Behavioral Ecology Vol. 10 No. 1: 73-79
© 1999 International Society for Behavioral Ecology
Male rank and optimal lek size
a School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK and b School of Biological Sciences, Woodland Road, University of Bristol, Bristol BS8 1UG, UK
Address correspondence to A. I. Houston. E-mail:A.I.Houston {at}bristol.ac.uk.
Received 1 October 1997; revised 12 March 1998; accepted 7 July 1998.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Widemo and Owens presented a model that calculates the expected copulation rates of males on leks of a range of sizes. They claim that a negative relationship between lek size and male mating skew will result in low-ranking males having greater optimal lek sizes than higher ranking rivals. Widemo and Owens offered no proof of their claim, and their model assumes that the rank of a male does not change as lek size increases, whereas in reality, rank may change as more males arrive. We present a general model that allows rank to change as lek size increases. We show that the crucial determinant of whether optimal lek size increases with male rank is whether relative competitive differences increase with lek size. Contrary to the claim of Widemo and Owens, the relationship between skew and lek size has no direct bearing on the optimal levels of aggregation of males of different rank. We show that a negative relationship between skew and lek size can exist even when high-ranking males have the greatest optimal lek sizes.
Key words: optimal lek size, reference rank, relative competitive difference.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Leks (see Höglund and Alatalo, 1995
; Wiley, 1991, for
reviews) are communal display grounds on which males defend small, clustered
territories that receptive females visit purely to select mating partners. In
lek-breeding species, mating success is typically variable, with some males
achieving a high proportion of the copulations
(Alatalo et al., 1992;
Mackenzie et al., 1995). We refer to the
more successful males as being of high rank, with the highest ranked male
labeled 1, the second highest ranked male labeled 2, and so on, where the rank
of each male could depend on his attractiveness or ability to dominate other
males.
There has been considerable interest in the evolution of lekking and in
particular into why low-ranking males aggregate on leks, despite the fact they
often perform so poorly in comparison with higher ranking rivals. Widemo and
Owens (1995) introduced an interesting
model examining the fitness consequences to males of differing rank in
displaying on leks of different size. In their model leks are regarded as
patches that differ in female encounter rate, where this encounter rate is
assumed to be an increasing function of current aggregation size. Widemo and
Owens used encounter rates consistent with empirical data from the ruff,
Philomachus pugnax, and calculate the expected copulation rates of
different ranks of male on leks of a range of sizes.
Widemo and Owens claim that the relationship between optimal lek size and
male rank is determined by how mating skew (Keller and
Vargo, 1993; Reeve and Ratnieks,
1993) and female visitation rate depend on lek size. In
particular, they state that a negative relationship between lek size and male
mating skew will result in low-ranking males having greater optimal lek sizes
than high-ranking males.
Widemo and Owens use reproductive skew in an inconsistent manner in their model. They assume that females arrive at the lek sequentially and visit the males in order of their position in the dominance hierarchy of the lek, mating with each with a probability proportional to the mating skew at that lek size. The problem with this is the skew generated by this mating pattern is in general inconsistent with the original skew used as an input to the model to determine the mating probability of each male on the lek.
Widemo and Owens use reproductive skew merely for the purpose of attempting to justify the mating probability they use in their model. This mating probability also relies on females visiting males sequentially in order of their position in the dominance hierarchy of the lek, something for which no evidence is given. These features are not essential components of the model. The basis of the model is the way in which lek size determines female visitation rate and the way in which lek size and male rank determine the probability that a given male mates with a visiting female. Widemo and Owens give no general account of how these two functions influence the relationship between male rank and optimal lek size, but analyze a specific case without any indication of robustness.
Furthermore, their approach amounts to assuming that the rank of each male
remains fixed as the aggregation size increases. However, in reality male rank
could decrease as lek size increases if more attractive or competitive males
enter the aggregation and displace resident males from their positions in the
success hierarchy. Admittedly, instances of this may be rare but in the ruff,
Hogan-Warburg (1966) and van Rhijn
(1991) observed new birds taking over
territories on leks. In addition, Lank and Smith
(1987) report considerable movement of
successful males between leks, with two males being observed to mate at more
than one lek. It is also worth noting that holding the top position on a lek
is energetically expensive
(Höglund and Alatalo,
1995; Wiley, 1991), and
so there is likely to be a turnover of males on the lek. Clutton-Brock et al.
(1993) review data from ungulates showing
that successful males do not hold their territories for very long.
A detailed model of the behavior of lekking males would have to include the choice of lek site by each male. Thus a game theoretic approach would be necessary. We believe, however, that the first step in an analysis of leks should be a general and complete account of a non—game-theoretic model based on the trade-off between increased male—male competition and increased female visitation rate with increased lek size.
Modeling male fitness and optimal lek size when success rank is not
fixed
We consider each male in the lekking population to have reference rank,
i0, which is an intrinsic measure of his success rank in
some reference population. We let r(i0,
n) denote the actual rank such a male will achieve when displaying on
a lek of size n. This rank will vary between courtship periods as a
result of stochasticity in the quality or competitive ability of other males
displaying on the lek at any one time. We denote by
R(i0, n) the average value of
r(i0, n), where this average is taken
over temporal variation in r(i0, n).
We will associate low i0 values with the most
attractive, competitive males in the population, who will attain high success
ranks whenever present on the lek, and we refer to such males as being of high
reference rank. Therefore 
. In
addition, for males of all reference ranks, we allow
R(i0, n) to increase with lek size
n, i.e., 
0, a consequence of more
attractive or competitive males entering the lek and relagating resident males
to lower positions in the success heirarchy.
The female visitation rate on a lek of size n is denoted by
f(n), and the expected mating success (or fitness) of a male
of reference rank i0 on a lek of size n is given
by F(i0, n). In general
 | (1) |
where expr(i0, n) denotes the average
probability that a male of reference rank i0 mates with
each female visiting a lek of size n. If P(r, n) is
the probability that a male of rank r mates with each female on the
lek, then
 |
By performing a second-order Taylor series expansion of P(r,
n) about R(i0, n), we have
 |
Thus if either (1) P(r, n) is a linear function of
r, or (2) Var[r(i0,
n)] = 0 so that E{[r(i0,
n) - R(i0, n)]p} = 0
for p = 2, 3,... then
 | (2) |
and
 | (3) |
otherwise these relations are only approximately true.
A typical feature of lekking species is that female visitation rate
increases with lek size (see Alatalo et al.,
1992;
Höglund et al.,
1993 for examples). This could be a consequence of females showing
an active preference for mating within large aggregations of males. A number
of hypotheses have been proposed to explain this possible female
preference; for example, females may gain a selective advantage through
the increased mate choice (Bradbury,
1981; Kokko, 1997) and
the increased protection from predators and nonterritorial males that large
aggregations afford (Clutton-Brock et al.,
1993; Wittenberger,
1981).
As a consequence we expect f(n) to be a monotone
increasing function of n. However, f(n) will not
increase indefinitely with n but will asymptote at a level controlled
by local female density, and so for each i0,
F(i0, n) will reach a strict optimum at
an aggregation of a certain size. We let n*
(i0) denote the optimal lek size of a male of reference
rank i0. Thus for each i0,
 | (4) |
Then n*(i0) = 1 is the trivial optimum of having maximum fitness when displaying alone (i.e., not lekking).
By definition, P(r, n) will be a decreasing function of
(i.e., 
), and, further, increased
male—male competition and the increased mate choice available to females
on larger leks ensures that P(r, n) will also be a
decreasing function of n. Thus Equation 3 clearly illustrates the
trade-off between the costs [decreasing mating probability, through increasing
n and increasing R(i0, n)] and
benefits (increasing female encounter rate) that accrue to males of all
reference ranks i0 as lek size increases.
How will the predictions of the general model differ from those of
one in which ranks are fixed?
We now explore the sensitivity of
n*(i0) to variation in
i0. We only search for general trends in the case of males
of reference rank i0 who have nontrivial optimal lek
sizes. For such males n*(i0) satisfies
 | (5) |
We take a logarithmic transformation of F(i0,
n) and denote log{F(i0,
n)} by A(i0, n). Note that the
nontrivial optimum n*(i0), which
maximizes F(i0, n) in the range
n > 1, also maximizes A(i0,
n) in the range n > 1, thus
 | (6) |
subject to
 | (7) |
We now differentiate Equation 6 with respect to i0,
using the chain rule for functions of two variables. This gives us
 | (8) |
Equations 7 and 8 show us that the direction in which optimal lek size,
n*, changes with male reference rank,
i0, is independent of the female visitation rate,
f(n). Of course, the magnitude of
dn*/di0 (for each
i0) is itself not independent of f(n) as
both A(i0, n) and
n*(i0) are functions of the female
visitation rate. If we use our approximation 2 and expand the partial
derivatives, then
 | (9) |
The model of Widemo and Owens (1995)
is a special case of this more general model, in which the rank of each male
is fixed, and so is independent of aggregation size; i.e.,
(and so
).
We have argued that ,
, and . Further, the ranks of
males associated with high i0 values will worsen as lek
size increases at a greater rate than the ranks of males associated with low
i0 values, a consequence of males of intermediate
attractiveness or competitive ability entering the lek and coming between them
in the success hierarchy. Thus 
will be a monotone
increasing function of 
, and so
. Therefore, the two
expressions on the second line of Equation 9 can never be positive, and it is
these expressions which are absent from the model of Widemo and Owens.
Thus, the general model will generate dn*/di0 values that are smaller than those produced by the model of Widemo and Owens. As a consequence, the general model is less likely (all else being equal) to reproduce the Widemo and Owens prediction of low-ranking males having greater optimal lek sizes than high-ranking males. In fact, this behavioral trend can be shown to go either way (Figure 1), depending on the nature of the relationships between P and r and R and (i0, n).
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We also note that the prediction of Widemo and Owens is based on a specific
form of mating probability, P(r, n), and is not even general
under the assumption of fixed ranks. In the fixed-rank model, high-ranking
males can have greater optimal lek sizes than lower ranking rivals, provided
is positive and sufficiently large
to make the first line of Equation 9 negative.
Under what circumstances should low-ranking males promote
aggregation?
We say relative competitive differences between individuals of different
reference rank decrease as lek size increases if, given any
i0, i'0 with
i'0 > i0, the ratio of
average mating probabilities of males of these reference ranks is a decreasing
function of n, for all n. If we use our approximation 2,
this ratio equals P[R(i0, n),
n]/P[R(i'0, n),
n].
It can be shown that, if relative competitive differences between
individuals of different reference rank decrease as lek size increases, then
 | (10) |
Applying inequalities 7 and 10 to Equation 8 gives us: relative
competitive differences between individuals of different reference rank
decrease as lek size increases
 | (11) |
By a similar deduction we can also prove the converse; that is,
relative competitive differences between individuals of different reference
rank increase as lek size increases
 | (12) |
These behavioral trends hold for all female visitation rates, f(n), and are not based on the approximation 2. If competitive differences between males decrease as aggregation size increases, then Equation 11 predicts that males of low reference rank (high i0) will have greater optimal lek sizes than males of high reference rank (low i0). Conversely, Equation 12 illustrates that males of high reference rank will have the greatest optimal lek sizes if competitive differences increase with aggregation size (for example, see Figure 1).
Does empirical data refute the possibility of relative competitive
differences increasing with lek size?
Two commonly used measures of reproductive variability on leks are
reproductive skew (Keller and Vargo,
1993; Reeve and Ratnieks,
1993) and the proportion of copulation accruing to top-ranking
males. We now examine the relationship between the way in which relative
competitive differences change as lek size increases and the direction in
which these two indices change as lek size increases.
Reproductive skew was originally designed only as a measure of average departure from equitable reproduction within cooperative breeding groups. This measure was fine for tests of optimal skew theory for cooperative groups and gives us the prediction that breeding asymmetry should increase with increasing within-group relatedness. However, we show that the relationship between skew and lek size provides no indication of how optimal lek size will vary with male rank.
It has been reported that reproductive skew decreases with increasing
aggregation size (see Widemo and Owens,
1995, and references therein). However, the direction in which
reproductive skew changes as aggregation size increases provides no indication
of how relative competitive differences between males will be affected. In the
example presented in this paper (Figure 1, 2, 3), skew decreases with lek size both when relative competitive
differences decrease with lek size [Figure
3 (squares)] as well as when relative competitive differences
increase with lek size [Figure 3
(triangles)].
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The proportion of copulations achieved by the highest ranking males has
been observed to decrease with lek size (Alatalo et
al., 1992). This again provides no indication of how relative
competitive differences change as aggregation size increases. Simple
hypothetical examples can be created (see appendix) to demonstrate that
relative competitive differences may increase with lek size even though the
proportion of copulations accruing to the highest ranking males decreases.
Relative competitive differences and previous models relating to lek
evolution
Ideal free distribution theory has been used to model the distribution of
unequal competitors between patches that differ in the input rate of valuable
resources (Parker and Sutherland,
1986; Sutherland and Parker,
1985). These models have been used in the context of leks
(Höglund and Alatalo,
1995; Sutherland, 1996)
to model the distribution of unequal males between leks that differ in female
encounter rate. Within this context the interaction between males on a lek has
been modeled in various ways. We focus on the interference model with unequal
competitors (Parker and Sutherland,
1986; Sutherland and Parker,
1985). We show that the mating probabilities used in this model
are such that they can produce relative competitive differences (between
males), which increase as aggregation size increases.
Allowing for a slight change of notation from Parker and Sutherland
(1986), in the interference model with
unequal competitors, the pay-off F(i, n) to a male of
reference rank i on a lek of size n is given by
 |
is the
average competitive weight of the males on a lek of size n, and
m is a constant measuring the severity of the interference between
rival males (0 < m 
1). We consider two males of respective
reference ranks i and j, where i < j
(male i is of higher reference rank than male j), and so by
definition Ki > Kj. the relative
competitive difference between these two males RCD. (i, j, n) on a
lek of size n is
 | (13) |
If we take the logarithm of Equation 13 and differentiate with respect to
n, then
 | (14) |
Thus, for RCD between the two males to increase as lek size increases, we
must have,
 | (15) |
In other words, provided that the rate of change (with lek size n)
of the average competitive weight of the lek is sufficiently large to satisfy
inequality 15, RCDs will increase with lek size. This will certainly happen if
males of above average competitive weight (and thus above average reference
rank) join the aggregation, but it can also happen if males of below average
competitive weight enter the lek, provided
is less negative than
. Indeed for the
interference model with unequal competitors, RCDs between lekking males can
only decrease with lek size when males of relatively low competitive weight
(and relatively low reference rank) join the aggregation. It is only in these
circumstances that inequality 15 can be violated (for an example, see
Figure 4). Van der Meer
(1997) has expressed reservations as to
the "biological realism" of the interference model with unequal
competitors. However, RCDs can be shown to increase in other models as
well.
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at each lek size is
given in parentheses (when this term is positive condition, Equation 15 is
satisfied, and the RCD will increase with lek size). In this model
.
In the kleptoparasitism model (triangles) G(i, ns,
n) = ans, L(i, nd, n) =
and (from Parker and Sutherland,
1986
In the kleptoparasitism model (Parker and
Sutherland, 1986), the pay-off F(i0,
n) to a male of reference rank i0 on a lek of
size n is given by
 | (16) |
G(i0, ns, n) represents the gains, in terms of copulations stolen from subordinates, that the male of reference rank i0 achieves on the lek of size n on which there are ns, lower ranking rivals. L(i0, ns, n) similarly defines the same male's losses to higher ranking rivals, and f(n) again represents the female input rate on the lek.
In the kleptoparatism model, it can be shown that the RCD between unequal
males of reference ranks i, j, i < j will increase with
lek size if either (1) males of higher reference rank than i(i.e.,
males of reference rank r, r < i) join the aggregation,
or (2) males of intermediate reference rank (i.e., males of rank r, i
r j) join the aggregation. Only when males whose
reference rank is lower than j enter the lek will RCDs between the
males of reference ranks i and j decrease with aggregation
size (see Figure 4).
The model of Kokko (1997) also uses a
mating probability than can result in competitive differences between males
increasing as aggregation size increases, although this result strongly relies
on males of intermediate reference rank (in the sense that each arriving male
takes a median position in the dominance heirarchy) joining the aggregation.
To see this, we consider two males of reference ranks i, j, and
i > j [reference rank is referred to as male quality in
the model of Kokko (1997)]. The ranks of
these two males on a lek of size n will be r(i, n),
r(j, n) with r(i, n) <
r(j, n), and, from Kokko
(1997), the RCD between them will reduce
to
 |
where 
is a skew parameter in the range 0 < 1. If
males of intermediate quality join the lek, then r(j, n)
> r(i, n) will increase with n, and RCDs between
males will increase with aggregation size; otherwise r(j,
n) — r(i, n) will be constant, and RCDs will
remain constant.
Van der Meer (1997) presents a general
matrix approach to modeling the level of interference between males who differ
in competitive ability. In a special case of this model, the level of
interference given to male i by male j is the ratio of their
respective competitive weights Ki/Kj, and the
RCD between males i and j is
{Ki/Kj}m, regardless of lek
size, where m is a constant scaling the severity of the interference.
Clearly, in this case the competitive difference between any two males remains
constant as lek size increases, and our model predicts that all males should
have the same optimal lek size, irrespective of their underlying competitive
ability. A slight adjustment to this special case could offset the competitive
differences between males and cause RCDs to increase (or decrease) as
aggregation size increases.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Widemo and Owens (1995)
present a
model to determine the expected copulation rates of different ranks of male on
leks of a range of sizes. They claim that a negative relationship between lek
size and male mating skew results in high-ranking males having intermediate
optimal lek sizes, unless the effect of the decreasing skew is more than
offset by a concomitant increase in overall copulation rate with lek size.
Furthermore, in such cases they predict that low-ranking males will have
larger optimal lek sizes than high-ranking males. They claim this effect is
dependent on the relative shapes of the relationship between lek size and
overall copulation rate and between lek size and mating skew. Our model goes beyond that of Widemo and Owens in that we establish a general condition for low-ranking males to have greater optimal lek sizes than higher ranking rivals. Furthermore, we allow male rank to worsen as lek size increases. We predict that it is the way in which relative competitive differences change as aggregation size increases that determines whether high- or low-ranking males have the greatest aggregation sizes. If RCDs decrease with lek size, then (by Expression 11) low-ranking males will have the greatest optimal lek sizes (for an example, see Figure 1A). Conversely, if RCDs increase with aggregation size, we predict (Expression 12) that males of high reference rank will have the greatest optimal lek sizes (Figure 1B). Thus, if RCDs decrease as aggregation size increases, leks should form with low-ranking males accumulating around higher ranking rivals. The converse would be true if RCDs increase with aggregation size.
Contrary to the claim of Widemo and Owens, the relationship between skew and lek size has no direct bearing on the optimal levels of aggregation of males of different success rank. A negative relationship between skew and lek size can be produced either if RCDs increase or decrease with aggregation size (see Figures 1 and 3). Thus a negative relationship between skew and lek size can be produced both when optimal lek sizes increase and when they decrease with male reference rank. Furthermore, although the visitation rate (or "copulation rate" in the terminology of Widemo and Owens) influences the optimal lek size, it does not affect the direction in which optimal lek size changes with male reference rank (see Expressions 11 and 12).
Empirical data are unable to refute the possibility of RCDs increasing in
nature. In addition, commonly used ideal free distribution models such as the
interference model with unequal competitors (Parker
and Sutherland, 1986; Sutherland
and Parker, 1985) and the kleptoparasitism model
(Parker and Sutherland, 1986) use mating
probabilities that can produce RCDs (between males) that increase with
aggregation size.
In conclusion, we present a model that shows the fitness consequences of displaying on leks of different size to males whose rank can vary. This model is robust enough for its predictions to hold for all mating probabilities and all female visitation rates, and it enables us to highlight the conflict of interests that can exist between lekking males who differ in rank.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Here we demonstrate with a simple example that relative competitive differences may increase as lek size increases, while the proportion of copulations achieved by the top male decreases.
Consider a lek of size n = 3 and suppose the reproductive success
of the three males is as follows:
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We denote the rank 1 male on this lek of size 3 "male 1,"
etc.; we will follow the fortunes of individual males as the lek size
increases so that on a lek of size n = 4, male 1 will still refer to
this same male, although his rank may now have worsened. Let us consider the
relative competitive differences (RCDs) between the three males on the lek of
size 3 and how RCDs change as lek size increases, say, to size n = 4.
On the lek of size n = 3:
 |
Suppose now that another male joins the aggregation, and suppose the
reproductive success on the lek then becomes:
|
In principle, the fourth male (referred to as male 4) may have joined the
lek at any position in the dominance hierarchy, and so may be ranked 1, 2, 3,
or 4. There thus exist the following four possible permutations of the ranks
of males 1-4 on the lek of size 4:
|
We now compute the RCDs between males 1-3, for the 4 possible permutations
of their new ranks on the lek.
 |
thus,
 |
The terms in parentheses are the permutation numbers for which the RCDs take the values given. Clearly, as the lek size increases from 3 to 4 males, RCDs between the males increase, regardless of the rank taken by male 4.
However, on the lek of size 3:
 |
and on the lek size 4:
 |
Thus we have a simple case in which the proportion of copulations accruing to the top-ranked male has decreased (as the lek size has increased), but RCDs between males have increased with increasing lek size.
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TOPABSTRACTINTRODUCTIONDISCUSSIONAPPENDIXREFERENCES |
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Alatalo RV, Höglund J, Lundberg A,1992 . Evolution of black grouse leks: female preferences benefit males in larger leks. Behav Ecol 3: 53-59.
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Reeve HK, Ratnieks FLW, 1993. Queen-queen conflicts in polygynous societies: mutual tolerance and reproductive skew. In: Queen number and sociality in insects (Keller L, ed). Oxford: Oxford University Press; 45-85.
Sutherland WJ, Parker GA, 1985. Distribution of unequal competitors. In: Behavioral ecology: ecological consequences of adaptive behavior (Sibly RM, Smith RH, eds) Oxford: Blackwell; 255-274.
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Van der Meer J, 1997. The ideal free distribution when predators differ in competitive abilities. Oikos 80:301-310.
Van Rhijn JG, 1991. The ruff. London: T & AD Poyser.
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