I present equations for calculating the annual reproductive success (ARS) of females engaged in typically monogamous, rapid double-clutch monogamous, typically polyandrous, and polygynous-polyandrous relationships. With appropriate redefinition of terms, these equations can be used to calculate ARS of males. I calculate annual reproductive success for female Prairie Warblers (Dendroica discolor), female Florida Scrub Jays (Aphelocoma c. coerulescens), and male and female Spotted Sandpipers (Actitis macularia). I show that the ratio of ARS of all females of breeding age to ARS of all males of breeding age equals the sex ratio (all males of breeding age to all females of breeding age).
With data on the Spotted Sandpiper, I illustrate the usefulness of the equations for evaluating hypotheses regarding the evolution of mating systems. Given that a caring parent can rear no more than one brood in a breeding season, and given a low probability of clutch success, I evaluate the potential ARS for hypothetical female Spotted Sandpipers in monogamous, rapid double-clutch monogamous, and polygynous-polyandrous relationships. The maximum ARS is the ARS actually achieved by the Spotted Sandpiper with typical polyandry. Received 9 April 1990, accepted 28 March 1991.
Department of Biological Sciences, Rutgers University, Piscataway, New Jersey 08855 USA
THE AVERAGE annual reproductive success
(ARS) of a group of female birds is the number
of offspring produced by that group divided by
the number of individuals in the group. The
reproductive behavior of birds, however, varies
greatly, and reproductive success of females is
affected by such factors as clutch size, number
of clutches laid, probability of rearing a brood
from a clutch, the number of broods reared, and
the number of mates obtained.
Most females are monogamous (Lack 1968).
Typically, a female mates with one male
throughout at least a single reproductive cycle
until she is successful or fails. Subsequently, she
may make a second effort with the same male,
separate and mate with a different male, or cease
reproductive activity for the season. In several
species, a female may lay two or more clutches
in rapid succession (Pitelka et al. 1974; Emlen
and Oring 1977; Oring 1982, 1986). In some of
these species, the male and female maintain a
monogamous pair bond, but each tends a dif-
ferent clutch, as in the Red-legged Partridge
(Alectoris rufa) (Green 1984). In a few species,
such as Temminck's Stint (Calidris temminckii)
(Hildn 1975), the sexes regularly switch mates
between clutches; a female's first mate tends her
first clutch, and she usually tends her second.
If her second mate accepts responsibility for
tending her second clutch, she may obtain a
third mate and produce a third clutch. In this
system, males often fertilize two clutches, each
942
for a different female, and are thus polygynous.
In more typical polyandrous relationships,
called "classical polyandry" (Oring 1986), a fe-
male switches mates between clutches and the
males fertilize only the clutches they are tend-
ing (e.g. Spotted Sandpiper, Actitis macularia;
Hays 1972, Oring and Knudson 1972), or she
may mate with several males simultaneously
but lay a separate clutch for each of them (e.g.
Northern Jacana, Jacana spinosa; Jenni and Col-
lier 1972, Jenni 1974).
My purpose is to present equations for cal-
culating the annual reproductive success of fe-
males with different reproductive behavior (i.e.
monogamy, rapid double-clutch monogamy,
polyandry, and polygyny-polyandry) and for
evaluating the contributions to reproductive
success of clutch size, number of broods at-
tempted, success in rearing broods, and number
of mates obtained. With appropriate redefini-
tion of parameters, these equations can be used
to calculate reproductive success of males. Fur-
thermore, these equations should allow us to
compare ARS of males with ARS of females
within and among species more rigorously when
evaluating hypotheses accounting for the evo-
lution of reproductive behavior.
MEASUREMENTS OF REPRODUCTIVE SUCCESS
The reproductive success of a female depends
on the number of clutches she lays, the number
of offspring produced from each clutch, and the
number of broods attempted. The number of
broods attempted will depend on the length of
the breeding season compared with the time
required to rear young to independence and on
the number of mates she obtains. The numerical
value of reproductive success will also depend
on whether one counts success in terms of num-
ber of broods, fledglings, independent young,
or yearlings reared.
In this paper, I present two sets of equations.
The first set refers to calculation of ARS in terms
of number of broods reared because this mea-
sure appears in the denominator of the Murray-
Nolan clutch size equation (Murray and Nolan
1989),
a+l
CS = , (1)
xx(P + P2 + ... + P,)
where CS is the average clutch size, a is the
primary sex ratio (males/females), Xx is the
probability that an individual from an egg laid
in a successful clutch survives to age class x, a
is the average age class of first breeding, co is
the age class of last breeding, P is the proba-
bility of rearing a first brood, P2 is the proba-
bility of rearing a second brood, and P,, is the
probability of rearing an nth brood. In this
equation, ARS is measured as the sum of prob-
abilities of rearing at least one, two, or more
broods, that is
ARS = P + P2 + ... P,,. (2)
This sum of probabilities translates into the av-
erage number of broods reared by females. For
example, if P + P + ... P, = 0.77, then the
average female should have reared 0.77 broods
per breeding season.
Equation 1 is applicable to either sex. For fe-
males, CS refers to the number of eggs laid in
a clutch; for males, CS refers to the number of
eggs fertilized in a clutch. Note that the average
clutch size should be identical for males and
females in the same population. Thus, differ-
ences between the sexes in one parameter (e.g.
survivorship, age of first breeding, or proba-
bilities of rearing one, two, or more broods suc-
cessfully) should be balanced by differences in
one or more of the other parameters.
The second set of equations is presented be-
cause ornithologists normally measure repro-
ductive success in terms of the number of off-
spring produced.
The equations for ARS vary with the mating
system, the number of broods or offspring
reared, and whether success is counted through
rearing young to fledging, independence, or
some other stage. Thus, the notation of ARS
requires elaboration. For example, ARS(m, b, i)
would refer to the annual reproductive success
of monogamous (m) females rearing broods (b)
through independence (i). Other specific defi-
nitions are introduced at the appropriate places
below. The form of the equations does not differ
with regard to whether success is counted
through nest-leaving, fledging, independence,
or some other stage. Thus, the third letter mod-
ifying ARS only indicates the stage through
which success is counted. In these equations, I
use success through independence because I
think this is probably most important (Murray
and Nolan 1989). Often, investigators are un-
able to obtain such data, and they may wish to
measure success through nest-leaving. The
equation is the same, but ARS would have a
different modifier. Thus, ARS(m, b, nl) refers to
the annual reproductive success of monoga-
mous females rearing broods through nest-leav-
ing. Furthermore, the reader should be aware
that the equations presented below do not rep-
resent an exhaustive list that accounts for all of
the variations in reproductive behavior. Some
of these variations will be mentioned below.
Finally, it should be emphasized that female
(or male) ARS refers to the success of all the
females (or males) in the breeding population,
which is defined as all females (or males) of
breeding age, including experienced but non-
breeding "loafers" and excluding inexperi-
enced birds that are not breeding and have nev-
er made a breeding attempt. To account for
nonbreeding experienced birds, I use the pa-
rameter Q, which is the proportion of the breed-
ing population of females (or males) that ac-
tually breed.
EQUATIONS FOR CALCULATING ANNUAL
REPRODUCTIVE SUCCESS
Monogamy.--Typically, a monogamous fe-
male lays a clutch of eggs and tends it, with or
without assistance from her mate, until she ei-
ther successfully rears her brood or fails. In some
species, one or the other member of the pair
may desert before the young reach indepen-
dence. Often no more than a single brood can
be reared during a breeding season because the
time required for development of the young to
independence precludes rearing two. If devel-
opment time of young is short relative to the
length of the breeding season, more than one
brood may be reared. After either success or
failure of a nesting attempt, a female (if she has
time) may attempt a second brood with her first
mate or she may mate with a different male.
The ARS for monogamous (m) females in terms
of number of successful broods (at least one
young reared to independence) is
A_RS(m, b, i) = Q c,s, (3)
where cn is the average number of clutches laid
per female in producing brood n to indepen-
dence, and s,is the probability of a brood n
clutch producing at least one young to inde-
pendence, regardless of whether or not the mate
is the same male for each clutch, and n refers
to the first, second, or later brood.
The ARS of monogamous (m) females in terms
of the number of young (k) reared to indepen-
dence (i) is
ARS(m, k, i) = Q c,s,,, (4)
where k, is the average number of young reared
per successful brood n.
Rapid double-clutch monogamy.--If develop-
ment time is long relative to the length of the
breeding season, monogamous females, as de-
scribed above, can produce no more than one
brood during a breeding season. A pair-bonded
monogamous female of a species whose young
have a long development time may improve her
reproductive success during a breeding season
if she rapidly lays two clutches, one to be tend-
ed by her mate, the other by herself. The success
of this behavior depends on the ability of a lone
parent to rear a brood successfully. The maxi-
mum number of broods that can be reared by
a female and her mate with rapid double-clutch
monogamy is two, one by each member of the
pair. Because replacing failed clutches is still a
possibility, the annual reproductive success in
terms of broods reared to independence of dou-
ble-clutch monogamous (dcm) females is
ARS(dcm, b, i) = Q(cmsm + csf), (5)
where cm and cf are the average number of
clutches tended by males and females, respec-
tively, and sm and sf are the probabilities that a
clutch tended by a lone male or female, re-
spectively, will produce young to indepen-
dence.
The ARS of rapid double-clutching females
in terms of young reared to independence is
ARS(dcm, k, i) = Q(csk + csk), (6)
where k and kr are the average number of young
reared per successful clutch tended by a male
or a female, respectively.
Classical polyandry.--Rarely, females may ob-
tain more than one mate. In species such as the
Spotted Sandpiper, females often desert their
mates. The males care for the eggs and young,
and the females obtain additional mates. In spe-
cies such as the Northern Jacana (Jenni and Col-
lier 1972, Jenni 1974), a female may copulate
with several mates on a single day, but lay a
separate clutch for each of them. If the caring
parent can rear no more than one brood, the
maximum number of broods a female can pro-
duce is the number of mates she is able to ob-
tain.
In classical polyandry (pa), the females' A_RS
in terms of broods reared through indepen-
dence is given by
ARS(pa, b, i) = Q (Fg crs), (7)
g=l r=l
where Fg is the frequency of females mated to
g number of mates, cr is the average number of
clutches laid for the rth mate, and s, is the prob-
ability that a clutch laid for the rth mate is suc-
cessful. In terms of number of independent
young reared, the ARS is
ARS(pa, k, i) = Q (Fg crskr), (8)
where k, is the mean number of fledglings per
successful clutch of the rth mate.
Cooperative polyandry.--In species with co-
operative polyandry, such as the Galapagos
Hawk (Buteo galapagoensis), two or more males
and a female form a group that attends one
clutch or brood at a time (Faaborg and Patterson
1981). The calculation of ARS for females is no
different from that for monogamous females.
Thus, Eqs. 3 and 4 are appropriate for calculat-
ing ARS for females that participate in coop-
erative polyandrous systems.
Polygyny-polyandry.--In some species, both
sexes may have more than one mate. In the
Temminck's Stint, for example, after a female
lays her first clutch, she leaves her mate, but he
delays incubation and solicits a second female.
The female tends her second clutch, unless her
second mate does, in which case she solicits a
third male. In any case, she tends her final clutch
alone. Although an individual, as the lone car-
ing parent, can rear only one brood during a
breeding season, each male and female may be
the genetic parent of more than one brood. In
polygynous-polyandrous systems the success of
females in rearing young must be added to the
success of their mates. Thus, the ARS for fe-
males in polygynous-polyandrous (pp) systems
in terms of broods reared to independence is
ARS(pp, b, i) = Q (Fg crsrl + cfs f (9)
g=l r=l
or, in terms of young reared,
ARS(pp, k, i) = Q (Fg crs, kr)
g=l
+ cfsk. (10)
In some polygynous-polyandrous species,
such as the Greater Rhea (Rhea americana), a
group of females visits individual males in se-
quence and provides each with a clutch (Bru-
ning 1975). Females do not incubate eggs or rear
young. Equations 7 and 8 are appropriate for
females participating in this kind of polygy-
nous-polyandrous system.
ILLUSTRATIONS WITH FIELD DATA
Despite a voluminous literature reporting
"hatching success," "fledging success," or
"nesting success," there are few published field
data adequate for performing computations of
ARS as proposed here. Indeed, until recently
(e.g. Oring et al. 1983, Woolfenden and Fitz-
patrick 1984, Grant and Grant 1989, and indi-
vidual papers in Clutton-Brock 1988 and New-
ton 1989), few studies have reported the number
of young reared per female during a breeding
season. Even fewer studies have reported either
the number of clutches laid per brood (c, or
the probability that a brood n clutch will be
successful (s,).
We have available for analysis only two pub-
lished studies of monogamous species (i.e. Prai-
rie Warbler, Dendroica discolor, and Florida Scrub
Jay, ApheIocoma c. coeruIescens) that provide suf-
ficient data for illustrative purposes. In addi-
tion, Lew Oring has generously provided me
with data from his study of the polyandrous
Spotted Sandpiper. There are no adequate data
for rapid double-clutch monogamous females
or for females in polygynous-polyandrous sys-
tems.
Prairie WarbIer.--For monogamous species,
ARS is calculated with Eq. 3 or 4. In a sample
of 70 female Prairie Warblers (Nolan 1978), fol-
lowed throughout most of a breeding season,
49 successfully reared one brood and 5 reared
a second brood through nest-leaving (nl). In the
rearing of the 49 first broods, 203 clutches were
laid (Murray and Nolan 1989). Thus, cx, the
number of clutches laid per female in produc-
ing a first brood, is 203/70 = 2.900, and s is 49/
203 = 0.241. In rearing 5 second broods, 15 ad-
ditional clutches were laid (Murray and Nolan
1989). Thus, c2, the average number of clutches
laid per female in producing a second brood, is
15/70 = 0.214, and s2 is 5/15 = 0.333. The ARS(m,
b, nl) of female Prairie Warblers is cs + c2s2 =
(2.900 x 0.241) + (0.214 x 0.333) = 0.770. This
means that the probability of successful repro-
duction is 0.770. Alternatively, one could say
that on average females produce 0.770 success-
ful broods in a breeding season.
Murray and Nolan (1989) measured ARS with
Eq. 2. Of the 70 females, 49 reared at least one
brood and 5 reared two broods successfully
through nest-leaving (Murray and Nolan 1989).
Thus, P, the probability of rearing at least one
brood, is 49/70 = 0.700, and P2, the probability
of rearing two broods, is 5/70 = 0.071; P + P2
= 0.771. Because P,and c,,. are mathematically
equivalent, ARS = P + P2 + ß ß ß P = cs + c2s2
-I- . . .
Florida Scrub/ay.--Murray et al. (1989) pre-
sented similar data for 495 breeding female sea-
sons of 143 female Florida Scrub Jays. The num-
ber of first-brood clutches per female per year
was 644/495 = 1.301, and the number of second-
brood clutches per breeding female per year
was 33/495 -- 0.0667. The probability of success
(through nest-leaving) of first brood clutches
was 348/644 = 0.5404, and of second brood
clutches, 16/33 = 0.4848. Thus, ARS(m, b, nl) is
(1.3010 x 0.5404) + (0.0667 x 0.4848) = 0.7031
+ 0.0323 = 0.7354. Murray et al. (1989) calcu-
lated P + P2 = 348/495 + 16/495 = 0.7030 +
0.0323 = 0.7353 for the reproductive success
through nest-leaving of the Florida Scrub Jay.
TABLE I. Breeding success of female Spotted Sandpipers engaged in different mating relationships. C is the
number of clutches laid. S is the number of clutches producing at least one fledgling. F is the number of
fledglings reared.
First Mate Second Mate Third Mate Fourth Mate
Clutch
number C S F C S F C S F C S F
1 100 31 88
2 32 9 18
3 17 6 19
4 4 0 0
5 2 I 2
155 47 127
Monogamous Females (F = 0.4587)
ARS(broods, monogamous females) = c,s = (155/100)(47/155) = 0.4700
ARS(fledglings, monogamous females) = c,s,k = (155/100)(47/155)(127/47) = 1.2700
Bigamous Females (F2 = 0.3532)
I 77 26 73 -- --
2 II 7 15 66 25 84
3 I I 2 24 9 29
4 5 I 1
5 I 1 3
89 34 90 96 36 117
ARS(broods, bigamous females) = c,s + c2s2 = (89/77)(34/89) + (96/77)(36/96) = 0.9091
ARS(fledglings, bigamous females) = c,s,k, + c2s2k2 = (89/77)(34/89)(90/34) + (96/77)(36/96)(117/36) = 2.6883
Trigamous Females (F3 = 0.1697)
1 37 5 12 ......
2 8 I 3 29 14 41 -- -- --
3 2 1 4 8 3 8 27 8 22
4 2 2 4 12 3 5
5 3 I 2
47 7 19 39 19 53 42 12 29
ARS(broods, trigamous females) = cs + c2s2 + c3s = (47/37)(7/47) + (39/37)(19/39) + (42/37)(12/42)= 1.0270
ARS(fledglings, trigamous females) = csk + c2s2k2 + csk = (47/37)(7/47)(19/7) + (39/37)(19/39)(53/19) +
(42/37)(12/42)(29/12) = 2.7297
Quadrigamous Females (F = 0.0183)
I 4 0 0 .........
2 2 I 2 2 0 0 ......
3 2 1 4 2 0 0 -- -- --
4 2 0 0 2 2 3
5 2 0 0
6 I 2 4 I 4 4 0 0 4 2 3
ARS(broods, quadrigamous females) = cs + cs + cs + cs = (6/4)(I/6) + (4/4)(1/4) + (4/4)(0/4)
+ (4/4)(2/4) = 1.0000
ARS(fledglings, quadrigamous females) = csk + c2s2k + csk + csk = (6/4)(1/6)(2/1) + (4/4)(1/4)(4/1)
+ (4/4)(0/4)(0/0) + (4/4)(2/4)(3/2) = 2.2500
ARS(pa, b, f) = Q (F c,s,) = 0.7293
g=l r=l
ARS(pa, k, f) = Q (F c,s,k,) = 2.0365
TABLE 2. Breeding success of male Spotted Sandpipers mated to monogamous, bigamous, trigamous, and
quadrigamous females. C is the number of clutches laid. S is the number of clutches producing at least one
fledgling. F is the number of fledglings reared.
Clutch
Mates
Monogamous Bigamous Trigamous Quadrigamous All males
number C S F C S F C S F C S F C S F
1 92 31 88 130 48 151 77 25 67 14 2 6 313 106 312
2 36 7 14 43 17 47 40 10 27 4 2 3 123 36 91
3 19 6 19 11 5 9 8 3 7 0 0 0 38 14 35
4 6 2 4 1 0 0 2 0 0 0 0 0 9 2 4
5 2 1 2 0 0 0 1 0 0 0 0 0 3 1 2
155 47 127 185 70 207 128 38 101 18 4 9 486 159 444
c 1.6848 1.4321 1.6623 1.2857 1.5527
p 0.3032 0.3784 0.2969 0.2222 0.3272
k 2.7021 2.9571 2.6579 2.2500 2.7924
cp 0.5108 0.5419 0.4935 0.2857 0.5080
cpk 1.3803 1.6025 1.3118 0.6428 1.4187
Spotted Sandpiper.--The Spotted Sandpiper is
a typically polyandrous species. A female often
switches mates between laying clutches (Oring
1986). For calculating ARS, Eqs. 7 and 8 are
appropriate. The Spotted Sandpiper has been
studied in detail, and reproductive data were
reported in Oring (1982) and Oring et al. (1983).
In the following analysis, I have used Oring's
data from 1973 through 1989 (except 1984 and
1985, when female reproduction was experi-
mentally interrupted).
Females may lay up to five clutches for one,
two, three, or four males (Table 1). Annual re-
productive success is calculated for monoga-
mous, bigamous, trigamous, and quadrigamous
females separately and, then, for all females with
Eq. 7 (Table 1). The ARS(pa, b, f) is 0.7293.
The number of fledglings produced per suc-
cessful brood varied with clutch sequence num-
ber and the number of males per female (Table
1). From Eq. 7, ARS(pa, k, f) for all females is
2.0 (Table 1). The ARS(pa, k, f) can be calculated
separately for females with different numbers
of mates. It is 1.27 for monogamous females,
2.69 for bigamous females, 2.73 for trigamous
females, and 2.25 for quadrigamous females
(calculated from data in Table 1).
I have used Eq. 3 to calculate ARS(m, b, f) for
male Spotted Sandpipers mated to monoga-
mous, bigamous, trigamous, and quadrigamous
females, separately, and for all males (Table 2).
Males mated to bigamous females were most
successful, and males mated to quadrigamous
females were least successful. Inasmuch as
quadrigamy occurred in breeding seasons char-
acterized by high rates of clutch loss, the low
success of males mated to quadrigamous fe-
males may have been a seasonal effect rather
than an effect of its mate being quadrigamous.
Indeed, trigamy and quadrigamy may be a con-
sequence of high nest failure in bad years, which
makes more males available to females for a
third or fourth breeding attempt. If there were
high nest success and if the sex ratio were 1.4,
few females could obtain even two mates. The
overall ARS(m, b, f) for males is 0.5080.
Finally, consider the ratio of the ARS of fe-
males of the breeding population to the ARS of
males of the breeding population. For the Spot-
ted Sandpiper, ARS(pa, b, f)/ARS(m, b, f) =
0.7293/0.5080 = 1.44, and ARS(pa, k, f)/ARS(m,
k, f) = 2.0365/1.4187 = 1.44. This is the sex ratio
(males/females) of the breeding population (i.e.
313/218).
DISCUSSION
The evolution of reproductive behavior.--Several
factors constrain successful reproduction. One
constraint is the availability of mates. In my
hypothetico-deductive theory on the evolution
of mating systems (Murray 1984, 1985b), I ar-
gued that males and females shared mates
(forming polyandrous and polygynous rela-
tionships, respectively) when the availability of
mates was limited and individuals could not
afford to wait for a monogamous relationship.
Waiting always results in increasing the annual
replacement fecundity. If the annual replace-
ment fecundity were greater than the number
of eggs that a female could lay or a male fertilize
in monogamous relationships during a breed-
ing season, then polygyny or polyandry should
evolve.
A second constraint is the loss of eggs or young
in nests, mainly to predation and inclement
weather. Although there is ample time for rear-
ing two broods during a breeding season in the
Prairie Warbler, only 70% of females rear even
one because of high rates of predation on eggs
and young in the nest (Nolan 1978, Murray and
Nolan 1989). Female Prairie Warblers may lay
up to seven clutches in a breeding season. If
the breeding season were longer, more clutches
could be laid (higher cn) in producing a suc-
cessful brood, and we might expect a higher
ARS; if shorter, we might expect a lower c, and
lower ARS.
A third constraint is the length of the breed-
ing season relative to the length of time re-
quired to rear independent young. I have dis-
cussed this constraint with regard to the
evolution of clutch size (Murray 1979, 1985a,
1991). Here, I want to address this constraint
with regard to the evolution of mating systems.
If the time required to rear independent
young is short relative to the length of the
breeding season, then monogamous pairs can
rear more than one brood in a breeding season.
If it is long, then no more than one brood can
be reared by a pair in a typical monogamous
relationship. If it is long, however, more than
one brood could be reared if each parent cared
for one clutch and brood (rapid double-clutch
monogamy) or if one or both parents obtained
more than one mate (polygamy). Let us now
examine the possibilities open to female Spot-
ted Sandpipers by applying the ARS equations
to empirically determined constraints on Spot-
ted Sandpiper reproduction.
The constraints on Spotted Sandpiper repro-
duction are (1) that the time required to rear a
brood successfully, relative to the length of the
breeding season, precludes a second successful
attempt by the parent caring for the eggs and
young and (2) that the probability of nest suc-
cess (s) is 0.327 (i.e. 159 successes/486 clutches;
Table 1). Given these constraints, female Spot-
ted Sandpipers achieve an empirically deter-
mined ARS(pa, b, f) of 0.729 (Table 1). We may
now consider whether it is possible for female
Spotted Sandpipers to achieve an ARS of 0.729
broods as typically monogamous females, as
rapid double-clutch monogamous females, or as
polyandrous females mated to polygynous
males.
First consider typical monogamy. To achieve
an ARS(m, b, f) of 0.729, the average female would
have to lay 2.23 clutches during the breeding
season (Eq. 3). Whereas some females may be
successful with their first clutch, others could
be unsuccessful with their first, second, third,
and fourth. We want to know the expected suc-
cess of the average monogamous female Spotted
Sandpiper during an egg-laying period in Min-
nesota, which varies among years from 34 to 48
days (mean = 42.2; Lank et al. 1985). Consid-
ering the length of incubation (ca. 21 days), the
length of time from hatching to fledging (14-
18 days), the time between failure and comple-
tion of the next clutch (8-10 days; Lank et al.
1985), and the fact that many females do not
arrive at the beginning of the egg-laying period
(Oring et al. 1983, Lank et al. 1985), it seems
unlikely that the average monogamous female
Spotted Sandpiper could produce as many as
two clutches (even as failures) per season. In
fact, the average monogamous female Spotted
Sandpiper lays only !.55 clutches (i.e. 155/100;
Table 1), owing in part to the later arrival of
the females that are monogamous. Seemingly,
typical monogamy seems a poor alternative to
polyandry for the Spotted Sandpiper. Of course,
a pair of Spotted Sandpipers together might
have greater success rearing a brood than either
male or female as a single parent (i.e. s in Eq.
3 could be > 0.327). In fact, lone males are more
successful than males assisted by mates (Oring
et al. 1983, Oring pers. comm.). Typical monog-
amy does not seem to be an option open to
female Spotted Sandpipers.
Perhaps, our hypothetical Spotted Sandpiper
could increase its probability of success by en-
gaging in rapid double-clutch monogamy. As-
suming that the probability of success for
clutches reared by either a male or female is
0.327, we can determine (Eq. 5) that the average
female would still have to lay 2.23 clutches if
it were to achieve an ARS(dcm, b, f) of 0.729.
Conceivably, females could do this during the
34 to 48-day egg-laying period. On average, only
23% of females would have to lay one replace-
ment clutch (i.e. a total of three clutches). One
constraint on success would be how readily an
incubating or brooding male or female could
be seduced into courtship by its mate for the
laying of the replacement clutch. In the Red-
legged Partridge (the best known rapid double-
clutching monogamous species), females lay a
replacement clutch only if a clutch is lost before
incubation starts. Once incubation starts, re-
placement of a lost clutch can occur only after
both mates fail (with s = 0.327, the probability
of both failing is 0.107). Furthermore, few year-
ling females laid two clutches, and only 60-80%
of experienced females attempted two clutches
(Green 1984). Thus, on average, female Red-
legged Partridges lay fewer than two clutches
per season. If the same behavioral restrictions
were placed on female Spotted Sandpipers, they
could not reproduce as well with rapid double-
clutch monogamy as they do with typical poly-
andry.
A second problem, the effect of the sex ratio,
is probably more serious. If clutches and broods
could be successfully tended by one parent, as
in rapid double-clutch monogamy, and if one
sex significantly outnumbered the other, it is
unclear why members of the rarer sex should
maintain pair bonds at all. Seemingly, if males
were the more numerous sex, females could do
better by courting additional mates rather than
by maintaining monogamous pair bonds. If this
were so, then rapid double-clutch monogamy
should be limited to species in which the num-
bers of males and females were about equal. It
seems not coincidental that in the only species
for which we have good information, the Red-
legged Partridge, the sex ratio is near unity
(Green 1984). When the sex ratio is near unity,
an individual improves its chance of producing
two broods by maintaining the pair bond be-
cause, if it deserted, the probability of finding
a second mate would be low. If, on the other
hand, males outnumbered females, and males
alone could successfully rear young, then poly-
andry should be expected.
The monogamous male-polyandrous female
relationships, characteristic of the Spotted
Sandpiper, could occur only if single-brooded
males were reproducing at least at the replace-
ment rate. If single-brooded males were not re-
placing themselves, then their multibrooded fe-
males could not be replacing themselves either
(Murray 1984, 1990).
If male mortality were sufficiently high, males
would have to have a greater clutch size, earlier
age of first breeding, a greater probability of
rearing one, two, or more broods, or some com-
bination of these than with lower mortality (Eq.
1). Given the constraints listed above, males
would have to be multibrooded, increasing their
P2. Males could do this either by rapid double-
clutch monogamy or by obtaining additional
mates. But, if males outnumber females, and if
females desert their mates and are polyandrous,
precluding rapid double-clutch monogamy, it
is difficult to see how males could be multi-
brooded. They could possibly postpone incu-
bation of the first clutch and solicit second fe-
males, as occurs in the Temminck's Stint (I-Iildn
1975). After the male Temminck's Stint fertil-
izes his second female's eggs, he incubates his
first clutch, leaving the second clutch for the
female to incubate. If a female's second mate is
mating for the first time, she may desert him
and solicit a third mate. In such a system, males
have a chance to produce two broods, one by
each of two females, and females could produce
two or more broods, each fertilized by a differ-
ent male.
The demography of polygynous-polyan-
drous systems (such as those of the Temminck's
Stint, Greater Rhea, and some of the tinamous)
and rapid double-clutch monogamous systems
(such as that of the Red-legged Partridge) has
not been described sufficiently well (or at all)
to test these interpretations of the possible cause-
and-effect relationships between demographic
parameters and mating relationships. Never-
theless, we know that rapid double-clutch mo-
nogamy and polygyny-polyandry occur, and it
seems worthwhile to propose what the demog-
raphy of species engaged in these different mat-
ing systems might be.
Assuming that the primary sex ratio remains
constant, Eq. 1 indicates that an increase in mor-
tality must be balanced by an increase in clutch
size, an earlier age of first breeding, an increase
in successfully rearing a first brood (P), an in-
crease in successfully rearing additional broods
(P2, P3, etc.), or some combination of these. In
some species (such as in many Galliformes, rat-
ites, and other ground living forms), the evo-
lutionary response to high mortality is larger
clutches or an early age of first breeding, or
both. But, in shorebirds the evolutionary re-
sponse seems to have been an increase in the
number of broods reared. The only way to in-
crease the number of broods reared, if a brood-
tending parent is limited to one successful brood
per year, is for each member of a pair to care
for a separate brood (rapid double-clutch mo-
nogamy) or for individuals to obtain additional
TABLE 3. Relationship between relative survivorship, sex ratio, and mating relationship of males and females
("high," "moderate," and "low" are relative rates) when the clutch and brood caring parent is limited to
successfully rearing no more than one brood per year. When annual survivorship is either "moderate" or
"low" a maximum of one brood per year is not sufficient to assure replacement.
Relative annual survivorship
Mode of reproduction Males Females Sex ratio
Monogamy "high .... high" 1
Rapid double-clutch monogamy "moderate .... moderate" 1
Polyandry "high .... low" >> 1
Polygyny-polyandry "moderate .... low" > 1
mates that accept rearing the offspring with lit-
tle or no help (polygamy). Which of these pos-
sible mating relationships occurs seems to be a
function of the absolute and relative survivor-
ship of males and females (Table 3).
Seemingly, for any set of circumstances (e.g.
survivorship of males and females, age of first
breeding, length of time required to rear young
to independence relative to the length of the
breeding season, and the probability of rearing
young from a clutch), there seems to be only
one combination of features (e.g. clutch size,
mating system) that could prevail in a "steady-
state" population (one whose numbers are
fluctuating about a mean value). These combi-
nations are those expected from my hypothe-
tico-deductive theories on the evolution of
clutch size (Murray 1979, 1985b, 1991) and mat-
ing systems (Murray 1984, 1985a). Unfortu-
nately, quality demographic data are scarce.
Nevertheless, the general pattern of relation-
ships predicted by the theories are those that
are observed (Murray 1979, 1984, 1985a, b, 1991).
With regard to the evolution of polyandry in
the Spotted Sandpiper, comparative data on its
close relative, the Common Sandpiper (Actitis
hypoleucos), which is monogamous (Holland et
al. 1982), would be useful. What is known is
suggestive. The Common Sandpiper has a rel-
atively high adult survivorship (ca. 90%; Hol-
land et al. 1982) compared with annual return
rates of 60% for experienced females and 62%
for experienced males in the Spotted Sandpiper
(Oring et al. 1983). Furthermore, the Common
Sandpiper has a much higher survival of clutch-
es through hatching (80-84%; Cramp et al. 1983)
and through fledging (ca. 35%; Holland et al.
1982), than the Spotted Sandpiper has (44%
through hatching and 22.4% through fledging;
Oring et al. 1983). Both high survivorship
through hatching and fledging (i.e. high P) and
high annual adult survivorship tend to favor
monogamy (Table 3). More accurately, the ab-
sence of high mortality at these stages does not
enhance selection for multibroodedness or
polyandry (Murray 1984).
Alternative hypotheses.--The alternative hy-
potheses proposed to account for classical poly-
andry in birds (reviewed by Oring 1986) are
nonquantitative, often conflicting, and difficult
to test. The equations I offer here should be
useful in sorting out the many ideas proposed
to account for the various mating relationships.
For example, the equations and analysis support
the views that (1) a sex ratio that favors males
is a precondition for the evolution of polyandry
(Maynard Smith and Ridpath 1972), (2) "clas-
sical" polyandry is favored when females must
produce many eggs because of high predation
rates on clutches (Jenni 1974; Oring 1982, 1986),
and (3) polygyny-polyandry (what Oring calls
"multi-clutch polygamy") is not an evolution-
ary stage between monogamy and polyandry
(Oring 1982, 1986). These ad hoc hypotheses
are consistent with my general theory on the
evolution of mating systems (Murray 1984,
1985a).
Most discussions regarding the evolution of
polyandry refer to such factors as the avail-
ability of resources (mainly food) in time and
space, relative efficiency of males and females
in coping with uniparental care, genetic relat-
edness of the individuals sharing a mate, and
relative abilities of males and females in as-
sessing alternative reproductive opportunities
(reviewed by Oring 1986). Although some of
these factors may be necessary conditions for
the evolution of polyandry, such as successful
uniparental care, none of them is a sufficient
condition, and others may be neither necessary
nor sufficient. For example, for uniparental male
care and polyandry to evolve, it is not necessary
for uniparental male care to be more successful,
or even as successful, as biparental care or even
uniparental female care. It is only necessary that
a female be able to rear more offspring with two
or more males than she could as a monogamous
female. If she were to participate in parenting
because she was the better parent, she would
be unable to replace herself. Thus, failure to
desert her mate and court additional males
would result in extinction. Furthermore, if food
resources precluded successful uniparental
rearing of young, or if the sex ratio did not favor
males, the population would also become ex-
tinct. Thus, polyandry is found in species in
which males outnumber females and males are
able to rear the young unaided (Murray 1984).
The need for reporting the new demographic pa-
rameters.--As indicated above, ARS = P + P2
+ ... P, = cs + c2s2 + ß .. c,s,. Unfortunately,
the values for these parameters are not extract-
able from the data reported in the literature.
The usual report on reproductive success pro-
vides data on clutch size, the number of clutches
started, the fraction of eggs that hatch or of
clutches that produce hatchlings, and the frac-
tion of hatchlings that survive to fledging or
the fraction of clutches that produce fledglings.
Increasingly, with long-term studies of distinc-
tively marked individuals, the average number
of young (e.g. fledglings, yearlings, or breeders)
per female (or pair) is reported (e.g. studies in
Clutton-Brock 1988, Newton 1989). Neverthe-
less, despite the seeming fullness of the re-
ported data, none of the parameters of the ARS
equations can be determined from them.
Measuring ARS rather than clutch (or egg)
success and measuring ARS in terms of broods
reared rather than young reared are essential
for comparing reproductive success within, be-
tween, and among populations (Murray 1991).
For example, I have proposed that small clutch
sizes occur in the tropics because the females'
probabilities of rearing one, two, or more broods
are greater there than at higher latitudes (Mur-
ray 1991). Even though the probability of suc-
cess of a clutch (s,) is low in the tropics, the
number of clutches started (c,) for each brood
and the average number of broods reared could
be larger than in temperate regions, because of
the longer breeding season in the tropics, re-
suiting in a greater ARS(broods) (i.e. P + P2 +
ß.. P,). If ARS(broods) is greater in the tropics
than at higher latitudes, then clutch sizes should
be smaller in the tropics (Eq. 1).
In the only comparison with fairly good data
that I have found, the House Sparrow (Passer
domesticus) has longer breeding seasons and
more but smaller clutches at lower latitudes than
at higher latitudes (Summers-Smith 1988). The
effect of a larger clutch size on reproductive
success at higher latitudes is offset by the short-
er breeding season and fewer clutches laid there,
such that the average number of fledglings pro-
duced per pair per year is virtually the same at
all latitudes. Although ARS in terms of broods
is greater in the tropics, ARS in terms of fledg-
lings reared is approximately the same in trop-
ical and temperate regions.
As a result of this analysis, I urge that the
number of clutches started of first, second, and
later broods (G, c2, etc.) and the probabilities of
first, second, and later brood clutches (s, sv etc.)
successfully producing young to nest-leaving,
if not independence, be reported in future stud-
ies (for examples, see table 2 in Murray and
Nolan [1989], table 3 in Murray et al. [1989], and
Table 1 in this paper).
ACKNOWLEDGMENTS
I am most grateful to Lewis W. Oring who gener-
ously allowed me to examine his data on the Spotted
Sandpiper and permitted me to report the results of
my analysis. I thank David J. T. Hussell, L. W. Oring,
and William M. Shields for commenting on earlier
drafts of the manuscript.
LITERATURE CITED
BRUNING, D.F. 1975. Social structure and reproduc-
tive behavior in the Greater Rhea. Living Bird
13: 251-294.
CLUTTON-BROCK, T. H. (Ed.). 1988. Reproductive suc-
cess. Chicago, Univ. Chicago Press.
CP,aP, S., ET AL. (Eds.) 1983. Handbook of the birds
of Europe, the Middle East, and North Africa,
vol. III. Oxford, Oxford Univ. Press.
EMLEN, S. T., & L. W. ORING. 1977. Ecology, sexual
selection, and the evolution of mating systems.
Science 187: 215-223.
FAnORG, I., & C. B. PATTERSON. 1981. The charac-
teristics and occurrence of cooperative polyan-
dry. Ibis 123: 477-484.
GRANT, B. R., & P. R. GRANT. 1989. Evolutionary
dynamics of a natural population: the Large Cac-
tus Finch of the Gahipagos. Chicago, Univ. Chi-
cago Press.
GREEN, R.E. 1984. Double nesting in the Red-legged
Partridge Alectoris rufa. Ibis 126: 332-345.
HAYS, H. 1972. Polyandry in the Spotted Sandpiper.
Living Bird 11: 43-57.
HILDIN, O. 1975ß Breeding system of Temminck's
Stint Calidris temmincki. Ornis Fennica 52: 117-
146.
HOLLAND, P. K., J. E. ROBSON, & D. W. YALDEN. 1982.
The breeding biology of the Common Sandpiper
Actiris hypoleucos in the Peak District. Bird Study
29: 99-110.
JENNI, D.A. 1974. Evolution of polyandry in birdsß
Am. Zool. 14: 129-144.
--, & G. COHaER. 1972. Polyandry in the Amer-
ican Jacana (Jacana spinosa). Auk 89: 743-765.
LACK, D. 1968. Ecological adaptations for breeding
in birds. London, Methuen.
LANK, D. B., L, W. ORING, & $. J. MAXSON. 1985. Mate
and nutrient limitation of egg-laying in a poly-
androus shorebird. Ecology 66: 1513-1524.
MAYNARD SMITH, J., & M. G. RIDPATH. 1972. Wife
sharing in the TasmanJan Native Hen, Tribonyx
mortierii: a case of kin selection? Am. Nat. 106:
447-452.
MURRAY, B. G., JR. 1979. Population dynamics: al-
ternative models. New York, Academic Press.
1984. A demographic theory on the evolu-
tion of mating systems as exemplified by birds.
Pp. 71-140 in Evolutionary biology (M. Hecht, B.
Wallace and G. Prance, Eds.). New York, Plenum.
1985a. The influence of demography on the
evolution of monogamy. Pp. 100-107 in Avian
monogamy (P. A. Gowaty and D. W. Mock, Eds.).
Ornithol. Monogr. 37. Washington, D.C., Am.
Ornithol. Union.
1985b. Evolution of clutch size in tropical
species of birds. Pp. 505-519 in Neotropical or-
nithology (P. A. Buckley, M. S. Foster, E. S. Mor-
ton, R. S. Ridgely, and F. G. Buckley, Eds.). Or-
nithol. Monogr. 36. Washington, D.Cß, Am.
Ornithol. Union.
ß 1990. Population dynamics, genetic change,
and the measurement of fitness. Oikos 59: 189-
199.
--. 1991. Sir Isaac Newton and the evolution of
clutch size in birds: a defense of the hypothetico-
deductive method in ecology and evolutionary
biology. Pp. 143-180 in Beyond belief: random-
ness, prediction, and explanation in science (J. L.
Casti and A. Karlqvist, Eds.). Boca Raton, Florida,
CRC Press.
--, & V. NOLAN JR. 1989. The evolution of clutch
size. I. An equation for predicting clutch size.
Evolution 43: 1699-1705.
-, J. W. FITZPATRICK, & G. E. WOOLFENDEN. 1989.
The evolution of clutch size. II. A test of the
Murray-Nolan equation. Evolution 43: 1706-1711.
NEWTON, I.(Ed.) 1989. Lifetime reproduction in birds.
New York, Academic Press.
NOLAN, V., JR. 1978. The ecology and behavior of
the Prairie Warbler Dendroica discolor. Ornithol.
Monogr. 26. Washington, D.C., Am. Ornithol.
Union.
ORING, L. W. 1982. Avian mating systems. Pp. 1-92
in Avian biology, vol. 6 (D. S. Farner, J. R. King
and K. C. Parkes, Eds.). New York, Academic Press.
1986. Avian polyandry. Pp. 309-352 in Cur-
rent Ornithology, vol. 3 (R. F. Johnston, Ed.).
New York, Plenum.
--, & M. L. KNUDSON. 1972. Monogamy and
polyandry in the Spotted Sandpiper. Living Bird
11: 59-73.
--, D. B. LANK, & $. J. MAXSON. 1983. Population
studies of the polyandrous Spotted Sandpiper.
Auk 100: 272-285.
PITELKA, F. A., R. T. HOLMES, & $. F. MACLEAN JR.
1974. Ecology and evolution of social organi-
zation in arctic sandpipers. Am. Zool. 14: 185-
204.
SUMMERS-SMITH, J.D. 1988. The sparrows: a study
of the genus Passer: Calton, England, Poyser.
WOOLFENDEN, G. E., & J. W. FITZPATRICK. 1984. The
Florida Scrub Jay: demography of a cooperative-
breeding bird. Princeton, Princeton Univ. Press.