Behavioral Ecology Vol. 9 No. 4: 376-383
Queuing for territory positions in the lekking
black grouse (Tetrao tetrix)
Hanna Rokko,* Jan Iindstrom,' Rauno V. Alatalo,b and Pekka T. Rintamaki0
•Department of Ecology and Systematics, Division of Population Biology, P.O. Box 17, FIN-00014
University of Helsinki, Finland, b Department of Biological and Environmental Science, University of
Jyvaskyla, P.O. Box 35, FIN-40351 Jyvaskyla, Finland, and department of Zoology, VUlavigen 9,
S-75236 Uppsala, Sweden
The importance of a central territory position as a determinant of male mating success in lekking species has been debated
The process by which a male can obtain a central territory has been especially lacking a quantitative analysis. We present a
stochastic queuing model describing territory succession toward the lek center and apply it to a 8-year sequence of territory
centrality measures on a black grouse Tetrao tetrix lek. The lek shows a value of intermediate queue discipline, which deviates
significantly both from strict orderly queuing and from random ranking of males. Thus, high-ranking males are partly able to
maintain their superior position over years, but queue-jumping is not excluded, especially because highly successful males do
not attempt to change their territory position toward the lek center. As a result of stochastic queuing, a central territory indicates
an older than average male, as well as a male with a history of high fighting rates. These results are consistent with the hypotheses
that territory position is an honest signal of male quality and that the long-term fighting effort and survival required to acquire
a central territory may increase the reliability of die signal over that of short-term display effort The impartial queue discipline,
however, also leaves room for other male characteristics to play a role in determining individual mating success. Key words:
black grouse, honest signaling, lekking, queuing theory, territory succession, Tetrao tetrix. [Behav Ecol 9:376-383 (1998)]
N
umerous studies have been devoted to detecting correlates of mating success of lekking males (reviewed in
Hoglund and Alatalo, 1995). Although in some species clear
morphological characters have been found to influence female choice, such as the long tail in Jackson's widowbird Euplectes jaduoni (Andersson, 1989,1992), such evidence is lacking in other species. Another major group of traits that have
often been reported to correlate with mating success are behavioral cues such as display rates (e.g., Balmford et aL, 1992;
Gibson and Bradbury, 1985). The reliability of such traits as
cues for quality is supposed to be based on costs such as energy expenditure (Haiti day, 1978). However, die reliability of
short-term observations of male behavioral characters may suffer from males adjusting their advertisement to a higher level
when females are present (Wiley, 1991). Using such traits as
cues requires balancing the costs of spending time to assess
males and the expected benefit obtained (see also Luttbeg,
1996).
In some species, females also seem to pay attention to territory features, such as the position within a lek (e.g., Balmford et aL, 1992; Chitton-Brock et aL, 1988). Sometimes it is
easy to recognize the proximate benefits of occupying a specific territory, like one located next to the path used by females that enter the lek (Apollonio et aL, 1990). However, a
more commonly reported correlate of mating success is territory centrality. Several studies claim to show centrality effects
(e.g., Balmford et aL, 1992; Clutton-Brock et al., 1988; Gosling
and Petrie, 1990; Gratson et al., 1991; Hoglund and Lundberg, 1987; Trail and Adams, 1989; van Rhijn, 1991; Wiley,
1973), but others have been unsuccessful in finding an effect
or show in«t«H other links between female choice and spatial
Address correspondence to H. Kokko, Department of Zoology, University of Cambridge, Downing Street, Cambridge CBS 3EJ, UK.
E-mail; h tnfctottrrtn r-atn m- n|r
Received 5 December 1996; first revision 19 August 1997; second
revision 6 January 1998; accepted 7 January 1998.
O 1998 International Society for Behavioral Ecology
characteristics of territories (e.g., Apollonio et al., 1990; Qutton-Brock et al., 1989; Gibson and Bradbury, 1985; Hill, 1991).
Often central males are reported to be heavier and/or older
than peripheral ones (Balmford et aL, 1992; Hoglund and
Lundberg, 1987; Tsuji et al., 1992, 1994), and they may also
show higher testosterone levels (Alatalo et aL, 1996). Hence,
centrality has been suggested to function as a cue for high
quality of the male (e.g., Alexander, 1975; lill, 1974). Other
alternatives include the possibility that centrality becomes correlated with mating success only because less attractive males
gather around more attractive ones (Beehler and Foster, 1988;
Bradbury and Gibson, 1983; Hoglund and Robertson, 1990)
or that females prefer to mate on positions that are safe from
prcdation (Gosling and Petrie, 1990).
Territory position is a phenotypic property of an individual
that is Ukely to undergo less rapid changes than momentary
behavior (though probably more rapid than morphological
traits); this could increase reliability in terms of repeatability.
On the other hand territory movements may function to pick
out the most viable males, if changes occur in a successive
manner such that only die most viable males will live long
enough to arrive at the center. Such a correlation, however,
is presumed to be weak due to the stochastic nature of the
"queuing" process (Wiley, 1991). The lek system of longtailed manalcin« Chiroxiphia Unearis resembles most prominently a queue toward increased mating success (McDonald
1993). Only males of alpha status copulate, and the transition
to alpha status requires queuing through a beta status after
several predefinitive plumages. In the m^natin system, however, territory position is not an indication of alpha status.
In other species, much research has focused on die quesa»a ef whether lekkisg males move their territories toward
the lek center if possible. Here again, results are mixed (Apollonio et aL, 1989; Avery, 1984; Fryxell, 1987; Hoglund and
Robertson, 1990; Knapton, 1985; Movies and Boag, 1981; Rintamaki et aL, 1995a; Rippin and Boag, 1974; Stiles and Wolf,
1979; Wiley, 1973, 1991). To reach a better resolution of the
problem, there is a need to quantify a supposed queuelike
Kokko et al. • Queuing for lek territories
377
1987
1990
a
<D
1991
1992
1993
1994
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 18 17 18 19 20 21 22 23
Current distance rank
Figure 1
The observed development of distance ranks on the Periijavesi black grouse lek over years 1987-1994. Each male is presented either with
"n," denoting a newcomer, or with a number that gives his expected distance rank according to his previous distance rank and the number
of more central males that are no longer present on the lek. For 1987, previous attendances are not known. Observations of the same male
are connected with thin lines, and tied ranks (equal distances to lek center) are represented by with bold lines.
behavior of the lek and provide a significance test for the
measure obtained. Up to now, studies describing queuelike
successions have been lacking a rigorous model to characterize the stochasticity of the system, so the presence and degree
of true queuing and its importance has been hard to judge.
Here we present a method of assessing impartial queue discipline (c£ Kingman, 1982) of a lek in the presence of stochasticity and apply it to a 8-year data set of black grouse
Tetrao tetrix territory positions on a lek. The black grouse is
an example of a species in which morphological cues seem to
play only a minor role, despite an intensive long-term study
seeking for such correlates (reviewed in Hoghind and Alatalo,
1995). TniffM^, females seem to favor dominant males that
fight much yet keep their plumages intact (Alatalo et aL,
1991). Also, both observations of natural black grouse leks
(Hovi et aL, 1994; Rintamaki et aL, 1995ajb) and an experi-
mental study (Hovi et aL, 1994) suggest that females show a
preference for mating on central territories in this species.
However, because black grouse females also seem to favor
large leks (Alatalo et al., 1992), it is difficult to distinguish
between effects of group size and centrality with this kind of
experimental approach where group size is manipulated. In
any case, as both territory location and fighting may contribute to the mating success of black grouse males, the black
grouse system should provide an ideal test for evaluating the
possibility of queuing and the implications it has for femalechoice cues.
METHODS
A black grouse lek was observed from 1987 to 1994 in Teerijarvensuo, Petajavesi, central Finland (62°10' N, 25°05' E).
Behavioral Ecology Vol. 9 No. 4
378
Males were captured, weighed, and color ringed at a feeding
site during winter. We recorded male display behavior, mating
success, and territory use during approximately 10 days including peak mating activity each spring. The positions and
activity of each male on the arena were measured to within 1
m with the help of a 10 m X 10 m grid marked with sticks.
The center of die lek was denned as the median for all male
observations along two perpendicular axes of the grid. The
center of each male's territory was similarly derived as the
median of male positions in each year. For a more detailed
description of the observation methods, see Rintamiki et al.
(1995b).
In all following analyses, the male nearest the lek center is
given a rank of 1. Rank progress is defined as the difference
in rank number in subsequent years (positive values denoting
progress toward central positions). The data indicate that
changes in rank toward die lek center are significantly more
prevalent than vice versa (mean rank progress • 3.6 ± 2.0;
progress more often positive than negative, binomial ttstlp <
.001), However, such a result can be expected if the lek has
been decreasing in size over years, which is true for die lek
in question (Figure 1). Detecting queue discipline therefore
requires a more detailed analysis of die rank progress of individuals.
Measuring the queue discipline of a lek
We model die spatial structure of die lek as a queue, where
males ideally move in orderly rank toward central positions.
The lek center is defined yearly by die horizontal and vertical
median of all male observations, and males are ordered according to die distance of dieir territory location (median of
male positions) from die lek center of that year. Ranks are
averaged over several males if they possess territories at equal
distance from die lek center.
Queuing dieory (e.g., Bunday, 1996) is a subject of operations research that has been applied, for example, to teletraffic dieory, where predictions like average waiting times are
derived for a customer-server system with a given "queue discipline." The queue discipline specifies die order in which
customers are served, and die strictest queue is described with
a FIFO, "first in, first out" discipline (e.g., Medhi, 1994).
Equating customers widi lekking males and die goal (die
"server") as die lek center, die interpretation of a FIFO
queue for a lek system is "first in, first to center (if still alive)."
A pure FIFO queue would behave in a weD-disciplined way.
surviving males move toward central territories in an orderly
manner, and new males always obtain the most distant territories. Given diat territories o n a lek are located o n a twodimensional ground and do not have fixed boundaries over
years, it is not surprising that die observed pattern of black
grouse does not correspond to a strict FIFO queue (Figure
1). The question then becomes whether a certain amount of
order is still maintained, so d m males increase their chances
of obtaining good territories in die future if diey have maintained territories in a previous year.
In a FIFO queue with strict discipline, an open position in
die queue will always be occupied by die male next in die
rank order. Therefore, given that deaths of males are known,
an expected distance rank can be calculated for all die remaining males (Figure 1). Far oxample, die male widi rank 5
in 1987 had die expected rank value 2 in 1988 because three
of his predecessors had died. Due to high success of some of
die newcomers, however, his actual distance rank had a value
of 6 in diat year. T h e stricter die queue discipline, die smaller
and less likely such deviations between expected and observed
rank should be.
That some degree of orderliness is maintained in die queue
Table 1
Spearman's rank correlations (r,) between the expected and
observed nnk dbtance and between rank «
d
*
mccen in the black grouse lek data
Year
Expected
and
observed
distance
rank
1987
1988
1989
1990
1991
1992
1993
1994
.40
30
.89
.46
36
30
.30
p
Distance
and matins
success
p
<.O5
<.O5
<.01
ns
<.O5
ns
ns
-38
-.48
-.62
-.73
-.46
-30
-.83
-.09
<.O5
<.O5
<.01
<.01
ns
<.O5
<.01
ns
is seen by comparing observed and expected ranks of die
black grouse queue (Figure 1). Giving newcomer males a
greater expected distance rank value than any of die males
widi previous attendance. Spearman rank correlations between observed and expected ranks are positive in all years,
though not significant in some years widi fewer than 10 males
on die lek (Table 1). Also, die distance of die territory from
die center is negatively correlated widi mating success (Table
1). However, these tests may lack sufficient power because diey
use data from single years only, and diey do not allow two
possibly different processes diat contribute to die queue discipline to be distinguished: (1) whether males widi previous
attendance have a relative benefit over newcomer males, and
(2) whether die queue is maintained among previously territorial males also, such diat die future success of a male depends on his previous rank.
Kingman (1982) describes die general setting of stochastic
queues, ranging from ROS ("random order of service"; Carter and Cooper, 1972; Kingman, 1962), to queues widi impartial queue disciplines. An impartial queue discipline is defined
by specifying die probabilities, pb of a male of rank i being
"served" (Le., acquiring a territory). The degree of discipline
can be quantified by die parameters describing die distribution of pj, such diat die ROS queue corresponds to a uniform
distribution.
Because die queue discipline on a lek should depend both
on die difference between newcomer versus previously territorial males and on die inequality of probabilities, pb among
territorial males, we chose the combined usage of two discipline indices, P and Q, to define die distribution of pi.
p, ~ £,«'<«-»
for males for which die expected rank, E»
can be calculated, and
pj = Pp~i
for newcomer males, where p~{ is die average
of p, values for males widi previous territories.
The newcomer priority index, P, defines die disadvantage diat
a newcomer male suffers compared to die average previously
territorial male. The extreme P ~ 0 implies diat no newcomer
male can ever bypass a previously territorial male in die
queue, P » 1 means that uicre is no disadvantage for newcomers, and P > 1 implies diat a male benefits from not having had a territory previously. The rank priority parameter, Q,
defines die shape of die distribution for previously ranked
males: Q • 0 leads to no effect of previous distance rank, and
values approaching 1 imply ever-stricter priority of die male
diat is "in line" to obtain a territory according to a FIFO
Kokko et al. • Queuing for lek territories
1
379
1
(A)
0.8
0.8
* °-6
0.6
O
S
<D
C
OJ
0.2
0.2
TTTTTTTTTT
0
1 2 3 4 5 n n n n n
1 2 3 4 5 n n n n n
o
5
figure 2
Examples of the priority distributions (P, Q) for a lek of five
males with previous rank distances |1, 2, 3, 4, 5), and five
newcomer males (n). (A) Random order of service queue,
(P. Q) - (1,0): all males have
an equal probability of obtaining a desired territory. (B)
First-in, first-out queue, (P, 0
*• (0, 1): males obtain territories in a strict orderly manner,
the best-ranking male obtaining the best territory with
probability 1. (C) No queue
discipline among previously
territorial males, but strongly
reduced success for newcomers, (P. Q) - (0, 0). (D) An
intermediate value (P, Q) "
(0.28, 0J5S), equal to the estimated value for the black
grouse data set. In each case,
when the first territory has
been assigned to a male, the
ranks are formed again for
opting for the next territory,
until all territories are occupied.
0.4
0.4
a
(B)
1
(C)
0.8
0.8
0.6
0.6
0.4
0.4
(D)
2
0.2
0
inn
0.2
i T t T t t t f t
0
1 2 3 4 5 n n n n n
1 2 3 4 5 n n n n n
Male rank
discipline. Examples of distributions specified by Pand Qare
given in Figure 2.
We next seek the maximum likelihood estimates of P and
Q for the black grouse queue. The likelihood of an observed
occupancy pattern, such as [4, n, 1, 2, 4, 4, 5, n] in the year
1994 is obtained through the following procedure:
1. For the previously territorial males with expected ranks
[4, 1, 2, 4, 4, 5] (note the ties arising through equal distances
in the previous year 1993), the relative probabilities of success
are [4<y«H>, l<y<e-», 2<y<G-"( 4<y<G-1>, 4 <#«*•», 5 <y<G-»]. The
probability of success for each newcomer is a fraction, P, of
the average success of established males: P{4<*«i-»+lW<*-»+2<*
<Qo(y(Ci>
<yen
yc<
2. The log-likelihood of obtaining a male with rank 4 to
position 1, as was observed in this sequence, is
1
FIFO
0.75
90
94
a.
c
0.5
92
<0
O 91
rr
0.25
pi of observed male
T pj of potential males
; ) •
3. The likelihood, L*, of the second position being occupied by male n is computed likewise. However, because the
male that was assigned to a territory in step 2 is not competing
for the current position, his probability, pb is first removed
from the denominator.
4. The process continues' until the likelihood, L» is computed for all positions. The total likelihood of the observed
sequence i s L c X Z « and it is a function of the parameters
P and Q. L values can be evaluated numerically for a grid of
P and Q vahies, or evaluated analytically with a suitable computer program such as Mathematica. The point (P, Q) that
maximizes L forms the mairimiim likelihood estimate for the
observed queue.
For years with ties, we calculated the (P, Q) pair for males
ROS
0.25
0.5
0.75
Priority of newcomer males, P
Figure 3
Estimated queue discipline (P, 0 pairs for the black grouse lek.
Upper left corner (P ~ 0, Q ~ 1) represents completely strict
queue discipline, first-in, first-out (FIFO), whereas queue discipline
is completely lyrVing (random order service, ROS) in the lower
right corner (P » 1, Q - 0). Fuled squares denote the (P, Qf
values estimated for the whole sequence of observations; open
circles respresent the ^ttimaf^ for individual years. Connected
markers give two estimates if the data include tied ranks.
Behavioral Ecology Vol. 9 No. 4
380
1988
1989
0.1 -
0.1 -
1991
0.1 -
syO
0.1 -
o
cz
a>
3
CX
1992
0.1 -
0.1 -
-1
1994
-0.5
0
0.5
1
Spearman's rank correlation, r,
0.1 -
0.5
Spearman's rank correlation, r,
Figure 4
Verification of the performance of the estimated queue model. The model arranges a known number of newcomer males and males with
previous territories into a distance sequence according to the probability distributions (see Figure 1) using either the null model (P = 1, Q
= 0; dotted line) or the model estimated for the black grouse lek (P = 0.28, Q •» 0.53; solid line). This arrangement is repeated 10,000
times for each year, collecting the emerging rank correlation values, rr The observed rank correlations of the black grouse lek are indicated
by arrows.
ordered both according to the most favorable interpretation
for queiiing (as in Figure 1) and by reversing the order within
tied ranks that leads to a least favorable interpretation. We
also used this approach to estimate the single (P, Q) value
that maximized the likelihood of observing the whole 8-year
data set The UkeHhood ratio test (e.g., EHason, 1993) was
then used to test for a significant deviation from the null modr
e l / & (i> 0 - (1.0).
RESULTS
The discipline parameter pair (P, Qj that is found to best
describe die whole sequence of black grouse lek observations
hes between (0.25, 035) (most favorable ties) and (030,030)
(least favorable ties). Both estimates are highly significant (p
< .001), and we use the average value (P, Q) =» (0.28, 0.53)
in further analysis.
Estimating (P, Q) values by using only single-year data gives
yearly fluctuations that are fairly consistently located around
this value (Figure 3). The estimated value of the newcomer
priority, P, lies dearly below 1 in all years. This reveals the
disadvantage of being a newcomer on a lek. Likewise, the estimated rank priority value, Q, is positive in each year, such
that territories are distributed nonrandomh/ among previous
territory owners. These short-term results, however, become
significant only after pooling data for at least 2 or 3 years.
An additional test for the queue model is to verify that it
can explain the ebsorved rank correlations between expected
and observed distance ranks (Table 1). If correct, the model
should give correlations that resemble those of the observed
data. Here, the maximum-likelihood discipline parameters (P,
Q} « (0.28, 033) perform well in reproducing die observed
correlations in each year (Figure 4). The estimated model can
produce the observed values of r, in all years, which is not
381
Kokko et al. • Queuingforlek territories
true for the null model of no queuing (likelihood ratio test
using estimated r, distributions, p < .0001). dearly, while far
from being a perfect queue, the black grouse lek system shows
a tendency toward disciplined queuing, indicated both by bad
performance of newcomer males and by high rank priority of
established males. Both components contribute to the discipline of the queue, so that queue jumping occurs significantly
less often in the black grouse queue than in a ROS queue.
There remains the question of what factors determine the
success of individual maU« Deviations both from strict queuing and random ranking have to be explained. Can a male
perform purposeful queue jumping, or do deviations from
orderliness merely indicate random effects of lacking queue
discipline? Further, if such purposeful jumping is possible,
why is it not sufficiently strong to destroy the remaining queue
discipline as well?
Highly successful males did not shift their territories between years even though the lek center moved (Rintamiki et
aL, 1995a), leaving the potential for younger males to enter
the central region of the lek. On the odier hand, fight rates
are generally higher at the center of the lek, where a male
has more neighbors (Spearman correlation betweenfightrate
and number of neighbors: r, = .56,p< .0001, n = 97; Spearman correlation between distance from lek center and number of neighbors: r, = -.79, p < .0001, n = 100; resulting
relationship between fight rate and distance, see Figure 5a).
There is a Tignifirent correlation between body weight and
fight rate in both newcomers and established males (newcomers: r, = .40, p = .0097, n = 34; established males: r, = .26, p
= .0305, n - 53; all males pooled: r, = .32, p = .0007, n =
101; all tests one-tailed). The weight of males also increases
with age for the first few years of life (Figure 5b), so that
increasing age may allow a male to become a more enduring
fighter.
0.6
(A)
o
0.5
UT 0.4
« 0.3
2=
0.1
* o o
0
10
30
40
Distance to lek center (m)
1.4
1.3
"3
2
J?
1.2
ID
1.1
1.0
2
DISCUSSION
Queuelike systems, where timing of setdement affects future
reproductive success or dominance relationships, are widespread in die field of behavioral ecology. The descriptions
range from short-term queues such as thirteen-Uned squirrels
Spermophihis tridtcsmHneatus lining up for raatings (Schwagmeyer and Parker, 1987), to persistent delays in reproduction,
such as deferred reproduction when waiting for suitable territories in oystercatchers Hamatopxu ostraJegus (Ens et al.,
1995), to die evolution of cooperative breeding by waiting for
favorable social positions (Wiley and Rabenold, 1984), and to
die formation of dominance hierarchies of hyenas Croatia
crocuta (Frank, 1986; Smale et aL, 1993) and boat-tailed grackles Quiscahu major (Poston, 1997).
Our results indicate diat die lek system of die black grouse
also has properties that can be described as a stochastic
queue. Why should lekking males obey queuing rules? We
illustrate die possibility of obtaining high disciplines by applying the idea of ownership games (Maynard Smith, 1982) to
display territories on a lek. Assume diat two males are competing for a specific rank position diat provides a fitness increase, V. Further, assume that diere is a prior ranking such
diat if no fight occurs, male 1 will gain die position. A male
may play eidier strategy A, "try to gain position no matter
what your previous rank," or B, "try to gain position only if
it is your turn in the queue." If bodi males try to gain die
position, afightoccurs. In a fight, bodi males suffer a viability
cost, Q which reduces dieirfitness,and diey have equal probabilities of achieving die desired position.
By seeking die evolutionarily stable strategy (ESS) solutions
of die game (Table 2), one can show diat all males will obey
the queue discipline if V/2 < C In diis case enhancing lon-
20
3
4
5
6
7
Minimum age (years)
Figure 5
Relationships of male character! in newcomer* (open circles) and
males with previously established territoriei (filled circles). (A)
Relationship between distance to lek center and fight rate;
Spearman rank correlation r, - - .51, p < .0001, n - 104. (B)
Relationship between minimum age of male (determined by yean
of observation, setting the minimum age to 2 yean for newcomer
males with fully developed plumage) and weight, records of the
same individual connected with lines; r, ** .31, p a .0006, n « 108.
Half-filled circles refer to the year 1987 with unknown previous
attendances.
gevity is die only option for increasing fitness. In contrast, a
fight always ensues if V/2 > C. This will destroy die queue
because die male winning the rank position is now randomly
determined; any increase in fighting success is now favored by
selection. Extending die pairwise contest to a group of N
males, a suffidendy high mating advantage (V/N > Q will
make it beneficial for all males to join in die fight, which
destroys any queue discipline if fighting capabilities are equal
(see also Maynard Smidi, 1983).
That die black grouse lek shows values of queue discipline
deviating from bodi 0 and 1 indicates diat neidier ESS is directly applicable. The two-dimensional structure of a lek arena
presents one possible cause of this: it is not hard to perform
queue jumps if die nearest competitor in die queue is, in fact,
located at a similar distance from die center but on die other
side of die lek. Estimates of a discipline value assuming a linear queue toward better territories are therefore bound to be
conservative. An additional factor diat may reduce queue dis-
582
Behavioral Ecology Vol. 9 No. 4
TableZ
Payoff matrix for the queue game (cf. Maynard Smith, 1982)
Male i, i i 2
Male 2
Male 1
ifN,- 0
(V,0)
if N, > 0,
(*-«•)
if/v>- 0
Left A fight occurs with a cost, Q to both players, if male 2 plays strategy A (attack); otherwise, male
1 obtains the mating advantage, V, with no cost The payoff rows are identical for male 1, so that male
2 will determine the outcome of the game. If V/2 > C, male 2 will always play A, and a fight emerges.
If V/2 < C, strategy B is dominant over A for male 2, and a strict queue discipline is established with
male 1 obtaining the advantage of first access to the rank position with die mating advantage V. Right:
In the payoff matrix for a game with N queuert, N, is the number of males fighting, and the benefit
of fighting decreases with increasing Np. If V/N > C, all males attempting to fight becomes an
evolutionarily stable strategy (strategy A is still beneficial if N, *° N).
ripline is that centrality is not the only criterion of success. In
the black grouse system this is shown in the fact that previously highly successful males do not attempt to further proceed in the queue. However, because the estimated (P, Q)
value still deviates significantly from a ROS queue, a component of queuelike behavior is present in the black grouse system.
The degree of queue discipline affects the optimality of
strategies of males waiting for success. It is plausible that longevity should be selected for in a queue with high discipline:
only males that are able to queue for a long time can successfully mate. On the other hand, low queue discipline suggests—all other factors being equal—higher importance of
competence in short-term competition, even at the expense
of future survival. A theoretical model shows that the reliability of viability indicators may be reduced if females can
assess short-term display effort only, whereas males adjust their
effort according to long-term trade-offs between display effort
and survival (Kokko, 1997). However, as die black grouse system deviates from both the FIFO and ROS extremes, a successful male strategy will have to combine both longevity and
a long-term effort to outcompete other individuals, assuming
that females use centrality as a choice criterion.
The constancy of the queuing rules, together with the increase of fight rates toward the lek center, mean that a central
position indicates that the male has experienced higher fight
rates in the past as well as in the present The finding that
males defend their territories in the autumn as vigorously as
during the mating period in spring and that territory centrality during the autumn correlates with copulatory success
the following spring (Rintamaki FT et al., unpublished manuscript) , adds further to the constancy requirement of male
effort. Additionally, as a consequence of a rigninrant queue
discipline, a central male will also be on average older than a
peripheral one. Thus, the position of a permanent territory
may indicate male age as an index of viability as well (Kokko
and Iindstrom, 1996).
Taken together, the reliability of observing territory position as a quality indicator is likely to be greater than that of
short-term male display or fight effort. According to the honesty principle of sexual advertisement, males should vary in
their advertisement according to their true quality or resources (Grafen, 1990a,b); however, the reliability of the signal can be limited if a female can assess the performance of
males only for a short time (Kokko. 1997; Luttbeg, 1996). Ul-
timately, whether a given signal is reliable depends on the
balance of costs and benefits that males face. Regarding fighting, the game theory model (Table 2) suggests limited fighting if it incurs high costs for the male; for males in weaker
body condition, the importance of such costs may be enlarged. Indeed, in the black grouse, male fight rates are related to viability (Alatalo et al., 1991). Viewing territory position on a lek as a potential signal for females, it becomes clear
that central males are those who are able to allocate enough
resources both to survival and to territory defense from year
to year. Thus, female preference for centrality can create a
competition where territory position itself becomes a viability
indicator—a sexually selected trait which becomes visible in
group displays only. Being related to success in male-male
competition, it is also an example of an "armament ornament" (Berghind et al., 1996). The resulting queue discipline
then indicates updates of a long-term trait value according to
the history and interaction of the males in question, not randomly determined changes in die spatial arrangement on die
lek.
We thank G. A. Parker, W. J. Sutherland, and three anonymous referees for comments on earlier versions of this manuscript, and the
HCM program of the European Community and the Academy of Finland forfinancialsupport
REFERENCES
Alatalo RV, Hogtund J, Lundberg A, 1991. Letting in the black
grouse—a test of male viability. Nature 352:155-156.
Alatalo RV, H&gtund J, Lundberg A, Rintamiki FT, SUverin B, 1996.
Testosterone and male mating success on the black grouse lets.
Proc R Soc Lond B 263:1697-1702.
Alatalo RV. Hdghind J, Lundberg A, Sutherland WJ, 1992. Evolution
of black grouse lets: female preferences benefit males in larger
leks. Behav Ecol 3:53-59.
Alexander RD, 1975. Natural selection and specialized chorusing behaviour in acoustical insects. In: Insects, science and society (Pimen tel D, ed). New 'Ybrk: Academic Press; 35-77.
Andcrsson S, 1989. Sexual selection and cues for female choice in
leks of Jackson's widowblrd EuptUUs jadaom, Behav Ecol Se«aobiol
25:403-410.
Andersson S, 1992. Female preference for long tails in letting Jackson's widowbirds: experimental evidence. ArUm Behav 43:379-388.
Apollonio M, Festa-Bianchet M, Man F, 1989. Effects of removal of
successful males in a fallow deer lek. Ethology 83:320-325.
Apollonio M, Festa-Bianchet M, Man F, Riva M, 1990. Site-specific
Kokko et al. • Queuing for lek territories
asymmetric* in male copulatory success in a fallow deer lek. Anim
Behav 39:205-212.
Avery MI, 1984. Lekking in birds choice, competition and reproductive constraints. Ibis 126:177-187.
Balmford A, Albon S, Blakeman S, 1992. Correlates of male mating
success and female choice in a lek-breeding antelope. Behav Ecol
3:112-123.
Beehler BM, Foster MS, 1988. Hotshots, hotspots, and female preference in the organization of lek mating systems. Am Nat 151:203219.
Bergtund A, Bisazza A, Pilastro A, 1996. Armaments and ornaments:
an evolutionary explanation of traits of dual utility. Biol J Linn Soc
58:385-399.
Bradbury JW, Gibson RM, 1983. Leks and mate choice. In: Mate
choice (Bateson P, ed). Cambridge: Cambridge University Press;
109-138.
Bunday BD, 1996. Introduction to queuing theory. London: Edward
Arnold.
Carter JS, Cooper RB, 1972. Queues with service in random order.
Operat Res 2O-J89-W5.
Qutton-Brock TH, Green D, Hiraiwa-Hasegawa M, Albon SD, 1988.
Passing the buck: resource defence, lek breeding and mate choice
in fallow deer. Behav Ecol Sociobiol 23:281-296.
Qutton-Brock TH, Hiraiwa-Hasegawa M, Robertson A, 1989. Mate
choice on fallow deer leks. Nature 340:463-465.
Eliason SR, 1993. Maximum likelihood estimation: logic and practice.
Sage University Paper series on Quantitative Applications in the
Social Sciences, 07-096. Newbury Park, California: Sage.
Ens BJ, Weissing FJ, Drent RH, 1995. The despotic distribution and
deferred maturity: two sides of the same coin. Am Nat 146:625-650.
Frank LG, 1986. Social organization in the spotted hyena (Crocuta
crocuta). n. Dominance and reproduction. Anim Behav 34:15101527.
Fryxell JM, 1987. Lek breeding and territorial aggression in whiteeared kob. Ethology 75:211-220.
Gibson RM, Bradbury JW, 1985. Sexual selection in lekking sage
grouse: phenotypic correlates of male mating success. Behav Ecol
Sociobiol 18:117-123.
Gosling LM, Petrie M, 1990. Lekking in topi: a consequence of satellite behaviour by small males at hotspots. Anim Behav 40:272287.
Grafen A, 1990a. Biological signals as handicaps. J Theor Biol 144:
517-546.
Grafen A, 1990b. Sexual selection unhandicapped by the Fisher process. J Theor Biol 144:475-516. ,
Gratson MW, Gratson GK, Bergemd AT, 1991. Male dominance and
copulation disruption do not explain variance in male mating success on sharp-tailed grouse {Tympanuckus phasiantUus) leks. Behaviour 118:187-213.
Halliday TR, 1978. Sexual selection and mate choice. In: Behavioral
ecology (Krebs JR. Davies NB, eds). London: Blackwell; 180-213.
Hill WL, 1991. Correlates of male mating success in the ruff Philomachxu pugnax, a lekking shorebird. Behav Ecol Sociobiol 29:367372.
Hoglund J, Alatalo RV, 1995. Leks. Princeton, New Jersey: Princeton
University Press.
Hoglund J, Lundberg A, 1987. Sexual selection in a monomorphic
lek-breeding bird: Correlates of male mating success in the great
snipe GaOmago media. Behav Ecol Sociobiol 21:211-216.
Hoglund J, Robertson JGM, 1990. Female preferences, male decision
rules and the evolution of leks in the great snipe GfilHvngo media.
Anim Behav 45:15-22.
Hovi M, Alatalo RV, Hoglund J, Lundberg A, Rintamaki FT, 1994. Lek
centre attracts black grouse females. Proc R Soc Lond B 258 J 0 3 305.
Kingman JFC, 1962. On queues in which customers are served in
random order. Proc Cambridge Phil Soc 58:79-91.
Kingman JFC, 1982. Queue disciplines in heavy traffic. Math Operat
Res 7:262-271.
Knapton RW, 1985. Lek structure and territoriality in the chryxus
arctic butterfly, Otneis chryxus (Satyridae). Behav Ecol Sociobiol 17:
389-395.
Kokko H, 1997. Evolutionarily stable strategies of age-dependent sexual advertisement. Behav Ecol Sociobiol 41.-99-107.
383
Kokko H, lindstrdm J, 19%. Evolution of female preference for old
mates. Proc R Soc Lond B 263:1533-1538.
Iill A, 1974. Sexual behavior of the lek-forming white-bearded manakin (Manacus monacal trinitatis Hartert). Z Tierpsychol 36:1-36.
Luttbeg B, 1996. A comparative Bayes tactic for mate assessment and
choice. Behav Ecol 7:451-460.
Maynard Smith J, 1982. Evolution and the theory of games. Cambridge: Cambridge University Press.
Maynard Smith J, 1983. Game theory and the evolution of cooperation. In: Evolution from molecules to men (Bendall DS, ed). Cambridge: Cambridge University Press; 445-456.
McDonald DB, 1993. Delayed plumage maturation and orderly
queues for status: a marnkin mannequin experiment. Ethology 94:
31-45.
Medhi J, 1994. Stochastic processes. New Delhi: Wiley.
Moyles DLJ, Boag DA, 1981. Where, when, and how male sharp-tailed
grouse establish territories on arenas. Can J Zool 59:1576-1581.
Poston JP, 1997. Dominance, access to colonies, and queues for mating opportunities by male boat-tailed grackles. Behav Ecol Sociobiol
41:89-98.
Rintamaki PT, Alatalo RV, Hoglund J, Lundberg A, 1995a. Male territoriality and female choice on black grouse leks. Anim Behav 49:
759-767.
Rintamaki PT, Alatalo RV, H&gtund J, Lundberg A, 1995b. Mate sampling behaviour of black grouse females (Tttrno Utrix). Behav Ecol
Sociobiol 37:209-215.
Rippin AB, Boag DA, 1974. Spacial organization among male sharptailed grouse on arenas. Can J Zool 52391-597.
Schwagmeyer PL, Parker GA, 1987. Queuing for males in thirteenlined squirrels. Anim Behav 35:1015-1025.
Smale L, Frank LG, Holekamp KE, 1993. Ontogeny of dominance in
free-living spotted hyenas: juvenile rank relations with adult females
and immigrant males. Anim Behav 46:467—477.
Stiles FG, Wolf LL, 1979. Ecology and evolution of lek mating behaviour in the long-tailed hermit hummingbird. Ornithol Monogr 27:
1-77.
Trail PW, Adams ES, 1989. Active mate choice at cock-of-the-rock leks:
tactics of sampling and comparison. Behav Ecol Sociobiol 25:283292.
Tsuji LJ, Kozlovic DR, Sokolowski MB, 1992. Territorial position in
sharp-tailed grouse leks: The probability of fertilization. Condor 94:
1030-1031.
Tsuji LJS, Kozlovic DR, Sokolowski MB, Hansel] RIC, 1994. Relationship of body size of male sharp-tailed grouse to location of individual territories on leks. Wilson Bull 106:329-537.
van RhijnJG, 1991. The ruff: individuality in a gregarious wading bird.
London: Poyser.
Wiley RH, 1973. Territoriality and non-random mating in sage grouse,
Ctntrocercus wrophasumus. Anim Behav Monogr 6:85-169.
Wiley RH, 1991. Lekking in birds and mammals behavioral and evolutionary issues. Adv Study Behav 20:201-291.
Wiley RH, Rabenold KN, 1984. The evolution of cooperative breeding
by delayed reciprocity and queuing for favorable social positions.
Evolution 38:609-621.