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Behavioral Ecology Vol. 13 No. 3: 337-343
© 2002 International Society for Behavioral Ecology
Protandry models and their application to salmon
Behavioural Ecology Research Group, Department of Biological Sciences, Simon Fraser University, Burnaby BC V5A 1S6, Canada
Address correspondence to Y.E. Morbey, who is now at the Department of Zoology, Ramsay Wright Zoological Laboratories, University of Toronto, 25 Harbord Street, Toronto, ON M5S 3G5, Canada. E-mail: morbey{at}zoo.utoronto.ca .
Received 8 October 2000; revised 25 June 2001; accepted 5 July 2001.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONDISCUSSIONREFERENCES |
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Mating systems characterized by restricted breeding seasons, male polygamy, and female monogamy are common among animals. In such systems (e.g., butterflies), the earlier emergence of males than females to breeding areas (protandry) is a typical phenological pattern. Protandry likely results from a timing strategy that maximizes mating opportunities by males. In Pacific salmon (Oncorhynchus spp.), males typically arrive at the spawning grounds in advance of females. Using arrival-timing models, I found that under the mate-opportunity hypothesis, the mating system of salmon favors protandry. Protandry is predicted under a range of competitive scenarios, and the degree of protandry is especially sensitive to the duration of male spawning activity. Greater protandry is expected with increasing population sex ratio (i.e., more males) when there is mate guarding, but lower protandry is expected with increasing population sex ratio when interference competition among males reduces male longevity. The timing of unequal competitors is expected to be similar, but among years, protandry may be less variable in the better competitor.
Key words: arrival timing, Onchorhynchus, protandry, salmon.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONREFERENCES |
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In mating systems characterized by a greater frequency of mating by males than females, males tend to arrive at breeding areas earlier on average than females (e.g., arthropods: Thornhill and Alcock, 1983
; ground squirrels:
Michener, 1984; Pacific
salmon: Morbey, 2000). This
form of sexbiased timing, called protandry, theoretically allows males to
maximize their mating opportunities with females (e.g.,
Botterweg, 1982;
Iwasa et al., 1983;
Wiklund and Fagerström,
1977). Protandry (and the less frequent, opposite pattern called
protogyny) may occur for different reasons in different taxa. For example, in
many migrant bird species, competition for territories may select for the
earlier arrival of the territorial sex (usually males)
(Ketterson and Nolan, 1976;
Myers, 1981). Most theoretical
work on protandry has focussed on the mate-opportunity hypothesis and how it
relates to protandry in arthropods. I assessed how well the mate-opportunity
hypothesis explains protandry in a different taxon, Pacific salmon
(Oncorhynchus spp.). The biology of salmon seems consistent with the
mate-opportunity hypothesis because reproduction is highly seasonal and
because male salmon mate with several females and have a longer period of
mating or spawning activity than females (e.g., chum salmon O. keta:
Schroder, 1982; sockeye salmon
O. nerka: McPhee and Quinn,
1998). Alternative hypotheses for protandry also are generally
inconsistent with salmonid biology
(Morbey, 2000).
I first determined whether the maximization of mating opportunities selects
for protandry in salmon by using biologically realistic parameters in a
modified version of an existing arrival-timing model. Competitive inequalities
are a feature of the salmonid breeding system, and so my second objective was
to incorporate competitive inequalities in arrival-timing models to quantify
their affect on protandry. For example, large male salmon often have an
advantage when competing against small males for access to females (coho
salmon O. kisutch: Fleming and
Gross, 1994; sockeye salmon:
Quinn and Foote, 1994). Males
in mate-guarding positions also may acquire familiarity with local habitat
features, which helps them during male—male competition (sockeye salmon:
Chebanov, 1997;
Foote, 1990). Previous models
of protandry in insects consider how the benefits of large male size, attained
through prolonged development, affect optimal protandry (Zonneveld
1996a,b).
I did not consider such a trade-off between early arrival and large size. My
third objective was to examine how well arrival-timing models predict
protandry in salmon by comparing the predictions to observed protandry from my
earlier study (Morbey,
2000).
The model
I followed the modeling approach of Iwasa et al.
(1983) and Parker and Courtney
(1983) and assumed that
selection acts on male arrival timing. The model is a game with n
players, and the strategy set contains a continuous set of timing options
(i.e., arrival days at breeding areas)
(Maynard Smith, 1982). This
type of model is often referred to as an ideal free distribution in time
because males are assumed to have complete knowledge of the female arrival
distribution and are free to arrive (i.e., are not prevented from arriving) on
any day of the season (Fretwell and Lucas,
1970). The evolutionarily stable strategy is a probability
distribution of arrival days, with the fitness of males among arrival days
equal.
I began with an equal competitors model with the same general assumptions
of Iwasa et al.'s (1983) and
Parker and Courtney's (1983)
models. Males were assumed to engage in scramble competition for unmated
females on each day while alive and were subject to both prearrival mortality
and postarrival mortality. My model differs in the form of postarrival
mortality and in the frequency of mating by females. I assumed that
postarrival mortality occurs mainly due to senescence (e.g.,
Groot et al., 1995) and that
male longevity (period between arrival and death) declines with arrival day
(English et al., 1992;
Hendry, 1998;
McPhee and Quinn, 1998;
Neilson and Banford, 1983;
Neilson and Geen, 1981;
Perrin and Irvine, 1990; but
see Fukushima and Smoker,
1997; van den Berghe and
Gross, 1986). Daily predation risk while spawning was not
considered because it is likely low in most populations. Salmon can use deep
areas as refuges, and important predators such as bears tend to be more
successful at capturing older, postspawning salmon (e.g., chum salmon:
Reimchen, 2000). However,
predation on spawning adults may be substantial in some years and in some
populations (e.g., sockeye salmon:
Ruggerone et al., 2000) and,
if present, would affect protandry in the same way as reduced longevity. I
assumed that females spawn once per day for 3 days, which is close to the
observed duration of spawning by female sockeye salmon
(McPhee and Quinn, 1998).
Following Iwasa et al.
(1983), the fitness of a
cohort of males arriving on day t, 
(t), is the
expected number of spawning events they participated in from day t
until death:
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; cf. the
two-patch continuous input model of habitat selection:
Sutherland and Parker, 1992).
An estimate of prearrival mortality (0.0012 per day) was based on Parker's
(1962) estimate of mortality
during the coastal phase of the return migration of pink salmon O.
gorbuscha. The daily mortality rate after arrival, µa,
equals 0 and the daily mortality rate before arrival, µb, equals
0.0012. Quantitative information on the relationship between longevity and
arrival day is available for male sockeye salmon
(Hendry, 1998). I used a range
of longevity functions, L(t), that approximated the seasonal
decline reported in Hendry's
(1998) study [shallow:
L(t) = 10.2 - 0.2t; intermediate:
L(t) = 15.3 - 0.3t; steep: L(t) =
20.4 - 0.4t].
Female arrival was modeled as a normal distribution with a mean of 15.5 and
SDs of 3, 5, or 7, which correspond to 10, 18, or 22 day periods,
respectively, during which 95% of females arrive. Although arrival can be
highly irregular, generally it is neither synchronous nor uniform (e.g.,
analysis of data used in Morbey,
2000); a normal distribution was used. Season length, T,
was set to 32 days to cover a 30-day female arrival period plus 2 days when
the last-arriving females were still actively spawning.
At the evolutionary stable strategy (ESS), (t) must be equal
for all t. The ESS was found using replicator dynamics programmed in
Quick Basic (see Parker and Courtney,
1983). The simulation began by randomly assigning arrival
probabilities for each timing option or cohort. During each iteration of the
simulation, (t) was calculated for each cohort t =
0,..., T, and then each cohort was replicated in direct proportion to
its relative fitness. If the longevity of a cohort had a noninteger value, the
survival of that cohort in its last day of life was set to the truncated
fraction. For example, if male longevity was 5.2 days, survival at age 6 days
was 0.2. The iterations were continued for 500 generations so that
(t) approximated a constant. Discretization produced
irregularities in the shape of the optimal male arrival distribution, but
these do not affect the interpretation of the results.
After the simulation was complete, the resultant protandry was calculated
as the area (in units of days) between the cumulative-percent distributions of
the arrival days of males and females:
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). In calculating
protandry, the expected arrival distribution after mortality is important, so
M(t) was adjusted by subtracting prearrival mortality from
the ESS. Protandry represents the number of days between the arrival of an
equal proportion of males and females, with positive values representing the
earlier arrival of males. This measure was used in my earlier descriptive
study (Morbey, 2000). The model just described effectively assumes all males are equal competitors (Model A). I also investigated how protandry was affected by interference competition among males, by mate guarding, and by phenotypic differences in competitive ability among males.
Model B: interference competition
Interference among male competitors may reduce longevity because males
compete more aggressively when sex ratios are male biased
(Fleming and Gross, 1994;
Schroder, 1982) and consume
energy reserves at a higher rate (Hendry A, personal communication). Breeding
density also can shorten male longevity
(Hendry, 1998;
van den Berghe and Gross,
1986). To model this scenario, I reduced male longevity by an
amount relative to the degree of interference (r) on each day
s, where:
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Model C: mate-guarding
Male salmon may guard a female for several days during her egg deposition
phase due to an acquired competitive advantage
(Chebanov, 1997;
Foote, 1990). This contrasts
with the ideal despotic distribution of Fretwell and Lucas
(1970), in which mate-guarding
males are intrinsically superior. I assumed that males pair with females and
continue mate guarding for an additional day (during a female's 3 days of egg
deposition) with probability pg. Without mate guarding,
pg = 0 and the model is equivalent to model A. Males
paired with females participated in spawning on the current day, and males who
mate guarded the female for an additional day participated in spawning on the
next day. Sneaking also was allowed as an alternative mating tactic (e.g.,
sockeye salmon: Foote et al.,
1997). The proportion of a clutch fertilized by a sneaking male
was set to 0.5 (see Chebanov et al.,
1983; Foote et al.,
1997; Maekawa and Onozato,
1986; Mjølnerød
et al., 1998; Schroder,
1982). On all days while alive, unpaired males either paired
randomly with unguarded females or sneaked. One unpaired male per spawning
event used the sneaking tactic to fertilize eggs. A male's lifetime
fertilization success was his accumulated daily fertilization success. The
model was run with the intermediate longevity function, two levels of
pg (0.5 or 1) and population sex ratio (0.75:1 or 1.25:1),
and with or without sneaking.
Model D: size-based inequalities
In salmon, larger males are competitively superior to smaller males (coho
salmon: Fleming and Gross,
1994; sockeye salmon: Quinn
and Foote, 1994) and tend to fertilize a greater proportion of the
clutch than smaller, subordinate males (chum salmon:
Schroder, 1982; sockeye
salmon: Chebanov et al., 1983;
charr Salvelinus malma: Maekawa
and Onozato, 1986; Atlantic salmon Salmo salar:
Mjølnerød et al.,
1998). (I do not consider the sneaking tactic here.) I assumed
that large males fertilize proportionately more eggs than small males (see
model V of Sutherland and Parker,
1985). Two male size-classes were considered with equal numbers of
each. I set the competitive weight (CW) of large males at 20, 2, or 1, so that
large males fertilized 20 times, 2 times, or as many eggs as small males. For
each timing option, t (indexed as y) and size class, i
(indexed as j), relative spawning success on day s is:
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Sensitivity analyses
Sensitivity analyses were conducted to assess the importance and robustness
of each parameter. Most parameters were varied independently. Testing every
parameter combination would have required an enormous amout of time, and I did
not detect any interactions among the parameters during preliminary testing.
Sensitivity was calculated as (protandry for the alternative
model—protandry for the basic model) / (protandry for the basic model).
The version of model A with the intermediate longevity function and 1:1
sex-ratio was used as the basic model.
Results
All the models predict protandry in salmon (Tables
1,2,3).
Predicted protandry ranges from 2.67 to 7.74 days and is more sensitive to
some parameters than others. For example, variation in male longevity affects
protandry much more than the details of male—male competition or the
pattern of female arrival (Tables
1 and
2). However, novel predictions
about protandry emerge when such details are considered.
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In the equal competitors model, greater protandry is favored when males live longer (Table 1). Compared to an intermediate male longevity at the beginning of the season, longer and shorter male longevity changes protandry by about 50%. This can be understood by considering a scenario in which males can spawn for only 3 days (the same period that females spend depositing eggs). Under this condition, male arrival should match the female arrival distribution, and no male could improve his spawning success by arriving earlier. But this would not be stable if males lived for 4 days because the peak of male presence would be shifted later, leading to greater competition among late males than among early males. The longer males live, the earlier they should arrive relative to females.
In contrast, the pattern of female arrival affects protandry only slightly (Table 1). Greater protandry is favored when female arrival is more protracted, possibly because of the increased availability of females earlier in the year. By arriving slightly earlier, males have opportunities to spawn with these early arriving females. The greater longevity of early-arriving males also allows them to closely match the longer period of concentrated female spawning activity.
Greater interference competition among males, caused by an increased cost of interference or an increased population sex ratio, reduces male longevity and so favors less protandry (Table 2). Compared to the equal competitors model, interference favors males who arrive and mate slightly later in the year, when there are fewer males per female (Figure 1). At the most extreme conditions for interference, protandry is 46% less than the basic equal competitors model.
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Mate guarding favors greater protandry than does the equal competitors model because earlier arrival allows males to pair with newly arriving females (Table 2). Newly arriving females are more valuable than previously arrived females because they offer more spawning opportunities for mate-guarding males and also are more likely to be unguarded. There was only a 1% reduction in protandry when sneaking was allowed (results not presented). Successful sneaking favors slightly less protandry because the availability of all females becomes important. An increase in male—male competition caused by an increase in the population sex ratio or an increase in the probability of continued mate guarding favors greater protandry because of increased selection to pair earlier and monopolize females. The temporary advantage held by mate-guarding males does not strongly affect protandry, however. When male—male competition is at its most extreme, protandry is only about 8% higher than the basic equal competitors model.
When large males fertilize more eggs than small males, an infinite number of arrival distributions are possible for each competitor type (represented by each line in Figure 2). At each equilibrium, large males have higher spawning success than small males (when the former are competitively superior), but spawning success is the same among the same-sized males. One possible equilibrium is for large and small males to have similar arrival timing and match the protandry predicted from the basic equal competitors model. Large males may arrive earlier than small males or vice versa. Average protandry is similar to that predicted by the basic equal competitors model, but small males show slightly less protandry than large males (Table 3).
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One interesting property of the unequal competitors model is the greater range of protandry equilibria for small males than for large males (Figure 2). This occurs because small males can be excluded from the mid-season period, when female availability is at its greatest, by the presence of large, competitively superior males. Small males cannot monopolize females during this period and therefore cannot exclude large males. The slope relating protandry among the two size classes represents the degree to which large males exclude small males from this period and is the ratio between the competitive weight of small males to that of large males.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONREFERENCES |
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Although the different protandry models used salmon-specific parameters, the predictions of the models are general and can be tested on a wide range of species with similar mating systems. The most important requirement is that males attempt to mate more often than females. Thus, these predictions apply less well to more monogamous species (e.g., birds) because the timing of males relative to females has less direct fitness consequences. The assumptions of each model also must be met for its predictions to be applicable. Support for the predictions would not only uphold the mate-opportunity hypothesis, but also would provide insight into how different selective factors affect male arrival timing. The models also suggest that some factors (e.g., male longevity) will affect mating opportunities, and therefore protandry, more strongly than other factors (e.g., mate guarding by males).
The effect of longevity
The key feature favoring protandry in salmon is the longer period of
spawning activity among males than females. Unlike previous models of
protandry in arthropods, I explicitly included senscence by constraining
longevity. Despite the difference in how longevity was modeled, my results are
consistent with Parker and Courtney's
(1983) prediction of increased
protandry with increased male longevity. If male salmon survive and spawn
throughout the entire spawning period, they all should arrive on the first
day. All else being equal, less protandry and greater tracking of female
availability by males is predicted when males expect to live and spawn for
fewer days. Analogously, in polygynous species returning to breed in
subsequent seasons (e.g., reptiles and amphibians), greater protandry is
predicted when males allocate more time to mating activity.
The modeling shows how protandry leads to equality of spawning success
despite seasonal differences in longevity. Intuitively, it would seem that
increased longevity would translate into greater spawning success, but this is
not necessary if shorter-lived males face fewer competitors but access to more
spawning females. However, one might question whether the seasonal pattern of
longevity assumed in the models is evolutionarily stable, because selection
could operate on male longevity as well as on arrival timing. Early in the
year, it makes sense for males to live longer because they could participate
in more spawning events (Hendry et al.,
1999). Presumably, the allocation of somatic energy stores to
longevity would be balanced by the costs of such allocation (e.g., smaller
gonads or secondary sexual characteristics). I propose that a combination of
protandry and a seasonal decline in longevity is evolutionarily stable,
perhaps even in arthropods, but this hypothesis awaits testing.
Contrasting effects of interference competition and mate
guarding
Protandry is less sensitive to the details of male—male competition
than to the longevity of males. At the equilibrium solution of the basic equal
competitors model, male—male competition is at its highest earlier in
the year (Figure 1). A seasonal
decline in male—male competition (at least during the male arrival
period) also occurs in species that experience greater prearrival mortality
for delaying arrival (e.g., butterflies:
Iwasa et al., 1983). If
interference competition among males reduces male longevity, males should
minimize male—male competition by arriving later and exhibiting less
protandry. Conditions that increase male—male competition should lead to
decreased protandry. This prediction may apply to other species with
interference competition among males. For example, in dense breeding
aggregations of amphibians, male—male competition for females may
increase the energy expenditure by males and, in turn, may reduce how long
males spend breeding. However, information about how male survival and the
duration of reproductive activity vary seasonally is needed to predict how
increased interference competition affects protandry.
The mate-guarding model represents a scenario where males benefit from
early arrival relative to both sexes. The model predicts greater protandry
with increasing population sex ratio (more males per female) because of
increasing selection to be early relative to other males (cf.
Kokko, 1999). Mate-guarding
prevents later-arriving males from participating in spawning events. By
arriving earlier, males have more opportunities to pair with and monopolize
newly settling females than by arriving later. The mate-guarding model can be
tested in other species with precopulatory mate guarding
(Ridley, 1983). It also can
apply to territorial species (including birds) if earlier arriving males
acquire higher quality territories that attract more mates, because the early
arrival of males relative to other males also would have direct fitness
consequences.
The observed relationship between population sex ratio and protandry could
provide insight into the relative importance of interference competition and
mate guarding on male-arrival timing. Depending on the strength of
interference and the probability of mate guarding, the relationship between
population sex ratio and protandry could be positive, negative, or flat.
However, a flat relationship also would result if neither factor affected
protandry. In Morbey's (2000)
study of protandry in Pacific salmon, the observed flat relationship in six of
seven populations suggests that both factors affect protandry equally or that
neither factor affects protandry (the observed positive correlation between
population sex ratio and protandry in Auke Creek pink salmon warrants further
study). Such counteracting effects make it difficult to assess whether males
can respond to factors affecting mating opportunities.
The effect of size-based competitive asymmetries
The inclusion of size-based inequalities in competitive ability has little
effect on the timing of different-sized males and on protandry. A range of
protandry equilibria is possible for each size class, and average predicted
protandry is slightly less for small males than large males. Among years or
populations, a greater range in protandry is predicted in small males than in
large males because the latter monopolize the mid-season period with the
greatest availability of females. The greater the relative competitive ability
of large males, the greater the exclusion. Although not stated explicitly, the
two-patch unequal competitor model of habitat selection also predicts a
negative correlation between the proportion of good and poor competitors in
the good patch that equals —CWpoor/CWgood, where
CW refers to the competitive weight of good or poor competitors
(Milinski and Parker, 1991;
Parker and Sutherland,
1986).
The effect of phenotypic differences in competitive ability on the timing
of different-sized males has been considered previously. Thornhill and Alcock
(1983) suggested that male
wasps with a short life span but superior ability to monopolize and mate with
females should arrive at the peak of female availability. Males with increased
longevity and lower competitive abilities should arrive when competition is
lower. Hastings (1989) also
suggested that small male wasps should avoid competition with large males, and
presumably delay emergence. In light of my modeling, males expecting to live
for a short period should arrive at peak female availability regardless of
their competitive ability. The effect of competitive ability depends on how
competitors differ. If the relative payoff among competitor types is constant
regardless of arrival day, they should exhibit similar protandry.
Extrapolating from Milinski and Parker's
(1991) model, in which
relative payoff among competitor types varies among patches, differential
arrival of different-sized males would be possible when large males achieve
higher spawning success than small males on days with high densities of
females.
In support of the unequal competitors model, younger age classes did not
arrive consistently earlier or later than older age classes within several
salmon populations (Morbey,
2000). The observation of similar arrival timing regardless of
competitive ability (or size) is unusual. In several species, time constraints
imposed by seasonality are expected to produce a seasonal decline in body
size. For example, in arthropods with overwintering larvae and rapid
development (months as opposed to years in Pacific salmon), greater time
pressure to reach sexual maturity or reduced food availability later in the
season may cause a seasonal decline in development time and therefore body
size (e.g., grasshoppers Sphenarium purpurascens:
Cueva del Castillo and
Núñez-Fárfán, 1999). In iteroparous
fish, a seasonal decline in body size may result because time constraints
cause smaller males to delay reproduction and preferentially allocate energy
to growth early in the season (e.g., smallmouth bass Micropterus
dolomieui: Ridgway et al.,
1991). In birds, males in better condition may be able to take
advantage of early arrival while suffering fewer costs than males in poor
condition (Kokko, 1999). In
Pacific salmon, time constraints may be less important because they do not
have to allocate energy for future reproduction, because seasonality in
feeding conditions is probably less severe than for bass breeding in northern
lakes, and because males are not territorial.
Observed versus predicted protandry
Depending on the assumptions made about male longevity and the nature of
male—male competition, predicted protandry (2.49-7.74 days) spans the
observed population average of 2.84 days (n = 7 populations of 4
species) but is slightly greater than the observed range of 0.90-5.18 days
(n = 105 years; analysis of data from
Morbey, 2000). Observed
protandry (2.84 days) differs statistically from the protandry predicted by
the basic model with the intermediate longevity function (4.96 days, one
sample t test: t6 = -3.65, p =.0108)
and, notably, 4.96 days far exceeds the average protandry observed in the two
sockeye salmon populations (about 1 day).
The different models offer potential explanations for why protandry was
overestimated. First, perhaps males spawn for fewer days than indicated by the
intermediate longevity function. This could happen if males senesce more
quickly than assumed, become weaker and less effective at spawning as they
age, or suffer predation risk on the spawning grounds. Significant prespawning
mortality also can occur when abiotic conditions are suboptimal (e.g.,
Groot and Margolis, 1991).
Second, interference competition may affect longevity more strongly than
assumed. Thus, the mismatch between observed and predicted protandry indicates
a need to identify the factors limiting the duration of male reproductive
activity in salmon.
Another explanation for the lower-than-predicted protandry is related to
postarrival waiting (Morbey,
2000). In the model, i assumed that males and females commence
spawning activities immediately after arrival. However, salmon may wait for
days or weeks before initiating spawning activities (e.g., pink salmon:
Mattson and Rowland, 1963;
sockeye salmon: Brett, 1995;
Hoopes, 1972; Morbey, personal
observation). Males do not begin mate searching until females have settled on
nest sites. Once the first female settles, all previously arrived males
suddenly can begin mate searching. Therefore, in populations with postarrival
waiting, protandry in arrival would underestimate protandry in the
commencement of spawning activites. This may explain why the model
overestimated protandry, particularly in sockeye salmon.
Conclusions
The biology of Pacific salmon is more consistent with the mate-opportunity
hypothesis than with alternative hypotheses
(Morbey, 2000), and the
modeling confirms that selection should favor protandry under a range of
competitive scenarios. However, observed and predicted protandry do not match
well, there is little support for the predictions of the different models, and
population differences in protandry are not yet understood. Further research
is needed to determine whether early arrival relative to females has direct
fitness consequences for males. A realistic goal is to first test for a
positive correlation between male longevity and protandry among years within a
population. This would require accurate data on annual and seasonal variation
in predation risk, the duration of male and female reproductive activity, and
prespawning waiting. It also would be essential to calculate protandry using
accurate measurements of male and female arrival timing.
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ACKNOWLEDGEMENTS |
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I thank Bernie Crespi, Larry Dill, Tom Quinn, Bernie Roitberg, David Westneat, Ron Ydenberg, and two anonymous reviewers for providing excellent comments on how to improve the manuscript. This research forms part of the Ph.D. dissertation of Y.E.M. Funding was provided by graduate fellowships from Garfield Weston Foundation and B.C. Packers Ltd., the Natural Sciences and Engineering Research Council of Canada, and Simon Fraser University.
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TOPABSTRACTINTRODUCTIONDISCUSSIONREFERENCES |
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