Behavioral Ecology Vol. 9 No. 2: 136-143
Optimal mating strategies in nonterritorial
ungulates: a general model tested on
muskoxen
Mads C Forchhammer and Jacobus !• Boomsma
Department of Ecology and Genetics, University of Aarhus, Ny Munkegade, Building 540, DK-8000
Aarhus C, Denmark
We present a marginal value model ^plaining mtraspecific and interspecific variation of mating systems in nonterritorial
ungulates. The model takes into account the simultaneous effects of spatial and temporal distribution of females, female group
size, male-male competition, female choice, and the operational sex ratio (i.e., the proportion of estrous females). The model
predicts that higher numbers of females per group increases the average exploitation time of such groups by males. An increase
in female group density, operational sex ratio, and age-specific fighting success of males are predicted to reduce the average
exploitation time of female groups, leading to roving of males (i.e., moving between female groups). In contrast, an increase
in die female rejection rate of males and in the time spent by males on nonmating activities (Le., foraging, lying down,
ruminating, migrating) are predicted to increase the average exploitation time of female groups and to favor staying behavior
of males (Le., defending a female group over a longer period of time). Consequently, die model predicts that young males will
tend to be "stayers," whereas middle-aged and old males are expected to be "rovers." Model predictions were tested widi field
data collected on muskoxen Ovibos moschatus in a natural population in west Greenland. Observed correlations between bull
exploitation times of cow groups and the six above-mentioned social and environmental parameters were all in die predicted
direction and statistically significant in five of die six cases. Overall, 69% of die observed variation in exploitation time of cow
groups by males was explained by die model predictions. Stepwise regression suggested that, of die six parameters, variation in
sex ratio (R1 =» .56) and time spent on nonmating activities (iP «• .35) had the largest effects on male exploitation time. Also,
die observed age-specific variation in bull exploitation time of cow groups was as predicted. Key words: female distribution,
muskox, nonterritorial ungulates, operational sex ratio, optimal mating strategies, Ovibos moschatus, sexual selection. [Behav
Ecol 9:136-143 (1998)]
M
ale reproductive strategies in mammal* are closely
linked to die spatial and temporal distribution of females. These distributions are ultimately determined by variation in environmental and social conditions of local populations (Clutton-Brock, 1989; Clutton-Brock and Harvey, 1978;
Davies, 1991; Emlen and Oring, 1977) so a wide range of male
reproductive strategies has been reported, bodi between species and among populations within species (Clutton-Brock,
1989; Gosling, 1986).
Theoretically, die interactions between die various selection
forces leading to differences in ungulate mating systems are
not well understood. In particular, die fitness consequences
of defense and monopolization of groups of females on one
hand, and die frequendy observed movements of males between female groups on die other, has been difficult to explain in quantitative terms (Gosling, 1986; Clutton-Brock,
1989). Previous evolutionary analyses of mating systems in ungulates have therefore been primarily based on classifications
and verbal predictions (e.g., Clutton-Brock and Harvey, 1978;
Davies, 1991; Emlen and Oring, 1977), which are difficult to
test widi specific empirical data.
In die only available model study, Sandell and Liberg
(1992) analyzed die fitness payoffs of "roving behavior" (i.e.,
moving between female groups in search of estrous females)
and "staying behavior" (i.e., staying and defending a series of
M. C Forchhammer is now at the Division of Zoology, Department
of Biology, University of Olio, PO Box 1050, I«)316 BUndern, Olio,
Norway.
Received 30 May 1996; fint revision 12 May 1997; second revision
26 August 1997; accepted 28 August 1997.
C 1998 International Society for Behavioral Ecology
single females or a group of females throughout die mating
season) of breeding males. The two alternative male mating
strategies were considered under a scenario where females
were assumed to be stationary and randomly distributed in
space. However, key reproductive parameters, such as die temporal distribution of females, active female choice of males,
and age-specific variation in male mating behavior, were not
considered in die model (Sandell and Liberg, 1992).
In diis study we apply a different approach and here present a theoretical framework using marginal value arguments
to predict male mating behavior in nonterritorial mating systems—Le., in species characterized by a homogeneous distribution of forage, indefensible female home ranges, and unpredictable female movements (sensu Clutton-Brock, 1989;
Gosling, 1986). In such situations, die mating behavior of
males can be divided into (1) movements between individual
females within female groups and (2) movements between
female groups (Gosling, 1986; Schwagmeyer and Parker,
1990). Depending on environmental and social conditions,
within-group mating behavior will include mate guarding of
one single estrous female at die time (e.g., bison Bison bisom
Komers et aL, 1992), whereas between-group mating behavior
is characterized by roving between female groups which are
exploited for variable periods of time. In roving behavior, one
extreme is reached when almost all time is allocated to roving
such as in Indian chital Axis axis populations (Schaller, 1967),
whereas- <h» •thw oxBomc occurs when almost all time is allocated to monopolization of a single female group as exemplified by red deer Cervus elaphus in Scodand (Clutton-Brock
et aL, 1982). Our objective is to present a realistic but quantitatively simple model incorporating variation in die spatiotemporal distribution of females, female group size, male-
Forchhammer and Boonuma • Mating itratcgies in ungulates
137
male competition, female choice, and the operational sex ratio. We focus on predicting the shifts of males between female
groups. By applying marginal value arguments, our model analyzes alternative male mating strategies as a continuum instead of considering roving and staying .as discrete alternatives, as was done by Sandell and Iiberg (1992).
We tested the model predictions with an extensive data set
on male mating behavior of muskoxen Ovibos wtoschatus in
west Greenland. The muskox is a nonterritorial ungulate
adapted to the extreme habitats in the Arctic (Forchhammer
and Boertmann, 1993; Forchhammer and Boomsma, 1995;
Klein and Bay, 1990; Klein, 1992; Thing et al., 1987; Wilson,
1992). Previously, the mating system of muskoxen has been
viewed as a traditional harem defense system with one male
defending the same group of females throughout most of the
mating season (Gray, 1987; Smith, 1976; Tener, 1965). Recently, however, it has been documented that muskox males
in populations in west and northeast Greenland move between female groups exploiting these for variable periods of
time (Forchhammer, 1997; Forchhammer and Boomsma,
1995).
RUTTING AREA
Ratting habitat A
•
•
•>
>
k.
i1
B
•
IV
The model
For the model, we have made the following assumptions: (1)
natural selection favors male mating behavior that maximizes
the number of matings per unit of time, (2) females come
into estrus independently (i.e., the proportion of estrus females in a given female group is equal to the proportion of
estrous females in the rutting habitat), (3) after successful
mating, females lose interest and do not mate again, and (4)
there is no significant emigration or immigration of individuals from or to the rutting area; consequently, both females
and males are present at the rutting area throughout the mating season.
Consider the scenario presented in Figure 1. During the
mating season of a given nonterritorial ungulate species, the
total time spent by a male can be divided into time allocated
to nonmating activities (7^ time spent to forage or to find and
gain access to female groups, for example) and time allocated
to mating activities (t, time spent exploiting a female group).
The number of matings that a male achieves in a given female
group is defined by the gain function N(Q. To our knowledge,
no empirical data exist on the specific shape of the function
N(() (i.e., on the temporal distribution of the cumulative
number of matings within a single female group). At the population level, however, such data are available for several nonterritorial ungulates (Clinton-Brock and Albon, 1989; Lott,
1981; Skogland, 1989) and suggest that the following family
of decelerating functions is appropriate: F\t, n) •» kt"/(c +
r ) , where k andcare constants. Stepwise nonlinear modeling
of the available population-level data (Clutton-Brock and Albon, 1989; Lott, 1981; Skogland, 1989) revealed that between
81% and 90% of the variation were explained by the model
given by F[t, \) (Forchhammer MC, unpublished data). Given
the high explanatory power for n = 1, we refrained from
increasing model dimensions. Thus, assuming that these population levels reflect the average temporal mating pattern
within single female groups, N(Q can be approximated by
AL(I
- r^).
da)
The parameter Nnax is the saturation value for N((), whereas
the shape of N(t) is determined by the constant a. Apart from
the time spent exploiting a female group (/), N(() depends
on (1) the number of females in a group (J), (2) the average
proportion of females in estrus in the rutting area («), and
(3) the proportion of males in the rutting area («). The up-
Figure 1
Graphical illustration of the hypothetical mating Kenario for a
nonterriotorial ungulate specie*. The distribution of female group*
(filled circle*) within the rutting area is random. Within the rutting
area, there are different rutting habitats (delimited by a black line)
synonymous with male home ranges during the rutting season
(exemplified by rutting habitats A, B, and Q . Each rutting habitat
is characterized by variation in female group size (size of filled
circles) and variation in female group density. Several males can
share a rutting habitat, and rutting habitats can overlap.
per limit of the number of group-specific matings achievable
by a male is equal to the number of females in a group (/),
whereas the relative availability of estrous females in a group
determines the rate at which matings are achieved per unit
of time. Also, we have to consider the effect of active female
choice of males, which is known to influence the reproductive
success of males. Even during estrus, females can still reject
certain individual males or entire age cohorts of males (e.g.,
Clutton-Brock et al., 1982). In our model, female choice will
influence the parameter t, as the proportion of estrous females in the rutting area will overestimate the mating opportunities of a given cohort of males (x) if female choice occurs.
To integrate female choice in the model, we therefore need
to weigh e with respect to male age cohorts. This is done by
multiplying t by (1 — T J , where T, is the average proportion
of males of age x rejected by females. Thus, setting JVTO =» /
and a = e(\ — TJ/IH, Equation la can be rewritten as
—-j
tm
•••
(lb)
The ratio e/m expresses the availability of estrous females per
adult male and is synonymous with the operational sex ratio
(Emlen and Oring, 1977).
Because we have assumed that the cumulative maximum
number of matings per unit of time is favored by natural selection, we need to include both time variables (7* and t) to
evaluate the effect of time allocation on the reproductive fitness of males throughout the mating season. Also, as a consequence of male-male competition, a male's relative fighting
ability is of relevance for his success in gaining access to a
female group (e.g., Dunbar et al., 1990; Geist 1971; Hirotani,
138
1994). In other words, a male's fighting ability will influence
the time he allocates to nonmating activities (T) relative to t
When a male's social rank increases, he spends less time acquiring control of female groups, and therefore his total time
allocated to T decreases. This effect of male-male competition on the number of matings per unit of time [n m N/
(T+()], can be incorporated in the present model by multiplying Tby (1 - 4>J, where 4>« »» the average proportion of
fights won by a male x years of age. By integrating 4>. in the
model, we imply that males have some knowledge of their
fighting ability before the mating season and adjust their behavior accordingly. This is a reasonable assumption because
males in several ungulate species have been reported to engage in fights outside the mating season, giving them the opportunity to assess their own relative season-specific fighting
ability (Dunbar et aL, 1990; Geist, 1971; Komers et aL 1992;
Owen-Smith, 1993a). Given these considerations, the number
of male matings in a given female group per time unit is then
given by
Behavioral Ecology Vol. 9 No. 2
thetical scenario, this means that naaM is constant for a male
within his rutting habitat. Between rutting habitats and males,
however, n ^ can vary (Figure 1).
Collection of mtulco
We collected data needed to test the predictions of our model
during a study of age-specific male reproductive strategies of
muskoxen in a natural population in Angujaartorfiup Nunaa,
west Greenland. Detailed information on topography of the
study area and local muskox population size are given by
Forchhammer (1995).
Behavioral data of muskox bulls and cows were collected
throughout the prerutting and rutting season covering six periods of 2 weeks: prerut (1-15 July), early rut (16-31 Jury and
1-15 August), peak rut (16-31 August and 1-15 September),
and late rut (16-30 September). The pooling of data into 2week periods allowed us to analyze also the seasonal variation
in male time budgets, aggressive interactions, and movements
between herds.
N
On a daily basis, two to four focal bulls were followed during
(2a)
4-23 h (mean: 6.9 h) by one or two observers using binoculars
(10X20) and spotting scopes (30X60) at a distance between
As documented in previous studies (e.g., Dunbar et aL, 1990;
200 and 1000 m. Adult muskox bulb were categorized into
Komers et al., 1992), female group size is usually variable. To
four age groups based on the procedure described by Smith
include this variability in our model, we consider a scenario
(1976): Bl bulls (4-6 years), B2 bulls (7-10 years), B3 bulls
with k different sizes of female groups in the rutting area. The
(11-15 years), and B4 bulls (16+ years). The age group clasfemale groups (1,2
k) are characterized by different gain
sification by Smith (1976) relies on horn development, wear
functions Nlt N,,..., Nk as a result of different group sizes/,
ft, • •., fk and group-specific operational sex ratios [«(1 — T J / of horn, and facial and body characteristics. This technique
has proven to be quite accurate in serial ranking by year of
"»3I> [*(1 ~ TJ/IH],, . . . , [«(1 — Tj/wt]*. Associated with each
muskox bulls of known age (Smith, 1976). In a trial covering
female group are the group-specific mating time allocations
all used age groups, the two field observers who collected all
*,, tf, ..., th by males. Due to the random distribution of fedata independently categorized bulls into the same age
male groups, the average time males spend on nonmating
groups
in 96% of the cases (r = .99, n - 25, p < .0001).
activities is assumed to be independent of group size. Defining
Overall, 208 focal muskox bulls (Bl: 57, B2: 54, B3: 62, and
gi as the number of female groups of size i a male encounters
B4: 35) were sampled during a total of 1131 h of observations.
during the mating season, the average number of matings per
During sampling of focal bulls, we recorded the time allounit of time taken over female groups becomes
cation to the following behaviors: (1) inactivity (resting, ruminating), (2) foraging (ingestion, searching), (3) aggression
toward other males (displacement, threat, and attack dis(2b)
plays), (4) impressive behavior (aggressive displays toward
cows), (5) sexual behavior (approaching, following, sniffing,
nosing, lip curl, head up, head over rump, foreleg kick, head
twist, rush, mounting, mating), and (6) other behaviors
From the marginal value theorem it follows that the maximum
value of n is given by nmmx ** dN^tf) / dt,* for any female group (standing, walking, bathing, playing). These standardized and
previously applied categories of behavior are described in dei of any size (Charnov, 1976). Combining this theorem with
tail by Gray (1973) and Smith (1976). All muskox bulls samthe expression for N, (Equation lb), the optimal'exploitation
pled
per day were different individuals. Because bulls were
time of female group i, tf, can be derived (see Appendix) as
not individually marked, some of the bulls sampled on consecutive days may have been identical. The shortest interval,
(S)
however, between two behavioral observations of muskoxen
' d - T.
that are statistically independent has earlier been shown to be
m
only 90 min (Jingfors, 1980). This combined with the fact that
Note that *,• has no positive solution w h e n / :£ ««„/[«( 1 — our daily behavioral recordings of bulls were separated by at
least 8-12 h at night makes it reasonable to assume statistical
Tj/m] t This "group acceptance" threshold defines the maxindependence between all behavioral data of sampled bulls.
imal size of female groups which always give a poorer payoff
Because within-bull age-group variances were generally larger
(i.e., number of matings per unit of time) than the alternative
than the between-bull age-group variances (Forchhammer
strategy of migrating to the next female group. Thus, when
MC, unpublished data), the pooling of data has no significant
nmmx is relatively high, the frequency of emigrations per unit
effect on the analyses between bull age-groups (Leger and
of time ("the emigration value"; Parker and Stuart, 1976) is
Didrichsons, 1994).
high and so is the group acceptance threshold and vice versa.
Our model is a long-term, rate-maxuniang a>*4ol and preCows encountered by focal bulls were identified and foldicts the optimal exploitation time that maximizes the rate of
lowed during the rest of the daily sampling of buHs. AddMenmatings across all female groups encountered by a male withally, we recorded age structure and sexual composition of
in his home range (Equation 3). Thus, irrespective of group
herds visited by the focal bulls, herd dynamics (leaving and
size, it follows that a male should use the same giving-up crijoining of bulls and cows), and the total number of cows in
terion (i.e., when the male's rate of mating decreases to the
focal and neighboring herds continuously throughout the daigroup acceptance threshold; cf. Equation 3). In our hypory sampling of focal bulls. The voluntary herd shift rate for
139
Forchharamer and Boonuma • Mating strategics in ungulates
Table 1
ITCQJCUU Q Q OOICTIM COfTBtmOllS D£f ween social ana D
n ana me • v n a p . nme
ipgul by onnkox bolls in groups of cowi•
Social and behavioral parameter
Predicted
correlation
with optimal
exploitation
time ((,•)
Observed correlation with
time spent in
female groups*
Partial correlation
coeffideno b
Number of females/group {/)
Female group density*
Operational sex ratio (t/vt)*
Female rejection of males ( T J
Male fighting success (4>J€
Time spent on nonraating activity (T) r
+
—
+
—
+
-50** (24)
-Si— (24)
-.74*** (16)
J0(ns) (24)
- . 9 0 * (4)
.61**» (24)
.34 (ns)
- . 4 3 (ns)
-.75»*»
.06 (ns)
—
J9*
Test for normality revealed that the observed male exploitation time, female rejection of males, and
time spent on nonmadng activity were significantly different from normal distribution (KolmogorovSmirnov tests with Lilliefors significance correction; p < .02; Sokal and Rohlf, 1995; SPSS, 1996).
Observed correlations are therefore Spearman rank coefficients between social and behavioral
parameters and the corresponding index of male staying time.
* Sample sizes [number of sampling periods (6) summed over all bull age-groups (4)] are given in
parentheses. Significance levels: ns, nonsignificant (p > .05), *p < .05, • • p < .01, ***p < .001.
b
Partial correlation coefficients are from a stepwise general linear model (entry tolerance = 0.05;
SYSTAT, 1992) with index of male staying time as dependent variable and the listed social and
behavioral parameters as independent factors. Due to low sample size, male fighting success was not
integrated in the model.
c
Female group density was expressed as an index: l/(time spent migrating between female groups).
d
Operational sex ratio could not be calculated for the first two 2-week periods, reducing the sample
size to 16.
* Sample size is the number of bull age group*.
' Nonmadng activities included foraging, lying down, rumination, and migration between female
groups.
each focal bull was calculated as the average number of herds
left and joined per hour. Bulls displaced from herds as a result
of aggressive interactions were not included in this rate. No
direct measurement of the staying time per herd was made.
The voluntary herd shift rate, however, is inversely correlated
to the staying time, i.e., the longer a bull stays per herd per
unit of time, the fewer herd shifts will be recorded per unit
of time. Therefore, the index of a bull's staying time was calculated as the reciprocal of his herd shift rate. Comparing
across bull age-groups within a population, this is a good approximation of the average exploitation time of female
groups.
Following previous studies (Berger, 1989; Komers et al.
1992; Lott, 1984), we used the tending (following) behavior
of muskox bulls as an indication of estrus status of cows. A
cow tended by a focal bull was defined as being in estrus if
and only if the cow reacted positively (i.e., by standing still)
to the approach/investigation of the focal bull. Estrous cows
can reject bulls, indicating that sexual selection through female choice is important for male mating success. The local
operational sex ratio for each focal bull was calculated as the
proportion of estrous cows present within a 2-km radius from
the focal bull. The average movement rate of muskoxen during this time of the year is up to 1.9 km/day (Forchhammer,
1995).
We quantified the degree of active cow preference for bulls
of a given age as the proportional difference between the total
number of mating attempts and the number of successful matings. This female rejection estimator is reasonable for two reasons. First, cows allowing bulls to mount are always in estrus,
whereas anestrous cows invariably reject bulls in the approaching phase of the long series of behavioral displays that eventually leads to mounting (Forchhammer MC, unpublished
data; Gray, 1973; Smith, 1984). Second, whether a mounting
results in a successful mating depends entirely on the cow, as
she can definitely interrupt a mating attempt by just stepping
forward. Due to their massive physical proportions, muskox
bulls are highly unstable when mounting a cow. This is probably the reason that forced copulations were never observed
during 3-4 years of intensive field work in west Greenland
(Forchhammer MC, personal observation).
Testing the model predictions on muskox
The predicted influence on optimal male exploitation time
((,*) by number of females per group (/"), female group density, operational sex ratio (t/m), female rejection of males by
females (TJ, male fighting success ( 4 0 , and time spent on
nonmating activities (T) are easily derived from Equations 2a
and 3 and are summarized in Table 1. Additionally, as demonstrated in muskoxen (see below) and other ungulates (e.g.,
dutton-Brock et al., 1982; Owen-Smith, 1993a; Skogtand,
1989), young, mature males usually can be characterized by
having a low average fighting success and a high mating rejection rate by females as compared to older, mature males.
Given these age-specific differences, our model predicts that
older males, characterized by a high 4>, and a low T 0 would
on average exploit a given female group for shorter periods
of time than young adult males (low <J>X and high t^ Equations
2,3).
As the rutting season progressed, muskox bulls responded
to the increasing numbers of estrous females by significantly
decreasing their average exploitation time of female groups
(Figure 2a). Furthermore, comparing bull age groups over all
seasons, there was a significant decrease in the average index
of exploitation time from young Bl bulls through middleaged B2 and B3 bulls to old B4 bulls (Figure 2b). Thus, as
predicted by the model, young bulls can be characterized as
Behavioral Ecology Vol. 9 No. 2
140
OJJ
1-13 JuJ
16-31 Jul 1-15 Aug 16-31 Aug 1-15 Sep 16-30Sep
Mating season
O23
Bl(4-6yn)
B2(7-10yrs)
B3 (11-13- yrs)
B4 (16+yn)
*
(b)
0.6
\
•
-
c"
Bl(4-6yn)
B2 (7-10yn)
B3(ll-15yn)
B4(16+yn)
BaO age-group
Figure 2
(a) Seasonal distribution of estrous cows (shaded bars) in relation
to seasonal variation in the index of staying time by bulls in cow
groups (mean values: data points) with error ban representing ±1
SD. The index of staying time was calculated across bull age groups
with sample sizes of 20, 32, 22, 54, 43, and 33 for the six
consecutive 2-week periods. No estrous females were observed in
the first 2-week period, 1-15 July, (b) The average index of staying
time for the different age groups of bulls with error bars of 1 SD.
Sample sizes are 57 (Bl bulb), 54 (B2 bulls). 62 (B3 bulls), and 35
(B4 bulls). Columns not sharing a letter are significantly different
(Games-Howell test, p <. 05; Games and Howell, 1976).
stayers (i.e., have a long exploitation time), whereas middleaged and older bulls can be characterized as rovers (i.e., have
a short exploitation time).
Bull age groups differed significantly with respect to fighting success (x1 = 182.9, n - 4, p < .001): Bl bulls won 0.0%
of all aggressive interactions with other age groups (n = 94),
B2 bull* won 38.7% (n - 129), B3 bull* 90.8% (n - 120),
and B4 bulb 60.0% (n - 65). In addition, Bl and B2 bulls
experienced significantly lower operational sex ratios in their
direct vicinity than did B3 and B4 bulls (Figure 3a). Age-specific differences were also apparent in the female rejection of
males, with a decreasing trend with increasing age of males
(Figure 3b).
In general, our model predictions were strongly supported
by die muskox data: increased density of herds, increased operational sex ratio, and increased fighting success of bulls significantly decreased the average exploitation time. In contrast,
increased number of cows per group and increasedtimespent
on non-mating activities (i.e., foraging, lying down, ruminating, migrating) significantly increased die average exploitation twao -by muskox bulls. Although not statistically significant, female rejection of bulls had die predicted positive effect on exploitation time (Table 1). Taken together, all the
above-mentioned parameters, except fighting success,
explained 69% of the variation in the exploitation time of
muskox bulls (multiple regression: R = .83; F(iw ,5) = 3.31, p
< .05). A stepwise regression procedure (entry tolerance "»
Bl(4-6yn)
B2(7-10yrs)
B3(ll-15-yn)
B4(16+yra)
Bull age-group
Figure 3
Age-specific variation experienced by muskox bulls in (a) the
average local operational sex ratio (OSR; % adult cows) and (b) the
average proportion of rejected mating attempts. Error bars
represent -1 SD. Sample sizes as in Figure 2b. Data points not
sharing a letter, are significantly different (Games-HoweU test, p <
.05).
0.05; SPSS, 1996) of bulls exploitation time on die same five
parameters revealed that variations in operational sex ratio
and time spent on non-mating activities explained 56% and
35% of the variation, respectively (Table 1). Fighting success
was not included in the above analyses due to die small sample size.
DISCUSSION
Optimal reproductive strategies in nontenitoibd nrtgnUtr*
At die intraspecific level, ungulate males of different age classes are under similar selection pressures to optimize their mating success but differ in dieir fighting success and attractiveness, resulting in different optimal reproductive strategies of
young, middle-aged, and old males.
The empirical data on muskoxen agree relatively well with
die predictions, bodi with regard to male exploitation time of
female groups and widi respect to age-specific variation in
male mating strategies (Table 1, Figure 2b). This suggests that
die roving of muskox bulls between cow groups in west Greenland can be explained to a considerable degree as a strategy
maximizing die number of matings per unit of time.
Within a given population, a gradual change in die proportion of estrous females oeeuw as the rutting taaon progresses, and thij change affects die optimal exploitation time
of female groups (/,•). If die temporal distribution of estrous
females is broadly bell shaped diroughout die rutting season
(e.g., Clutton-Brock et al., 1982; Prins, 1995), our model predicts that males on average will spend less time per female
Forchhammer and Boonuma * Mating strategies in ungulates
group during the peak rut (high e/m) than during early and
late rut (low e/m). In fact, this is what happens in the studied
muskox population, where the average voluntary herd shift
rate of bulls increases with e/m, reducing the overall time
spent per female group (Figure 2a).
In addition to the roving strategy of muskox bulls documented in the study population in west Greenland, similar
responses in reproductive decision making of muskox bulls
have recently been reported from another population in
northeast Greenland (Forchhammer, 1997). In both Greenland populations, bulls were shown to have a roving strategy,
challenging the previously accepted view that the muskox mating system is always a traditional harem defense system (Gray,
1987; Smith, 1976; Tener, 1965). Apparently, social and environmental conditions in the various populations throughout
the Arctic are sufficiently variable to select for different male
mating behavior as envisaged in our model.
Although not specifically collected to test our model, existing empirical data from other nonterritorial ungulate species
such as bighorn sheep Ovis canadmsis (Geist, 1971) and feral
goats Copra hrrctis (Dunbar et ah, 1990) do provide additional
support for the predicted intraspecific variation in male mating strategies. In feral goats, Dunbar et al. found a significantly positive correlation between staying time and number
of females per group. In bighorns, sex ratio had a negative
effect on time allocated to defending females, whereas time
allocated to nonmating activities increased the average time
of monopolizing females (Geist, 1971). We note, however, that
the predicted age-specific variation in male mating strategies
is to be expected only between the endpoints of the stayingroving continuum. When environmental and social conditions
unequivocally favor defense of a female group throughout the
mating season (i.e., a staying strategy), the age-specific variation in male mating strategies relates to the timing of monopolizing a female group; i.e., prime males gain and defend female groups during peak rut, and subordinate males exploit
female groups early and/or late in the rut, as reported in red
deer (Qutton-Brock et al., 1982).
Application of marginal value theory to reproductive
decision »wir"»g in i»fw^iyt^«
By using a theoretical approach based on marginal value theory (MVT) to analyze the effects of female group characteristics, sexual selection, and time allocation constraints on the
optimal mating strategies of males, we have implicitly assumed
that the frequency-dependent effects on the optimal exploitation time were minor. Such frequency-dependent effects are
explicitly taken into account in evolutionarily stable strategy
(ESS) models (Maynard Smith and Price, 1973). The degree
of deviation between the optimal solutions for an MVT and
ESS scenario depends on the relationship between the length
of the cycle duration (i.e., the average time to find and exploit
a female group) and the optimal exploitation time. If the cycle is relatively long compared to the optimal exploitation
time it makes little difference whether the MVT or the ESS
technique is employed (Parker et al., 1993). Empirical data
suggest that this is the case for time allocated to nonmating
behavior versus group exploitation time for mating in several
ungulate species (Geist, 1971; Gosling, 1986; Schaller, 1967;
Schaller and Mirza, 1971). In muskoxen, the ratio of the time
allocated to mating to the time allocated to nonmating activities ranges from 0.03 to 0.19 (Forchhammer, 1996), suggest-1
ing a high cycle-to-exploitation time ratio. Thus, the MVT-derived optimal times for the exploitation of female patches
should be reasonable approximations.
The ESS model presented by Sandell and liberg (1992)
analyzes a mating scenario somewhat different from the one
141
presented here, but both models agree on the predicted effect of allocation to nonmating activities on male reproductive
strategies (effect of 7] Table 1). However, Sandell and Iiberg's
model predicted, in contrast to ours (Table 1), that an increased female-biased sex ratio would favor a staying strategy.
This difference relates to the fact that Sandell and Liberg implicitly assumed that there is a nonvariable reproductive cost
attached to roving, independent of variation in the social environment of males. In that approach, a female-biased sex ratio has no effect on the tendency to adopt a roving strategy,
whereas a more female-biased sex ratio will eventually favor a
staying strategy (for details, cf. Sandberg and Liberg, 1992).
For muskoxen (and probably most animal^) _ we strongly believe that our assumption (i.e., variable cost associated with
roving) is the more reasonable one.
Except for the influence of age (Table 1, Figure 2) on optimal male mating strategy, the effect of variation in physical
condition (e.g., fat reserves, parasite load) of individuals has
not been considered in our interpretation of reproductive
strategies of muskox bulls. Condition is known to influence
behavioral actions (e.g., Houston et aL, 1988; Mangel and
Clark, 1988), and deterioration of the physical condition and
increasing parasite loads with age have been reported in older
muskoxen (Blake et aL, 1989; Korsholm and Olesen, 1993).
Whether this interacts with reproductive strategies remains to
be investigated.
Another parameter not considered in our model is predation. The risk of predation is known to have an effect on
group sizes (Hamilton, 1971; Owen-Smith, 1993b). In South
African populations of greater kudu Tragdaphus strtpsicews,
breeding is characterized by nonterritoriality, small female
groups, and an extremely female-biased operational sex ratio
(Owen-Smith, 1977, 1984). Such population structure is predicted to favor a roving strategy by males (Table 1). Kudu bulls
have indeed been reported to rove between female groups,
but not as much as expected a priori (Owen-Smith, 1984).
Further age-specific data on male mating behavior (e.g., time
spent between female groups, local sex ratio variation, female
group density) are therefore needed to fully evaluate the mating strategies of kudu bulls in relation to the present model.
Nevertheless, the reduced roving in kudu bulls could be partly
due to predation. Kudu bulls experience an exceptionally
high mortality rate after reaching sexual maturity (OwenSmith, 199Sa,b), and by leaving a group to search for new
females, a kudu bull would increase his predation risk considerably (Owen-Smith, 1993b). This is in contrast to the studied
muskox populations, where no predation occurs or where predation pressure is insignificant, like in the muskox populations in west and northeast Greenland.
Finally, a potential factor not included in our model is competition between sperm of different males mating with the
same females, which has been demonstrated in some nonterritorial ungulates (Meller and Birkhead, 1989). Present field
observations (Forchhammer, 1997; this study) suggest, however, that muskox cows only mate once and that sperm competition does not occur in muskoxen. We therefore refrained
from including any effects of sperm competition in our model. In general, we would expect that males would tend to be
rovers when they can detect that their sperm will have more
than average success, whereas they should adopt a staying
strategy when their sperm is unlikely to contribute to fertilization when moving away (Forchhammer, 1996). A qualitative
analogy to this situation has been created by Schwagraeyer and
Parker (1990).
Conclusions
Despite the above-mentioned, which indeed would make the
model much more complex and incomprehensible (see
Behavioral Ecology Vol. 9 No. 2
142
Forchhammer, 1996), our model explain* much of the variation in male reproductive strategies in nonterritorial ungulates. Based on simple theoretical and biological considerations, we have documented a considerable part of the variation in mating strategies of muskox bulls. The theoretical mating scenario presented is probably also relevant for other
nonterritorial animals, where groups of females represent
mating resources (see Davies, 1991, and references therein).
It may be that low female group density, low operational sex
ratio, low male fighting success, high female rejection rate,
and high environmentally or socially enforced time allocation
to nonmating activities will generally favor a staying strategy
of males (long exploitation time of female groups), whereas
the opposite conditions will favor a roving strategy (short exploitation time of fesmale groups). Our specific confirmation
of the model predictions in muskoxen emphasizes the importance of both social (especially sex ratio variation) and individual parameters and support, together with other studies
(e.g., Byers and Kitchen, 1988; Hogg, 1984; Gross, 1996; Koprowski, 1993), the notion that reproductive strategies are
flexible and conditional behavioral responses to the prevailing
environment
APPENDIX
Derivation of the optimal exploitation time t,*
Given the scenario leading to Equation 2b, Charnov (1976)
and Parker and Stuart (1976) proved, for any continuous
function of N^t), the maximum value of n ( n j , averaged
for all Nt(() in their respective frequencies, is given by
(Al)
In our model, which analyzes the reproductive decision making of nonterritorial ungulate males, we have approximated
the gain function Nt(t) with the decelerating function given
in Equation lb. Because this function is continuous for *, a
0 (cf. Equation lb), we can apply Equation Al to the present
ungulate scenario. Inserting Equation lb in Equation Al, we
thus get
'(1-T.)
m
which can be reduced to
"mx
(A2)
Isolating t," in Equation A2, we get the expression for the
optimal exploitation time of female group i in a variable female group environment (Le., Equation 3).
We thank J. C Deutsch, R. I. M. Dunbar, J. FryxelL B. Guldbrandtsen,
A. P. Meller, B. Roitberg, L. Sundstr6m, and an anonymous reviewer
for their critical and constructive comment* on earlier drafts of the
manuscript. We also thank the Department of Arctic Environmental
Research in Copenhagen and, in particular, D. Boenmann, for allowing us to use field equipment We are also grateful to H. B. Rasmuxsen
for his professional assistance and pleasant companionship in the
field. This work was supported by a grant from the Commission for
Scientific Research in Greenland to M.CF. and a grant from the Danish National Science Research Council toJ.J.B.
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