Behavioral Ecology Vol. 13 No. 1: 75-82
© 2002 International Society for Behavioral Ecology
Modeling spawning strategy for sex change under social control in haremic angelfishes
a Department of Information and Computer Sciences, Nara Women's University, Kita-Uoya Nishimachi, Nara 630-8506, Japan b Kyushu International University, Hirano, Yahata-Higashi-ku, Kitakyushu 805-8512, Japan
Address correspondence to N. Shigesada. E-mail: sigesada{at}ics.nara-wu.ac.jp . Y. Sakai is now at the Faculty of Applied Biological Science, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8528, Japan.
Received 1 May 2000; revised 22 March 2001; accepted 22 March 2001.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
|---|
In haremic angelfishes where protogynous (female to male) sex change is favored, females have been reported to adopt several tactics for earlier sex change on the basis of a trade-off between reproduction and growth, or survivorship. A recent field study on Centropyge ferrugatus revealed that females reduce spawning frequency in competition with similar-sized neighbors for social dominance. To evaluate the optimal spawning strategy taken by haremic fishes, we developed an evolutionarily stable strategy model that focuses on their life history and social structure based on field data of C. ferrugatus. The results of the analysis predict that the spawning frequency will be low when the mortality rate of females is high compared with males, the harem size is large, and there is a moderate degree of social control. Our model further predicts conditions under which females completely stop spawning, as if they have become bachelors. Thus, the regulated spawning frequency may be taken as a strategy to optimize the reproductive success of an individual in response to the available choices for sex change, social control, and environmental conditions. Social control would also play an important role in sex change in many other haremic species.
Key words: angelfish, bachelor sex change, Centropyge ferrugatus, harem-fission sex change, social control, spawning frequency, takeover sex change.
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
|---|
Sequential hermaphroditism (sex change) has been widely observed in marine fishes (Sadovy and Shapiro, 1987
; Yogo, 1987).
In particular, protogynous (female to male) sex change has been reported in
many haremic fishes with polygynous mating systems
(Moyer, 1990;
Robertson and Warner, 1978,
Ross, 1990;
Warner and Robertson, 1978).
According to the size advantage model, small males of haremic fish should have
a low fitness because they seldom would be able to acquire females in
competition with larger males (Charnov,
1982; Ghiselin,
1969; Warner,
1978,
1988b). In contrast, each
individual could maximize its fitness if it participates in reproduction first
as a female while it is small and later as a dominant male after it becomes
large enough. Thus, protogynous strategy would be selectively favored in
haremic fishes. However, more complex life-history pathways in relation to sex
change have recently been reported in many sequentially hermaphroditic fishes
(Aldenhoven, 1986;
Charnov, 1982;
Kuwamura and Nakashima, 1998;
Warner, 1984b,
1988a).
In marine fish harems, females often exhibit a body size-based linear
dominance relationship, which is closely reflected in the spatial pattern of
their home ranges (Hoffman,
1985; Kuwamura,
1984; Sakai and Kohda,
1997). For example, individuals of a similar body size
aggressively compete with each other to develop territorial relationships,
while individuals of different body sizes could cohabit in such a way that the
home ranges of smaller individuals are hierarchically included in those of
larger ones. Kuwamura (1984)
has noted two types of harems with slightly different spatial relationships: a
"linear harem" composed of one male and multiple females whose
home ranges overlap each other, and a "branching harem" composed
of two or more linear harems (subgroups) of females, each of which occupies a
separate range within the territory of one male (for terminology, see
Kuwamura, 1984; see also
Sakai and Kohda, 1997). Sex
change in these harems is generally under social control by the male
(Robertson, 1972;
Ross, 1990;
Warner, 1988b), so that the
dominant female can become a male and take over the harem only when the male
disappears ("takeover sex change";
Moyer, 1991;
Moyer and Nakazono, 1978;
Robertson, 1972;
Sakai, 1997).
In several haremic fishes, however, females may change sex even in the
presence of the dominant male, which has been called "early sex
change" in the general sense (Moyer
and Zaiser, 1984). Two contexts of early sex change can be
distinguished. One is the "bachelor sex change," whereby a female
becomes a bachelor male and spends a reproductively inactive period to wait
for a territory vacancy (Aldenhoven,
1984,
1986;
Moyer, 1987;
Moyer and Zaiser, 1984;
Warner, 1988a,
1991). The second is
"harem-fission sex change"
(Aldenhoven, 1986;
Lutnesky, 1994;
Moyer, 1991;
Robertson, 1974;
Sakai, 1997) as observed with
angelfish Centropyge ferrugatus. Angelfish usually form a linear
harem, where only the takeover sex change occurs. Occasionally, however,
angelfish also form a branching harem when the male disappears from one of two
adjacent harems. Then the two dominant females in the newly formed branching
harem begin to engage in direct competition, and one of them eventually
changes to male (Sakai, 1997).
This harem-fission sex change is presumed to take place because the single
preoccupant male would not be able to maintain sufficient social control over
the expanded harem. In the process of such sex change, females of C.
ferrugatus in adjacent harems tend to spawn less frequently and grow
faster than those in isolated harems, as though they make a trade-off between
reproduction and growth so as to gain competitive advantage. A similar
trade-off has generally been observed in other fishes
(Schultz and Warner, 1991;
Warner, 1984a;
Wootton, 1990). Thus, these
alternative spawning tactics may be developed for acquiring territories and
achieving mating success in the harem system.
In this study, we constructed mathematical models based on field data of
Sakai (1997) to evaluate the
reproductive success of primary females in the mode of the harem-fission sex
change and to determine the spawning frequency at an evolutionarily stable
state (ESS). Then we extended the model to a more generalized one to explain
the occurrence of other sex change modes and discuss the plasticity of sex
change tactics, based on data on several species.
Harem structure and spawning tactics of angelfish, C.
ferrugatus
Sakai (1997) studied harems
of C. ferrugatus at a coral reef on Sesoko Island, Okinawa, and
recognized two types of distribution of harems. One type contains two or three
harems adjacent to each other ("adjacent harems") and the other
consists of single harems spatially isolated about 10 m from the nearest harem
("isolated harems"). In either distribution, the most common type
was the linear harem, in which the females' home ranges hierarchically overlap
each other. In the isolated harems, takeover sex change occurred exclusively
in all cases observed. In adjacent harems, in contrast, upon disappearance of
one male, the remaining male expanded his territory to occupy the vacated
harem, thereby forming a branching harem
(Figure 1a,b). Subsequently,
one of the dominant females changed sex despite the presence of the occupant
male, presumably because the expanded harem was too large for a single male to
maintain sufficient social control. The newly converted male immediately went
on to acquire one of the subgroups of females (harem-fission sex change).
Consequently, each branching harem was reformed into two adjacent linear
harems (Figure 1c). Thus, the
branching harem is taken to represent a transitory phase in the re-formation
of adjacent linear harems: the mean duration of branching harems (39 days) is
much shorter than that of linear harems (213 days).
|
Mortality rates did not significantly differ either between males and females or among females with different spawning rates (ca. 0.387/year). Recruitment of males was rarely seen. Females spawn at around sunset, and the number of eggs per spawning is not correlated with the female's body size. The spawning frequency of females in adjacent harems (percentage of days on which spawning is observed = 79.6% ± 25.2 SD, n = 18) was lower than that in isolated harems (97.6% ± 5.8 SD, n = 12). In particular, females with neighbors similar in size tended to spawn less frequently. Furthermore, there was an inverse correlation between the spawning frequency of a female and her growth rate.
In the light of these observations, Sakai reasoned that females in adjacent harems might be driven to compete for a prospective vacancy in advance. This competition would be especially severe among similar-sized females, resulting in reduced spawning frequency. In the next section, we discuss optimal spawning tactics in the harem society of C. ferrugatus that involves a branching harem. We then extend the model to the case in which a male keeps the branching harem for a long time under his strict social control and examine how such control affects the spawning frequency.
A basic model
Here we focus on a simple case in which two harems are adjacent to each
other and each consists of one male and n females. The primary
females in the two harems spawn infrequently when they are similar in size. We
shall hereafter call one of the primary females the "focal female"
and the other the "opponent." In contrast to the primary females,
subordinate females spawn every day. We call these females "normal
females."
Let 
f denote the focal female's spawning rate (fertility)
per unit time, and o that of the opponent. Meanwhile, the
normal female's fertility per unit time, n, is
assumed to be constant, independent of body size
(Sakai, 1996). Once the
primary females begin to have lower spawning rates, they continue to keep
those rates. The mortality rate of a male, 
m, is assumed to
be constant, and that of a female, f, to be independent of
her social ranking.
When the primary females reject courtship, the energy saved by the reduced
spawning can be reinvested in their growth at rates proportional to
f = n - f and
o = n - o,
respectively. When one of the males in two adjacent harems disappears, the
focal female and the opponent immediately start a lottery competition in which
their winning rates are assumed to be proportional to the invested energies in
their growths to the power of 
:
f and
o, respectively. Thus, the winning
probability of the focal female is given by,
 | (1) |
We call (>0) the winning efficiency because, as
increases, the focal female's competitive ability is more effectively enhanced
by restrained spawning. Figure
2 illustrates the dependency of the winning probability on
f for = 0.5, 1 and 2, where
o is fixed. When = 1, the winning probability,
(f, o), increases as a parabolic function
of f. When > 1, (f,
o) rises in a sigmoidal curve with increasing
f. Conversely, when < 1, the two females
have a close game, in spite of the difference between their spawning
frequency. In two special situations, the winning rate becomes independent of
: 0.5 at f = o, where two
females adopt the same tactics, and zero at f = 0,
where the focal female always spawns.
|
In the case where the primary female in a harem dies or undergoes sex
change, the normal females in the harem are promoted by one rank each, and a
new female is recruited for the lowest rank. The new primary female is assumed
to spawn at rate o, according to the formality of the
subsequent ESS analysis in which all the dominant females except the focal
female have the same spawning rate. Because a male reproduces with both the
primary female and n - 1 normal females, the reproductive rate of the
focal female after sex change, m, is given by
 | (2) |
Given these basic assumptions, we now consider the possible life history of
the focal female starting with two complete adjacent harems as
diagrammatically illustrated in Figure
3. At time t = 0, the focal female and the opponent are
assumed to start a game by reducing their spawning rates to f
and o, respectively. First, one of the males or one of the
primary females will disappear, after which the whole system will move along
one of the six possible routes shown in
Figure 3. On each route, the
system progressively changes its state with the death of one individual at a
time. The game is over when the focal female dies.
|
When one of the males dies at first, the focal female may change sex to male either by outcompeting the opponent at the initial chance (route 1) or taking the next chance of automatically succeeding the death of the second male after her initial defeat by the opponent (route 2). Or else, the focal female dies in vain before that next chance comes, resulting in the end of the game (route 3). Route 4 is the case when the opponent and one male die successively, after which the focal female automatically changes to male. In route 5 or route 6, the focal female dies either second after the opponent or first among all, respectively.
We examine how frequently these routes occur and to what extent they
contribute to the reproductive success of the focal female. Thus we obtain the
average reproductive success, wi, for each of the
six routes as below (see Appendix A for derivation).
 | (3a) |
 | (3b) |
 | (3c) |
 | (3d) |
 | (3e) |
 | (3f) |
 |
f and o, respectively.
D is the ratio of the male's mortality relative to that of female,
and T is the normal female's spawning rate divided by its mortality. Summing
up all wi, we have the total reproductive success
of the focal female as
 | (4) |
*,
W(f, *) should have a maximum when
f = *
(Bulmer, 1994;
Maynard Smith, 1982).
Therefore, * that satisfies the following equation may be an
ESS frequency:
 | (5) |
Because Equation 5 includes parameters D, T, n, ,
and *, * should be given as a function of
D, T, n and . However, T is irrelevant to the
calculation of Equation 5, as W(f, *)
linearly depends on T. Therefore, * can be reduced to a
function of three parameters, D, n, and . We performed
numerical calculations of * for various values of these
parameters and confirmed that the resultant solutions for *
indeed give optimal values of W(f,
*).
|
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
|---|
We examined how the relative spawning rate at the ESS,
*,
depends on the male's mortality relative to that of female, D.
Figure 4a shows
* plotted against D for the cases of n = 3,
6, and 9, when = 1. The larger is D, the higher becomes
*. Thus, when the male's mortality is high, it should be more
advantageous for females to remain females and to directly invest in
reproduction by spawning frequently. Figure
4a also indicates that * decreases with increases
in the harem size, n, because the focal female could expect a large
reproductive gain by sex change. For n = 9, in particular, the focal
female does not spawn at all when D < 0.6. In contrast,
* increases to the maximum limit, * = 1, when
D > 3 at n = 3.
|
The mortality dependency of * for various combinations of
= 0.5, 1, 2 and n = 3 and 6 is illustrated in
Figure 4b. When D is
small, * becomes smaller with increases in . This means
that females tend to invest in growth when the winning efficiency is high.
However, when D becomes large enough (D 
2.9 for
n = 3 and D 6 for n = 6), *
converges to 1 regardless of . The result predicts that the best
strategy for females is to spawn at the maximal rate when the male's mortality
is sufficiently high.
Figure 5 illustrates the
relationship between * and harem size, n, for
= 0.5, 1, 2 at D = 1. * linearly decreases with
n, and the rate of decrease becomes higher as increases. For
example, with every increment of 1 in n, * decreases
by about 0.2, 0.1, and 0.05 for = 2, 0.1, and 0.5, respectively.
Incidentally, the harem size of C. ferrugatus is 2.3, and the average
spawning frequency of females in adjacent harems is 0.723. These data fit well
with the theoretical prediction for = 2, as shown in
Figure 5.
|
A generalized model
In this section, we extend the previous model to a more general one that
can be applied to other species in addition to C. ferrugatus. The
number of females that can be controlled by a male varies with haremic species
(e.g., Aldenhoven, 1984;
Robertson, 1974; Warner,
1988a,
1991). The primary female in a
harem will be able to change sex when the harem is too big for the male to
control. In fact, sex change has been induced by artificial addition of
females to isolated harems (Lutnesky,
1994). Conversely, if a male imposes strict social control, he
would be able to maintain even a branching harem, which holds twice as many
females as a linear harem.
Consider a situation in which a branching harem could persist for a certain
duration until it is terminated by one of two alternative events: (1) one of
the females undergoes sex change to male, resulting in harem-fission (see
Figure 6), or (2) either the
male or one of the dominant females dies (see
Figure 6). Note that the
previous model assumed that only case 1 occurs immediately. In case 1, the
focal female and the opponent engage in a lottery competition. Here we assume
that the rates of takeover by the focal female and the opponent per unit time,
a and a*, are proportional to the th power
of the energies invested in their growth, respectively:
 | (6) |
 | (7) |
represents the strength of social control by the male and is
referred to as the "social control strength." When is
large enough, takeover does not occur. However, as becomes smaller,
the rate of sex change increases. When goes to 0, the present model is
reduced to the previous one, in which takeover occurs instantaneously, where
the focal female wins at the probability (f,
o) as given by Equation 1. Note, however, that the social
control strength, , characterizes the situation specific to a branching
harem, in which the resident male has to control two dominant females.
Therefore even when becomes 0, a male can sufficiently control the
single dominant female in a linear harem regardless of the total number of
females.
|
If such diverging pathways for a branching harem are taken into account at each corresponding step in the basic model shown in Figure 3, there are 20 possible routes. For this generalized model, we calculate the reproductive success of the primary female at the ESS in the same way as in the previous model (see Appendix B for how to derive the probabilities of occurrence of the five alternative events in Figure 6).
We first examine how the social control strength, , relates to
*. Figure 7
shows * plotted against for all combinations of
= 1, 2 and n = 2, 3, 6, when D = 1. When is
zero, * coincides with the results from the basic model as
expected. As increases, * decreases at first and then
turns to increase monotonically until it reaches unity at a certain social
control strength. The initial decrease is enhanced with increase in either
n or . Note, however, that for the special case of n
= 6 and = 2, * starts with 0 so that there is no
minimum. The state of * = 1 reached at large values of
means that the female spawns maximally. The occurrence of this state is
readily understandable because, if social control strength is sufficiently
large, a female has scarce chances for takeover, so that she could optimize
her reproductive success by spawning at the maximal rate. But why there is an
initial decrease in * as the level of social control rises
above zero? The likely reason may be that as social control increases, so too
does the expected reproductive success of becoming a male, since the male has
the opportunity to maintain a branching harem. In addition, the generalized
model includes an extra state (marked as 4 in
Figure 6), at which a current
decrease in spawning of the focal female will open the way to greater
reproductive success by subsequent sex change. Furthermore, the enhanced
initial decrease in * with increases in n or
is explainable by the fact that sex change should be advantageous because of a
high reproductive rate as male or a high winning efficiency, respectively.
|
Figure 8 illustrates the
mortality dependency of * for n = 3 and 6, when
= 0.05 and = 1, the conditions under which a moderate degree
of social control is present. For comparison, the broken lines in
Figure 8 show the corresponding
results from the basic model, which is equivalent to the extreme case in the
generalized model for = 0 (no social control). Changes of
* with D when = 0.05 are more marked than in
the case of = 0. More specifically, * is smaller in
the generalized model than in the basic model when D is below a
certain value, but that relation becomes reversed at higher D. This
difference may again be explained by the same argument as above: Although
social control increases the benefit of becoming male, it simultaneously
increases the cost of competing, as it delays the opportunity for sex change.
When D is low, the benefit dominates to facilitate sex change.
Conversely, when D is high, the cost is more important, resulting in
repressed sex change.
|
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
|---|
Haremic fishes undergo sex change in three different modes, takeover sex change, harem-fission sex change and bachelor sex change, under various environmental and social contexts. C. ferrugatus has been reported to show variable spawning rates, depending on whether they undergo takeover sex change or harem-fission sex change (Sakai, 1997
). Here we developed a model to capture the life history of
protogynous haremic fish in which two primary females partake in a game to
gain the dominant status in the presence of varying social control by the
male. By incorporating a new conceptual parameter, , the social control
strength, this model allows us to quantitatively evaluate the spawning rate at
the ESS.
As a general trend, a high spawning rate of the dominant female is favored
when the social control strength () is strong, the relative mortality
rate of male to that of female (D) is high, the harem size
(n) is small, and the winning efficiency () is low (see
Figures 4,
5,
7, and
8). If these conditions are
satisfied in proper combinations and to sufficient degrees, a female will
continue to spawn maximally until a loss of the male opens the chance of
takeover sex change. Conversely, she will decrease her spawning rate as these
conditions are reversed. The tactics of restrained spawning may be adopted in
anticipation of harem-fission sex change in the near future. In extreme cases,
a female completely stops spawning in a way analogous to bachelor sex change
(Figures 4,
5,
7, and
8). Meanwhile, the model also
predicts an intriguing situation where the spawning rate attains a minimum at
an intermediate value of when the harem size is large
(Figure 7).
In the case of harem-fission sex change in C. ferrugatus studied
by Sakai (1997), the observed
spawning frequency fits reasonably well with the theoretical model when the
winning efficiency, , is assumed to be 2. This implies that the tactics
of restrained spawning may efficiently enhance the ability to compete for
dominant status. Harem-fission sex change has also been observed in C.
interruptus, C. tibicen (Moyer and
Zaiser, 1984), Labroides dimidiatus
(Robertson, 1974), and C.
ferrugatus at another location
(Moyer, 1987). Robertson
(1974) reported that the
harem-fission sex change occurred in Labroides dimidiatus when social
control by the male was weakened in either of the following situations: the
male's home range was shifted away, harem size was so large as to contain more
than two branches, and the female reduced interactions with the male by
staying on the border of the male's territory. Another example of weakened
social control was seen with C. tibicen in Miyake Island
(Moyer and Zaiser, 1984),
where the females' home ranges were sparsely distributed at distances of 20-30
m from each other and the male sequentially visited with females in his harem.
There the largest female was found to spawn at a markedly reduced rate of 33%,
while medium-sized females spawned at 87%. Furthermore, in enclosure
experiments of C. potteri, Lutnesky
(1994) demonstrated that a
larger harem size facilitates sex change in the presence of the male. These
observations seem to fit well with the predictions from our theoretical model.
The above noted fishes were also reported to undergo bachelor sex change. Our
model may be applicable to those cases as well, although sufficient data are
not yet available to perform fitting with the model.
Mathematical models for bachelor sex change have previously been proposed
from different angles by Aldenhoven
(1986) and Iwasa
(1991). Aldenhoven
(1986) found that C.
bicolor on the Great Barrier Reef is apt to undergo bachelor sex change
when the harem density is high (i.e., the number of adjacent harems is 5.1),
so that bachelors have a short waiting time before acquiring a harem.
Aldenhoven proposed a model to evaluate the reproductive value of staying
female or becoming a bachelor male under the assumption that bachelor males
randomly gain ownership of vacated harems. The study suggested that females
assess the number of bachelors and the harem density to decide whether they
should change sex. As a result, there is a frequency-dependent equilibrium
between takeover sex change and bachelor sex change. On the other hand,
bachelor sex change seldom occurs when the number of adjacent harems is less
than three, and hence the waiting time becomes long. This situation seemingly
resembles that of C. ferrugatus studied by Sakai
(1997), in which the number of
adjacent harems is two to three. Under such conditions, however, females of
C. bicolor spawn maximally, unlike C. ferrugatus, which
shows restrained spawning. This difference may be related to the fact that the
gonad weight of C. bicolor increases with its body size
(Aldenhoven, 1984), whereas
C. ferrugatus does not have such a tendency
(Sakai, 1996). Therefore, it
may be advantageous for a large, dominant female of C. bicolor, but
not C. ferrugatus, to maintain her full reproductive activity while
she is waiting for the chance of takeover sex change.
Iwasa (1991) used dynamic
programming to analyze the optimal sex change timing without spatial
restriction or social control. He showed that bachelor sex change may occur if
reproductively active individuals suffer from greater mortality or reduced
growth rate and found that there is a sexual difference in fertility increases
with size (age). Our result resembles his result despite the difference in the
assumed social structure.
The novelty of the present model is that it is a highly mechanistic one which focuses on the life history of the dominant females in flexible haremic structures and thus can estimate the ESS optimal spawning frequency as a continuous variable. With this framework, the model can explain how various modes of sex change arise depending on life-history parameters, the degree of social control, and the harem structure. Although the present model, either the basic model or the generalized version, contains a number of assumptions specific to C. ferrugatus, similar mechanistically based approaches with appropriate modifications would be extendable to a wider variety of haremic species.
|
APPENDIX A |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
|---|
Derivation of the average reproductive success as given by Equation 3
Here we first describe the detailed procedure for derivation of average reproductive success for route 1 of Figure 3. Let t1 and t2 denote the times of death of one male and the focal female after sex change, respectively. Then the following events occur successively at the respective probabilities indicated: (1) All the individuals (the two males and the focal female and the opponent) survive until t1, exp(-2
mt1)
exp(-2ft1); (2) one of the males dies
between t1 and
t1+dt1,
2mdt1; (3) the focal female wins,
(f, o); (4) the focal female newly
converted to male survives until t2,
mexp[-m(t2 -
t1)]; (5) the converted male dies between
t2 and t2 + dt2,
mdt2. Multiplying these probabilities
yields the overall probability of these events as:
 |
ft1 +
m(t2 - t1), where the
first and second terms represent gains as female and male, respectively. Thus,
we have the average reproductive success for route 1 by multiplying the
overall probability defined above and the corresponding reproductive success
followed by integrating the resultant product with respect to
t1 and t2 (t2 >
t1 >0):
 | (A1) |
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
|
APPENDIX B |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
|---|
A brief description of how to derive the probabilities of the five alternative events in Figure 6
In event 1 in Figure 6, the focal female wins and changes to male leading to harem fission. The probability for this event to occur between t and t + dt after formation of the branching harem is given by aexp(-
mt - 2ft
- at - a*t)dt, where the term
exp(-mt - 2ft -
at - a*t) represents the probability
that neither takeover nor any death occurs before t, and adt
is the probability that the focal female takes over between t and
t + dt. In a similar way, the corresponding probability for
the opponent to win (event 2) is given by
a*exp(-mt -
2ft - at -
a*t)dt.
In the remaining three events (3, 4, and 5 in
Figure 6), death occurs to
either the male, the opponent, or the focal female. The probability of these
events is given by the product of the mortality rate of each individual,
(mdt, fdt, or
fdt), and the probability of all three to survive
till t, exp(-mt -
2ff - at -
a*t).
|
ACKNOWLEDGEMENTS |
|---|
We are grateful to Y. Iwasa, K. Kawasaki, T. Kuwamura, H. Seno, and S. Takahashi for their fruitful discussions. This work was supported in part by the Grant-in-Aid for Scientific Research Fund from the Japan Ministry of Education, Science, Culture and Sports (no. 08640804 and no. 09NP1501).
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
|---|
Aldenhoven JM, 1984. Social organization and sex change in an angelfish, Centropyge bicolor on the Great Barrier Reef (PhD dissertation). New South Wales: Macquarie University.
Aldenhoven JM, 1986. Different reproductive strategies in a sex changing coral reef fish Centropyge bicolor (Pomacanthidae). Aust J Mar Freshwater Res 37: 353-360.
Bulmer M, 1994. Theoretical evolutionary ecology. Sunderland, Massachusetts: Sinauer Associates.
Charnov EL, 1982. The theory of sex allocation. Princeton, New Jersey: Princeton University Press.
Ghiselin MT, 1969. The evolution of hermaphroditism among animals. Q Rev Biol 44: 189-208.[Medline]
Hoffman SG, 1985. Effects of size and sex on the social organization of reef-associated hogfishes, Bodianus spp. Environ Biol Fish 14: 185-197.
Iwasa Y, 1991. Sex change evolution and cost of
reproduction. Behav Ecol 2:
56-68.
Kuwamura T, 1984. Social structure of the protogynous fish Labroides dimidiatus. Publ Seto Mar Biol Lab 29: 117-177.
Kuwamura T, Nakashima Y, 1998. New aspects of sex change among reef fishes: recent studies in Japan. Environ Biol Fish 52: 125-135.
Lutnesky MMF, 1994. Density-dependent protogynous sex
change in territorial haremic fishes: models and evidence. Behav
Ecol 5:
375-383.
Maynard Smith J, 1982. Evolution and the theory of games. Cambridge: Cambridge University Press.
Moyer JT, 1987. Social organization and protogynous hermaphroditism in marine angelfishes (Pomacanthidae) [in Japanese]. In: Sex change in fishes (Nakazono A, Kuwamura T, eds). Tokyo: Tokai University Press; 120-147.
Moyer JT, 1990. Social and reproductive behavior of Chaetodontoplus mesoleucus (Pomacanthidae) at Bantayan Island, Philippines, with notes on pomacanthid relationships. Jpn J Ichthyol 36: 459-467.
Moyer JT, 1991. Comparative mating strategies of labrid fishes. Watanabe Ichthyology Institute Monograph 1. Tokyo: Watanabe Ichthyological Institute.
Moyer JT, Nakazono A, 1978. Population structure, reproductive behavior and protogynous hermaphroditism in the angelfish Centropyge interruptus at Miyake-jima, Japan. Jpn J Ichthyol 25: 25-39.
Moyer JT, Zaiser MJ, 1984. Early sex change: a possible mating strategy of Centropyge angelfishes (Pisces: Pomacanthidae). J Ethol 2: 63-67.
Robertson DR, 1972. Social control of sex reversal in
a coral-reef fish. Science 177:
1007-1009.
Robertson DR, 1974. A study of the ethology and reproductive biology of the labrid fish, Labroides dimidiatus at Heron Island, Great Barrier Reef (PhD dissertation). Brisbane: University of Queensland.
Robertson DR, Warner RR, 1978. Sexual patterns in the labroid fishes of the Western Caribbean, II: The parrotfishes (Scaridae). Smithson Contrib Zool 255: 1-26.
Ross RM, 1990. The evolution of sex-change mechanisms in fishes. Environ Biol Fish 29: 81-93.
Sadovy Y, Shapiro DY, 1987. Criteria for the diagnosis of hermaphroditism in fishes. Copeia 1987: 136-156.
Sakai Y, 1996. Fecundity of female angelfish, Centropyge ferrugatus, independent of body size: field collection of spawned eggs. Ichthyol Res 43: 186-189
Sakai Y, 1997. Alternative spawning tactics of female
angelfish according to two different contexts of sex change. Behav
Ecol 8:
372-377.
Sakai Y, Kohda M, 1997. Harem structure of the protogynous angelfish, Centropyge ferrugatus (Pomacanthidae). Environ Biol Fish 49: 333-339.
Schultz ET, Warner RR, 1991. Phenotypic plasticity in life—history traits of female Thalassoma bifasciatum (Pisces: Labridae): 2. Correlation of fecundity and growth rate in comparative studies. Environ Biol Fish 30: 333-344.
Warner RR, 1978. The evolution of hermaphroditism and unisexuality in aquatic and terrestrial vertebrates. In: Contrasts in behavior (Reese ES, Lighter FJ, eds). New York: John Wiley and Sons; 77-101.
Warner RR, 1984a. Deferred reproduction as a response to sexual selection in a coral reef fish: a test of the life historical consequences. Evolution 38: 148-162.
Warner RR, 1984b. Mating behavior and hermaphroditism in coral reef fishes. Am Sci 72: 128-136.
Warner RR, 1988a. Sex change in fishes: hypotheses, evidence, and objections. Environ Biol Fish 22: 81-90.
Warner RR, 1988b. Sex change and the size-advantage model. Trends Ecol Evol 3: 133-136.
Warner RR, 1991. The use of phenotypic plasticity in coral reef fishes as tests of theory in evolutionary ecology. In: The ecology of fishes on coral reef (Sale PF, ed). San Diego, California: Academic Press: 387-398.
Warner RR, Robertson DR, 1978. Sexual patterns in the labroid fishes of the Western Caribbean, I: The wrasses (Labridae). Smithson Contrib Zool 254: 1-27.
Wootton RJ, 1990. Ecology of teleost fishes. New York: Chapman and Hall.
Yogo Y, 1987. Hermaphroditism and the evolutionary aspects of its occurrences in fishes [in Japanese]. In: Sex change in fishes (Nakazono A, Kuwamura T, eds). Tokyo: Tokai University Press: 1-47.
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||