF. 4. Theoretically suggested pattern of temporal division of labor of inside labor (A, B, C) and outside foraging labor. The switchfrom one inside labor to another is gradual while the switch from an inside labor to an outside labor is abrupt.

164

. .    .

comparable to Model 3. We suggest therefore that apattern of TDL as shown in Fig. 4 is theoreticallyreasonable. In fact, task switching between insidelabors has been observed to be more gradual than
the generally sudden change from inside labor tooutside labor (ants: Wilson, 1976; honeybees: Seeley,
1982).
We have assumed no variation among workers inthe same age class. However, this may not always be
the case (e.g. Corbara et al., 1989; Nakata, 1995).
Variation of the age at task switching can result in a
gradual change of labors in the whole colony(soft-AP) even when each individual switches labors
drastically. For example, in our model x2 = 0.8 meansthat all workers in age class 2 work inside with aprobability at 80% and outside with a probability at
20%. The same division of labor in age class 2 may
be realized in a way such that 80% of workers in age
class 2 labors only inside and 20% of workers labors
only outside. All workers enjoy a maximum
advantage of specialization when the specialization
effect occurs at the level of individuals. Our model canrepresent the latter case if we put a = 0 and take b to
be a base work-capacity plus additional workcapacity due to specialization. The latter case is
obviously more adaptive and this would be the reason
why variation among social insect workers of thesame age may be expected.
In general, division of labor is a productive systembecause the efficiency of each task is enhanced due tospecialization. However, this system is very sensitive
to the balance of labors. Lack of one task may imply
a bottle neck in the system and lower productivity of
the whole system. This situation is best described byusing a minimum function in a mathematical model.
This is why we applied this minimum function
technique in spite of the difficulty to analyse thesemodels analytically.
Patterns of AP can be regarded as life history
strategies which are selected at the colony level.
Although many studies on life history strategies ofanimals have analysed age-structured populations(Stearns, 1992), there are few studies on social
animals in this framework (but see Tsuji & Tsuji,1996). The present study using age structure dynamics
is the first attempt to do such explicit analysis ofadaptive strategies of caste organization in social
insects. This method would be applicable to various
other problems in the evolution of individual
behaviors and colony organization in social insects.
We thank M. Higashi for many useful suggestions andcooperation. We thank the members of the Center for
Ecological Research, especially J. Elser for improvement ofEnglish and valuable comments, A. Ushimaru for help withtables and figures and T. Abe for helpful suggestions. We
are grateful to T. Day and the other anonymous referees fortheir precious advice and for many careful corrections ofEnglish. We also thank K. Tsuji and E. Hasegawa for theirvaluable advice. We used gcc on Linux for computer
simulations and we thank all developers and programmers
who support Linux, the GNU project and other useful
freewares. This work is partly supported by a Grant-in-Aid
for Scientific Research from the Japan Ministry of
Education, Science, Sports and Culture (No. 0864810) andby a Inoue Research Award for Young Scientist.

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