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Keywords:

  • game theory;
  • mating opportunities;
  • Oncorhynchus;
  • protandry;
  • reproductive lifespan;
  • reproductive timing;
  • senescence

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

The timing and duration of reproductive activities are highly variable both at the individual and population level. Understanding how this variation evolved by natural selection is fundamental to understanding many important aspects of an organism's life history, ecology and behaviour. Here, we combine game theoretic principles governing reproductive timing and the evolutionary theory of senescence to study the interaction between protandry (the earlier arrival or emergence of males to breeding areas than females) and senescence in seasonal breeders. Our general model applies to males who are seeking to mate as frequently as possible over a relatively short period, and so is relevant to many organisms including annual insects and semelparous vertebrates. The model predicts that protandry and maximum reproductive lifespans should increase in environments characterized by high survival and by a low competitive cost of maintaining the somatic machinery necessary for survival. In relatively short seasons under these same conditions, seasonal declines in the reproductive lifespans of males of equivalent quality will be evolutionarily stable. However, over a broad range of potential values for daily survival and maintenance cost, reproductive lifespan is expected to be relatively short and constant throughout a large fraction of the season. We applied the model to sockeye (or kokanee) salmon Oncorhynchus nerka and show that pronounced seasonal declines in reproductive lifespan, a distinctive feature of semelparous Oncorhynchus spp., is likely part of a male mating strategy to maximize mating opportunities.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

Extreme reproductive tradeoffs occur in species with one reproductive season, such as annual insects and semelparous Pacific salmon in the genus Oncorhynchus. During the breeding season in these species, resources are allocated to reproductive functions at the expense of some somatic functions necessary for survival. Such organisms experience rapid physiological decline or senescence, during or after reproductive activities, and have a finite reproductive lifespan (Stearns, 1992). The range in reproductive lifespan among and within species breeding for one season is great. For example, lifespans of adult mayflies (Ephemeroptera) typically range between an hour and several days (Brittain, 1982) and lifespans of breeding Pacific salmon range between a few days and several weeks (Perrin & Irvine, 1990). Although some of this variation is a direct result of differences in extrinsic mortality (e.g. predators, temperature extremes), much is likely to be related to the amount individuals invest in mechanisms that maintain survival. Our general goal is to understand the variation in lifespan in such semelparous organisms, both within and between species.

Understanding variation in lifespan has typically been approached using the evolutionary theory of senescence (Rose, 1991). Williams (1957)‘antagonistic pleiotropy’ theory for the evolution of senescence, which assumes that late-acting deleterious alleles have early-acting benefits, is particularly relevant here. The generality of a physiological tradeoff between longevity and reproductive effort is supported by empirical research (e.g. Møller et al., 1989; Chippindale et al., 1993; Zwaan et al., 1995b). Given the short reproductive lifespans of semelparous organisms and the extreme tradeoffs they make between reproductive effort and survival, optimization of this tradeoff likely provides the major mechanism underlying variation in reproductive lifespan. Several life-history models investigate the optimal allocation of limited resources to repair mechanisms that promote intrinsic survival at the cost of decreased reproduction (e.g. Abrams & Ludwig, 1995; Cichoń, 1997; Cichoń & Kozlowski, 2000; Shanley & Kirkwood, 2000). However, these models have assumed an iteroparous life history, and do not consider any interaction between lifespan and temporal patterns of intraspecific competition that characterize semelparous organisms in seasonal environments. Additional theory is needed to understand the evolution of senescence in such organisms.

Semelparous organisms also frequently have the ability to adjust the starting date of their reproductive lifespan by maturing sooner or arriving at a breeding area earlier. The question of timing is particularly important for males in species in which individual females mate over a very short period. Earlier analyses of male arrival/emergence in organisms with fixed mortality rates predict males to arrive or emerge earlier on average than females to maximize their mating opportunities (Wiklund & Fagerström, 1977). Game-theoretic models have supported this prediction of protandry (e.g. Bulmer, 1983; Iwasa et al., 1983; Parker & Courtney, 1983). Protandry is a common feature of organisms breeding in seasonal environments, although it can evolve for a variety of reasons (Morbey & Ydenberg, 2003). The possibility of protandry must be taken into account in determining the adaptive adjustment of lifespan after arrival or emergence. All of the analyses of protandry have shown that the evolutionarily stable degree of protandry depends on male life expectancy. Conversely, the advantages of a longer life for males depend on mating opportunities later in life, which in turn, depends on protandry via the male arrival schedule. Thus, these problems of optimal arrival and optimal lifespan are intimately connected.

Here, we analyse the joint evolution of male arrival timing and reproductive lifespan using a game-theoretic approach. Based on an analytical version of the model, we describe some characteristics of the evolutionarily stable arrival schedule and allocation to lifespan that equalizes expected mating success, and present numerical results for a particular version. We also use a simulation model based on replicator dynamics (Taylor & Jonker, 1978) to examine more detailed features of the evolutionarily stable strategy (ESS) in populations with prolonged breeding periods. Similar models have been used to understand the activity schedules of animals living in seasonal environments (Iwasa & Odendaal, 1984; Iwasa & Obara, 1989; Iwasa et al., 1994). However, these models do not predict the duration of activity for individuals with different starting times, and cannot be applied directly to the problem examined here. Our approach makes predictions about the arrival schedule, the mortality schedule, and the seasonal variation in reproductive lifespan. Finally, we determine whether the model is consistent with seasonal variation in the reproductive lifespan of male kokanee salmon (nonanadromous Oncorhynchus nerka).

General considerations

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

In the simple evolutionary model we consider, male individuals are characterized by two traits: an arrival day and a lifespan. Males compete to mate with females, which are assumed to have a fixed arrival distribution over a finite season. For convenience, we consider the season to consist of a number of distinct ‘days’. As in most previous protandry models, we focus on a simple mating system with polygamous males and monogamous females. Males are subject to extrinsic mortality (cf. Iwasa et al., 1983) and have a maximum lifespan (cf. Morbey, 2002). Males are assumed to have equal amounts of capital resources that must be divided between reproductive effort (gonads, secondary sexual characteristics, or mating behaviours) and repair mechanisms that maintain survival. Allowing for differences in male condition may cause positive correlations between longevity and sexually attractive traits (Jennions et al., 2001), but this situation is not considered here. A male's investment in reproductive effort is assumed to affect his relative competitive ability; i.e. his ability to obtain mates and fertilize eggs, compared with other males, on all days of his life. On each day of the season, competition between males for unmated females is described by a weighted lottery competition model.

The general model makes the following assumptions:

  • 1
    Virgin females become available (‘arrive’) with a fixed distribution over a finite season of length T; the proportion of females arriving on day i is given by p(i). We assume that females mate once on their arrival day. However, the results are the same when females mate more than once on their arrival day if fertilization success is equal for all partners. Polyandry over several days can be handled by redefining the female arrival distribution to include all reproductively active females and extending T.
  • 2
    The male reproductive strategy is defined by an arrival date, i, and an associated lifespan, x(i), with the implicit assumption that the two traits can be coupled. This will be the case if the male types reproduce asexually. It will also be true for sexual species when the two traits are closely linked genetically or when one trait can be optimized by some form of phenotypic plasticity, given the value of the other trait.
  • 3
    The competitive ability of a male that has a lifespan of x days is c(x), which is a decreasing function of x with c(1) = 1. The expected number of matings obtained by a male with lifespan x is defined by a weighted lottery model. That is, a male alive on day t mates with a female with probability c(x)p(t)N/[ΣyQ(t,y)c(y)], where Q(t, y) is the number of males alive on day t with lifespan y. The total number of arriving individuals (N) is assumed to be the same for each sex.
  • 4
    Males have a fixed daily probability of surviving extrinsic mortality, denoted s. Senescence is reflected in death of all surviving individuals when they reach their intrinsic lifespan; before that age there is no effect of senescence on survival.
  • 5
    The evolutionary and population dynamical equilibrium is characterized by a male arrival distribution, q(i), and corresponding lifespans, x(i). At equilibrium, the expected lifetime number of matings for an individual male must be one for all strategies (combinations of arrival day and lifespan) present in the population.

The analytical model is analysed by searching for a set of lifespans, x(i), for males arriving on all days i, that will resist invasion by any alternative type of male. This requires determining the equilibrium frequencies, q(i), for each set of lifespans, x(i). These frequencies are determined by the condition that the expected number of matings for an individual male must be equal to 1 for all arrival days i, and that the arrival frequencies, q(i), must sum to 1. Given a potential set of lifespans, x(i), and the implied equilibrium frequencies, q(i), we can then test for whether a rare mutant type with a different lifespan, x′(i), is able to increase when rare on day i. If such a mutant does not exist for any day, the set of lifespans, x(i), and the associated equilibrium arrival frequencies, q(i), constitute an ESS. Protandry is achieved by males having an earlier mean arrival day than females.

The model can also be solved numerically using simulation based on replicator dynamics (Taylor & Jonker, 1978), as in a recent model used to predict female reproductive lifespan (Morbey & Ydenberg, 2003). The advantage of a numerical approach is its ability to accommodate larger strategy sets and alternative functional forms. The model tallies the presence of all males and their competitive abilities on each day of the season. The daily fertilization success of males using each strategy (i, x) is calculated according to the weighted lottery competition model. The lifetime reproductive success is then tallied for the males using each strategy, and this determines the frequencies of different types of males in the next generation (a very small quantity was first added to the fitness of strategies with zero fitness to prevent their extinction). The model was programmed in C and simulations were run in Microsoft Visual C++ (v. 6.0) for 2000 generations, or until an ESS was reached.

Here, we state some general analytical results regarding evolutionarily stable strategies. The evolutionarily stable set of x(i) and q(i) in general depends on the female arrival distribution, p(i), and the male competitive abilities, c[x(i)]. However, some of the properties of ESSs are independent of these details:

  • 1
    No males arrive before the first day. Doing so would only expose them to mortality before mating opportunities were available. In real systems, uncertainty in both male and female arrival schedules, as well as potential benefits from arriving before competitors, are likely to cause some males to arrive before any females (e.g. Iwasa & Haccou, 1994).
  • 2
    If males arrive on the last day of female arrival, T, their lifespan is 1 day. More generally, if males arrive y days from the end of the season, their lifespan is less than or equal to y. This is a consequence of the loss in competitive ability with increased lifespan, and the deterministic nature of the model. If there were uncertainty in the female arrival distribution, this result is also likely to change.
  • 3
    A sufficient reduction in competitive ability with lifespan or a low intrinsic survival rate results in a strategy where x(i) = 1 and p(i) = q(i) for all i. In other words, males do not exhibit protandry, and have the same reproductive lifespan (1 day) as females. This requires all possible longer lived males to have lower fitness when they are rare invaders: c(2) < 1/(1 + s), c(3) < 1/(1 + s + s2),…,c(j) < 1/(1 + s + s2 +⋯+ sj−1). These conditions follow from the fact that each 1-day male can expect one mating at the equilibrium, where p(i) = q(i). The expected lifetime mating success of a rare invader x-day male is c(x) + sc(x) +⋯+ sx−1c(x), and this must be greater than 1 for such a male to invade. These conditions apply to all arrival days until x days before T. Only males with lifespans of x ≤ k need be considered as potential invaders k days before T. These males would have higher competitive abilities than any longer-lived males but would have equal mating opportunities over the period until T. Relatively low survival probabilities, s, or a steeply declining c(x) function favour the strategy of no protandry.
  • 4
    Although a general proof for seasons of arbitrary length is elusive, protandry appears to characterize all male arrival distributions when at least some males live more than 1 day. Clearly, the frequency of males arriving on the first day, q(1), must be greater than the corresponding frequency of females, p(1), when male lifespan exceeds 1 day. If this were not the case, then those males would have an expected lifetime reproductive success greater than 1.
  • 5
    The most extreme form of protandry and the longest possible mean lifespan occurs when all males arrive on the first day, q(1) = 1, and have a lifespan equal to T. This requires a high daily survival probability and a weak tradeoff between long life and competitive ability [high c(i)’s]. If this were not the case, competitive, but shorter-lived forms could invade by arriving on later days, especially if the season is long. A general condition for this ESS can be derived if two simplifying assumptions are made (see Appendix 1). First, the female arrival distribution is assumed to be uniform with p(i) = 1/T for all days i. Secondly, competitive ability is assumed to decline geometrically with lifespan relative to the competitive ability of a 1-day form, c(1) = 1. Competitive ability is given by c(j) = Cj−1, where C is a constant less than 1. The ESS condition for maximal protandry under these conditions requires that sC > (T − 1)/T.

Results for a 3-day season

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

The exhaustive technique of determining the fitness of all potential invading strategies for all possible resident strategies is impractical, especially if the female distribution spans a large number of days and is nonuniform. Therefore, we illustrate the model using a 3-day female arrival period and assume that male competitive ability declines geometrically with lifespan; an exhaustive analysis is possible in this case. The results are shown for two different female arrival distributions in Fig. 1; the ordered triplet {ijk} denotes the lifespans of males arriving on days 1 (i), 2 (j) and 3 (k). Figure 1(a) is a case with a uniform female arrival distribution. The number of possible ESSs here is much smaller than the range of potential combinations of male lifespans and arrival days. An absence of protandry with no seasonal change in lifespan (i.e. the {1, 1, 1} strategy) is the ESS for the majority of potential parameter values. In addition, only a narrow range of parameters yields an optimal strategy having a continuous decrease in lifespan with arrival date (i.e. the {3, 2, 1} strategy). Maximal protandry, {3, 0, 0}, is restricted to high-survival rates in combination with low competitive costs (high s and C). A unimodal distribution of female arrivals with p(1) = 0.2 and p(2) = 0.6 (Fig. 1b) has slightly different conditions for the stability of different strategies, and a slightly different collection of strategies in the set of possible ESSs.

image

Figure 1. Results of the 3-day model as a function of daily survival s and competitive ability C for two different female arrival distributions [p(1) = p(2)  = 0.333; p(1)  = 0.2, p(2)  = 0.6]. Panels (a) and (b) show different lifespan strategies, which are characterized as the evolutionarily stable reproductive lifespan on days 1, 2 and 3 for each combination of s and C. The lines are the boundaries for the different lifespan strategies and were derived algebraically from the analytical version of the model. Increases in s and C favour a higher maximum lifespan, and if survival is high, increases in C favour truncated arrival. Panels (c) and (d) show the degree of protandry, calculated from the simulation model, for each combination of s and C. The contour lines, calculated using SigmaPlot 5.0, show combinations of s and C with equal protandry ranging from 0 to a maximum of 1 day. The magnitude of protandry increases with s and C.

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The evolutionarily stable degree of protandry can be derived for each reproductive lifespan strategy by calculating the evolutionarily stable proportion of males arriving on days 1, 2 and 3 [q(1), q(2) and q(3)] given their lifespans xi, competitive abilities ci, daily survival and the proportion of females arriving on days 1, 2 and 3 [p(1), p(2) and p(3)]. The equilibrium male arrival frequencies [q(1), q(2) and q(3)] are determined by the requirement that each male get an average of one mating. This can be determined algebraically for some lifespan strategies (e.g. {1, 1, 1}, {2, 2, 1}) but the solutions for other strategies are cumbersome. Therefore, we used the numerical approach to calculate the frequencies of different types of males and the resulting level of protandry at the evolutionary equilibrium. The level of protandry was calculated as the difference in mean arrival date between males and females, and is shown in Fig. 1(c,d) for the two female arrival distributions. The level of protandry increases with increases in either s or C for both arrival distributions examined. Protandry tends to be greater under uniform female arrival because the presence of more females early in the season provides mating benefits to early-arriving males.

Results for a longer season

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

To explore the possible ESSs for longer seasons and a wider range of female arrival distributions, it is necessary to use the numerical approach in which we adopt specific forms for the female arrival distribution, daily survival and competitive ability. The female arrival distribution over a season length T is specified as a truncated normal distribution with mean (T − 1)/2 and standard deviation (SD). The strategy set contains a matrix of arrival days (i = 0,…,T − 1) and intrinsic reproductive lifespans (x = 0,…,T − 1) for males. Males suffer instantaneous mortality μ and die if they exceed their intrinsic lifespan. Survival to age t is specified as exp(−μt), where exp(−μ) is equivalent to daily survival s in the previous section. A longer intrinsic lifespan trades off with competitive ability according to a geometric model, c = Cx, where C (C < 1) is the competitive ability of an x-day male relative to an x − 1-day male. In some simulations, the ES reproductive lifespan appeared to be a mixed strategy that included two or three adjacent values.

Many of the qualitative results for the 3-day season hold for longer seasons (results for T = 20 days shown in Fig. 2). Protandry and reproductive lifespans exceeding 1 day (the minimum) occur when C > 1/(1 + s). Greater protandry and greater maximum reproductive lifespans are favoured when daily survival is high and maintenance costs are low (high s and C). Here, maximum lifespans exceeding 10 days occur at relatively high-survival probabilities and competitive abilities. Under a broad range of potential values for s and C, maximum reproductive lifespans are short and protandry is small. When extrinsic survival is high, a low competitive cost of greater longevity increases the incidence of truncated male arrival, in which no males arrive during the latter part of the season. Uniform female arrival favours more protandry than unimodal female arrival because of the increased mating opportunities early in the season. Pronounced seasonal declines in reproductive lifespans require moderately high survival coupled with very low maintenance costs (i.e. high C). Very high-survival results in a truncated male arrival distribution, thereby reducing the duration of the seasonal decline.

image

Figure 2. Results of the 20-day simulation model for uniform (a) or unimodal (SD = 4) (b) female arrival. Each line is the result of one simulation and shows the evolutionarily stable male reproductive lifespan for each arrival day for a particular combination of daily survival s and competitive ability C. The three lines span the range of potential solutions. P is the difference between mean male and female arrival times (i.e. protandry) measured in days.

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One pattern that was difficult to discern in the 3-day example is clearer for the 20-day season. Provided the variance in female arrival is sufficiently low, there is a significant period in the middle of the season where the optimal male lifespan is relatively constant with respect to arrival day. Despite seasonal changes in female availability and selection on arrival timing, the predicted reproductive lifespan is approximately constant until the end of the season approaches. This is because the adjustment of the abundance of males arriving on different dates results in an approximately constant level of competition for females. This favours a lifespan that is independent of arrival day except at the beginning and end of the reproductive season. Consequently, seasonal declines in reproductive lifespan should be more apparent in short seasons.

The period of relative constancy in male lifespan often changes to an early season increase in male reproductive lifespan when female arrival becomes more uniform (Fig. 2). Female arrival schedules that are uniform or skewed towards late in the season make it advantageous for males to be abundant late, which tends to favour a relatively long period of decline in male lifespan. At the beginning of this period of decline, males will have relatively low competitive ability. In such cases, earlier-arriving males can best compete against later-arriving males by investing more in competitive ability and less in reproductive lifespan. An early season increase in male lifespan can also occur when female arrival is left skewed or bimodal or when females mate over multiple days.

Application of the model to male O. nerka

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

Pacific salmon (Oncorhynchus spp.) typically exhibit protandry (Morbey, 2000) and seasonal declines in reproductive lifespan (Perrin & Irvine, 1990; Hendry et al., 1999; Morbey, 2003). We are not aware of any reports of seasonal declines in reproductive lifespan in other taxa. Reasons for seasonal declines in female reproductive lifespan have been considered previously and relate to a nest defence strategy (Morbey & Ydenberg, 2003). For males, fewer mating opportunities late in the season and tradeoffs between somatic maintenance and reproductive effort have been invoked to explain seasonal declines in reproductive lifespan (Hendry et al., 1999). We assess whether this hypothesis is logically consistent and whether this pattern and protandry are evolutionarily stable by applying our model to O. nerka (sockeye salmon or kokanee). We parameterize the model using estimates for female availability and extrinsic mortality from the literature. Arrival of female salmon is described by a beta distribution (α = β = 1.5; based on analysis of sockeye salmon data in Hendry et al., 1999) over a season length of 30 days and females are reproductively active for 3 days.

Extrinsic mortality can be caused by predators or extreme environmental conditions. For male sockeye salmon in Pick Creek, Alaska, daily predation risk was 0.117 for a typical 35-day season, or equivalently, 0.137 for a 30-day season (A. Hendry & S. Gende, unpublished data). For kokanee in Meadow Creek, British Columbia, predation risk was close to zero but extreme environmental conditions can greatly increase mortality in some years (Morbey & Ydenberg, 2003). Based on an exponential survival model, daily mortality rate for kokanee was 0.08 in 1998 and 0.05 in 1999, making the difference as a result of extrinsic factors 0.03 for a season length of about 30 days. Thus, we experimented with daily survival values between 0.86 and 0.97. In contrast to the threshold function for intrinsic mortality assumed in the 3-day model, we allowed intrinsic mortality to accelerate with age to more realistically reflect the senescence that occurs in nature (Hendry et al., 2004). The strategy set contained a range of expected lifespans (x = 0,…,T − 1) and survival to time t was described by a Weibull model:

  • image

The scale parameter σ was estimated for the Meadow Creek kokanee data using PROC LIFEREG (SAS v. 8, Allison, 1995). Based on log-likelihood ratio tests, Weibull models explained the individual and seasonal variation in reproductive lifespan better than exponential models in 1998 (inline image = 1212.6, P < 0.05) and 1999 (inline image = 2161.4, P < 0.05), and could not be rejected for gamma models (P's > 0.2). Survival rates declined with arrival day and were greater in 1999 than in 1998 (Table 1). Plots of age-specific survival probabilities were created using the life table method in PROC LIFETEST for early (arrival days = 1–10) and late (arrival days = 11–21) kokanee. They show that seasonal declines in survival were due primarily to an earlier onset of senescence and not to a higher rate of senescence at all ages (Fig. 3, cf. Hendry et al., 2004). In general, hazard increased at an increasing rate with time since arrival. Based on the year with low extrinsic mortality (1999), σ = 0.2 best reflected the acceleration of intrinsic mortality. Male sockeye salmon from Pick Creek had similar rates of senescence with σ = 0.19 in 1995 and σ = 0.13 in 1996 (analysis of data in Hendry et al., 1999). Varying σ values between 0.13 and 0.2 made little difference to the predicted male strategies, so we set σ = 0.2 hereafter.

Table 1.  Parameter estimates for Weibull models describing survival probabilities as a function of age for male kokanee in 1998 and 1999.
ParameterCoefficientSEχ2
  1. *P < 0.05; **P < 0.0001.

1998
 Intercept3.230.11861.9**
 Arrival day−0.070.0146.0**
 Scale0.280.03 
1999
 Intercept3.230.081656.2**
 Arrival day−0.020.016.50*
 Scale0.200.02 
image

Figure 3. Survival distribution functions (SDF) for early (arrival days 1–10; solid lines) and late (arrival days 11–21; dashed lines) male kokanee from Meadow Creek in 1998 (top panel) and 1999 (bottom panel). Age-specific survival probability estimates are shown with upper and lower 95% confidence limits.

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Quantitative estimates of the tradeoff between lifespan and competitive ability are lacking for salmon, but the available data suggest that the cost of increasing lifespan is very low. Early- and late-arriving male sockeye salmon had similar somatic energy, testes size and testes energy content, with only mass-specific energy content of testes being higher in late-arriving males (Hendry et al., 1999). Similarly, body size, testes size and secondary sexual characteristics were similar between early- and late-arriving kokanee (Y. E. Morbey, unpublished data). Because early- and late-arriving males differ significantly in lifespan, this suggests that lifespan has at best a weak effect on competitive ability in O. nerka. However, we expect C to be less than 1; otherwise some long-lived males would be expected to arrive throughout the season, a pattern that is not observed in nature. We experimented with different values of C, and present results for values spanning the range that resulted in the closest match between predicted and observed reproductive lifespans for Meadow Creek kokanee.

The model predicts protandry and a seasonal decline in reproductive lifespan of similar magnitude to that observed when s = 0.97 and C = 0.97 or when s = 0.86 and C = 0.99 (Fig. 4). Predicted reproductive lifespans are shown as averages because the ESS sometimes comprised two to three adjacent lifespans. The reduced availability of females late in the year selects for shorter lifespans among the late-arriving males. Low maintenance costs allow a seasonal decline in reproductive lifespan to be part of an ESS because they favour long-lived, early-arriving males. Extrinsic mortality is sufficiently high to favour some later-arriving males, especially when s = 0.86.

image

Figure 4. Results of the simulation model predicting the reproductive phenology of male salmon Oncorhynchus nerka. The bottom panel shows the female arrival distribution (solid line) and predicted male arrival distributions when s = 0.97, C = 0.97 (without a late-breeding advantage: dashed-dotted line; with a late-breeding advantage: dashed line) and when s = 0.86, C = 0.99 (without a late-breeding advantage: dotted line). The irregularities of the male distributions arise from the discrete nature of the model but do not affect the interpretation of the results. The upper panel shows the corresponding predicted reproductive lifespans (dashed-dotted, dashed and dotted lines) with the observed reproductive lifespans of male kokanee from Meadow Creek arriving during the first 20 days of the 1999 field season (solid circles). Predicted protandry (the difference between the mean arrival date of males and females) is shown as P for each combination of parameters.

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As has been observed in some other game-theoretic models of protandry, our model predicts an earlier truncation date and greater protandry (range = 4.75–8.21 days) than observed (e.g. Morbey, 2002). In nature, more males than expected arrive late in the year, possibly for adaptive reasons not considered in our model. For example, stochasticity in female arrival timing is expected to broaden the optimal male arrival distribution (Iwasa & Haccou, 1994). Alternatively, early breeding can be costly for males if early females suffer higher egg mortality caused by the nest digging activity of later-arriving females (Morbey & Ydenberg, 2003). For the model with s = 0.97, allowing for a small benefit of late breeding by multiplying reproductive success on day t by (0.2t + 1) lowers protandry (3.73 days) and causes a later truncation date (Fig. 4). One feature that is common to both the data and all of the various parameterizations of the model is a pronounced seasonal decline in reproductive lifespan.

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

According to existing protandry models, the arrival or emergence timing of males is sensitive to extrinsic mortality, the schedule of female availability, and the timing of other males (Bulmer, 1983; Iwasa et al., 1983; Parker & Courtney, 1983). In addition to showing these effects, our model also shows that the costs of investing in somatic maintenance can affect protandry, primarily through its influence on reproductive lifespan. Somatic maintenance costs are critical factors affecting the evolution of reproductive lifespan according to existing models of senescence (e.g. Abrams & Ludwig, 1995; Cichoń, 1997). When maintenance costs reduce competitive ability or reproductive rate, males should sacrifice long life for short-term gain and consequently should show less protandry. In a model where competitive ability decreases geometrically with lifespan, protandry is predicted if C > 1/(1 + s), where C is the competitive ability of an x + 1-day male relative to an x-day male and s is the daily probability of surviving extrinsic mortality factors. The magnitude of protandry is expected to increase with C and s.

Many of our predictions are similar to those made by protandry models that lack any adjustment in lifespan, and by models of senescence that lack seasonality in male–male competition. Like protandry models, we predict males to have an earlier expected arrival date than females under a broad range of parameter values, when male survival is moderate or high. Like comparable models of senescence, we predict that greater reproductive costs of somatic maintenance and higher extrinsic mortality will select for a shorter expected lifespan.

A unique prediction of our model is the possibility of evolutionarily stable seasonal declines in lifespan for males that are equivalent in quality. This occurs because it is adaptive for late-arriving males to invest more in competitive ability than in survival, as mating opportunities will only be present for a short period of time. Selection imposed by limited mating opportunities is similar to the effect of short reproductive seasons, which also selects for reduced lifespans (Tatar et al., 1997; Gotthard et al., 2000). A seasonal decline in reproductive lifespan requires sufficiently low maintenance costs (i.e. high C) and sufficiently high extrinsic survival, because this combination favours allocation to lifespan early in the season. However, survival (s) must also be low enough to ensure a prolonged period of male arrival, a necessary condition to observe seasonal declines in reproductive lifespan. We recommend further research to quantify the maintenance costs in different animals. If somatic maintenance is inherently costly under certain conditions (e.g. intense sexual selection, small size), it may be possible to explain why seasonal declines in reproductive lifespan are common in some taxa (e.g. Oncorhynchus) but apparently not in others (e.g. annual insects).

Alternative explanations for seasonal declines in lifespan include the possibility that extrinsic mortality factors increase late in the season or that poorer quality males (having lower lifespans) arrive later in the season. These alternatives can be examined by quantifying seasonal variation in environmental or organismal condition. Our model also predicts male fitness to be independent of arrival date; this will not occur unless male lifespan can be adjusted adaptively based on arrival date.

The schedule of female availability affects protandry and reproductive lifespan in a nonintuitive manner. In contrast to unimodal female arrival, uniform female arrival favours more early-arriving males and, to a lesser extent, more late-arriving males. Whether this causes more protandry or a more pronounced period of seasonal decline depends on parameter values such as season length. For example, uniform female arrival resulted in more protandry than did unimodal female arrival in the 3-day model, but protandry was similar regardless of the female arrival schedule in the 20-day model with s = 0.75 and C = 0.99 (Figs 1 and 2).

Longer seasons increase the potential for greater protandry and greater reproductive lifespans, but under some conditions, reproductive lifespans will be short regardless of season length. Season length influences how early, in relation to the end of the season, the lifespan strategy converges to a constant value. A longer season drives this convergence date earlier, especially when maintenance costs and extrinsic mortality are very low. Beyond the influence of the end of the season, reproductive lifespans often appear insensitive to future mating opportunities.

Other factors not included in our model will influence the patterns of arrival and longevity seen in a given species. For instance, protandry can depend on pre-emergence mortality (e.g. Iwasa et al., 1983; Parker & Courtney, 1983) and on the nature of the competitive interactions between males (Morbey, 2002). We showed how a reproductive benefit of late breeding favours less protandry and later truncation of arrival in salmon; other costs and benefits of breeding timing relative to calendar date may also modulate male reproductive phenology. We have assumed that lifespan can be optimized based on arrival date. If this is not the case, male lifespan will not vary seasonally and optimal reproductive lifespan will be shorter, due to the cost of late-arriving males surviving past the end of the season.

The model suggests that in O. nerka, a seasonal decline in male reproductive lifespan and protandry comprise an ESS. Early-arriving males invest more in lifespan because it is cheap to do so, whereas late-arriving males invest less in lifespan because future mating opportunities are scarce. This explanation is consistent with part of Hendry's et al. (1999) hypothesis, but the tradeoff between somatic maintenance and reproductive effort must be weak to favour a seasonal decline in reproductive lifespan. Extrinsic mortality (caused by predators or adverse environmental conditions) was sufficiently high to favour some late arrival by males, although why more males than predicted arrive late in the season is still not fully understood.

Alternative hypotheses for seasonal declines in the reproductive lifespan of male O. nerka are not supported by empirical evidence. First, there is no evidence that late-arriving males are of poorer quality. Size (indicated by ocean age) of males in several Oncorhynchus spp. does not influence their relative arrival timing in any consistent way (Morbey, 2000). Early and late males differed in the onset of senescence but not in their initial survival rates. There was no evidence that senescence is entrained to calendar date regardless of individual phenology (cf. Hendry et al., 2004). Secondly, extrinsic mortality factors do not necessarily increase later in the season. In Meadow Creek, elevated water temperatures in 1998 were associated with higher mortality (cf. Brett, 1995), but water temperatures did not increase later in the season in either year and so could not account for the seasonal decline in lifespan (cf. Morbey & Ydenberg, 2003). Finally, sufficient genetic differentiation exists between early and late breeding salmon for adaptive adjustments in reproductive lifespan to evolve (Hendry et al., 2004).

Our models assume no constraints on the reproductive phenology of males, and thereby assume selection acts freely on arrival timing and intrinsic reproductive lifespan. In support of this view, variation in protandry and intrinsic lifespan are consistent with evolutionary hypotheses developed for each trait independently. There is growing evidence that lifespans evolve in response to ecological factors such as extrinsic mortality (Austad, 1993), season length (Tatar et al., 1997; Dudycha & Tessier, 1999) and mating opportunities (Sevenster & van Alphen, 1993; Promislow et al., 1998; Gotthard et al., 2000; Wiklund et al., 2003). Furthermore, selection at the adult stage can be decoupled from previous life-history stages (e.g. Zwaan et al., 1995a). Until discrepancies between theoretical predictions and empirical observations suggest otherwise, arrival timing and reproductive lifespan should be considered in a coevolutionary framework because they interact and both directly affect reproductive success.

Understanding the seasonal patterns in reproductive phenology has practical and ecological implications. From a practical point of view, laboratory and field studies to describe senescence and quantify intrinsic reproductive lifespans need to design sampling schedules when keeping in mind the possible differences in behaviour between early and late individuals. Similarly, certain population estimation techniques rely on accurate estimates of ‘survey life’ (the number of days an individual is active in a survey area) and can be biased by seasonal variation in reproductive lifespan (Perrin & Irvine, 1990; Asaro & Berisford, 2001). From an ecological point of view, seasonal patterns of reproductive phenology influence and are influenced by many aspects of mating systems (Odendaal et al., 1985; Wickman, 1992; Wiklund et al., 2003) and can affect competitive interactions with other species (Sevenster & van Alphen, 1993).

Our conclusions have implications for the evolution of semelparity in Pacific salmon. The ancestor to the semelparous Pacific salmon clade was likely iteroparous and anadromous, much like today's steelhead trout O. mykiss (for a recent review see Crespi & Teo, 2002). Anadromy would have had several direct consequences. First, individuals would have suffered high post-breeding mortality in association with migrations to and from breeding areas through unfamiliar, possibly resource-poor habitats. The lack of familiarity with breeding areas and low food availability would likely reduce territoriality, increase spawning densities and increase egg mortality through nest digup. As predicted by life-history theory, current breeding effort should be increased in response to high adult mortality and low juvenile survival. According to our model, male–male competition to mate as often as possible should also cause males to increase their reproductive effort at the expense of future survival, especially if arriving late. Thus, semelparity may have originally appeared in late-arriving males. Adaptations related to reproductive phenology may have been the first step in the evolution of semelparity at the population level. Reinforcement of semelparity through selection on increased reproductive effort has also been invoked in the woodrat Neotoma lepida inhabiting Death Valley, California (Smith & Charnov, 2001). In this case, selection for increased fecundity is expected to favour larger females, but larger females suffer high mortality during the annual period of severe heat and drought.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

We thank the Natural Sciences and Engineering Research Council of Canada for their financial support, Scott Gende and Andrew Hendry for providing data on sockeye salmon, and two anonymous reviewers for their constructive comments.

References

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  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix
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Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. Models
  5. General considerations
  6. Results for a 3-day season
  7. Results for a longer season
  8. Application of the model to male O. nerka
  9. Discussion
  10. Acknowledgments
  11. References
  12. Appendix

Appendix 1

The exact conditions for the ESS q(1) = 1 and x(1) = T require that forms with lifespan x = j cannot invade on day i for any i and j, where 0 < j < T − i + 1 and 1 < i < T. To derive a mathematical expression of the stability requirements, the fitness obtained by a (shorter-lived) j-day invader arriving on day i must be considered in relation to the fitness of the resident T-day form during the period when both are present, which is:

  • image

To obtain the invader's fitness, each term in this summation must be multiplied by the invader's mating advantage, c(j)/c(T), and its numerical advantage owing to its later arrival, 1/si−1. (The proportion of invaders remaining alive on a given day will exceed that of the residents by 1/si−1.) This leads to the following expression for the fitness of the invader:

  • image

This value must be greater than 1 (i.e. the resident fitness) for invasion which leads to the following general formula:

  • image

A concrete example is provided by making two assumptions that allow the summations in the preceding fitness expression to be simplified. First, we assume that competitive ability declines geometrically with lifespan relative to the competitive ability of a 1-day form, c(1) = 1. Competitive ability is given by c(j) = Cj−1, where C is a constant <1. In addition, we assume a uniform female arrival distribution p(i) = 1/T. With a uniform arrival distribution, an invader with lifespan j will always have the highest fitness if it arrives j + 1 days before the end of the season T. As a result, the greatest fitness of a j-day form for all possible arrival dates is (j/T)(sC)jT. If this expression has a maximum value greater than 1, then the arrival of all males on day 1 with a T-day lifespan is not an ESS.

If j were a continuous variable, calculus shows that the maximum invader fitness occurs when j = − 1/ln(sC), which is always greater than 1. However, this maximum value of j must occur for some integer value of j < T for a shorter-lived form to invade. The approximate maximum value of sC that allows invasion by forms living less than T days is thus exp{−[1/(T − 1)]}. If the season is reasonably long, this maximum sC is approximately equal to (T − 1)/T. Thus, sC must be close to one for an ESS consisting of all T-day males arriving on day 1.