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Keywords:

  • life history;
  • male allocation;
  • parental care;
  • reproductive strategies;
  • sexual selection

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

Recent theory predicted that male advertisement will reliably signal investment in paternal care in species where offspring survival requires paternal care and males allocate resources between advertisement and care. However, the predicted relationship between care and advertisement depended on the marginal gains from investment in current reproductive traits. Life history theory suggests that these fitness gains are also subject to a trade-off between current and future reproduction. Here, we investigate whether male signalling remains a reliable indicator of parental care when males allocate resources between current advertisement, paternal care and survival to future reproduction. We find that advertisement is predicted to remain a reliable signal of male care but that advertisement may cease to reliably indicate male quality because low-quality males are predicted to invest in current reproduction, whereas higher-quality males are able to invest in both current reproduction and survival to future reproduction.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

Understanding the myriad ways that animals advertise their attractiveness to potential mates and the responses of potential mates to such advertisement has been the subject of extensive theoretical and empirical research (Zahavi, 1975; Andersson, 1986; Wedekind, 1992; Price et al., 1993; Bakker & Mundwiler, 1994; Jennions & Petrie, 1997; Kokko, 1998; Houle & Kondrashov, 2002; McNamara et al., 2003; Coleman et al., 2004; Lopez & Martin, 2005; Griggio et al., 2009). In most species, it is males that attempt to attract females, and females that discriminate between the males that they choose as mates. The advertisements that males produce to attract females may be either physical (colourful plumage, elaborate ornaments), behavioural (mating dances, nest building, song production) or a combination of both (Andersson, 1994). The level of advertisement produced by a male (the intensity of colour, the length of an ornament, the complexity of a mating dance) is often argued in sexual signalling theory to evolve as a reliable indicator of a male’s quality as a mate; that is the indirect or direct fitness benefits that a female receives for choosing him (Zahavi, 1975; Grafen, 1990a; Andersson, 1994). In species where direct benefits such as male parental care or nuptial gifts are important to female fitness, we may expect to see strong selection on females to choose males that reliably advertise their ability to provide that benefit (Moller & Jennions, 2001). Some previous theoretical work has suggested that compared to species with indirect benefits, males in species with direct benefits should have more exaggerated traits, as female preference for reliable males will select for signals that are costly to produce or hard to falsify (Grafen, 1990a; Price et al., 1993). However, more recent theoretical work (Kokko, 1998; Kelly & Alonzo, 2009) has shown that in species where the costs of providing the direct benefit are high, males may produce less exaggerated signals, but that the ability of a male to produce any signal at all can itself act as a reliable indicator of male quality for females.

Central to signalling theory is the concept that there is a trade-off between the cost of producing a signal and the fitness benefits received for producing that signal. Grafen (1990b) assumed that the marginal costs of signal production were greater for signallers of low quality than for signallers of high quality. However, the differential costs of producing a signal depend on the resources available for a male to invest in signalling, which are subject to life history trade-offs. These life history trade-offs may be between reproduction and growth, or between reproduction and survival, and may influence the age of first reproduction. The benefits a male receives for allocating resources to reproduction must be balanced against those required for his own self-maintenance. Maximizing reproductive success at the cost of this self-maintenance may increase a male’s fitness in the short-term, but will decrease his lifetime fitness at the cost of all the possible future reproductive opportunities missed. Unless the chances of surviving to reproduce again are low, individuals are not expected to devote all their resources to reproduction (Williams, 1966). In iteroparous species, reproductive investment (the resources an organism allocates to reproduction) is expected to change with the probability of survival from one reproductive event to the next (Stearns, 1992). Kokko (1998) modelled the trade-off between investment in advertisement and male survival and predicted that a low-quality male will increase his advertisement as a terminal effort. When a trade-off with investment in parental care was included, Kokko’s (1998) model predicted that low-quality males in iteroparous species would either invest all of their resources in reproduction, essentially having semelparous life histories, or abandon any offspring to survive to the next mating event. The behaviour predicted depended on a male’s marginal benefits of investing in advertisement, parental care or self-maintenance.

A more recent theoretical model showed that when males allocate resources between current advertisement and care, whether male parental care is required for offspring survival affects the reliability of male signalling as an indicator of his ability to provide care (Kelly & Alonzo, 2009). When offspring survival requires either male-only or obligate bi-parental care, then male advertisement was always predicted to be a reliable indicator of male parental care. Although this model did not explore the trade-off between current and future reproduction, it did show that the relationship between the fitness components of current reproduction (advertisement and care) affected the predicted allocation of resources (and therefore the expected empirical observations) for species where offspring survival requires male care. Although all males were predicted to provide some care in these species, the amount of care provided was not always closely correlated with the level of male display seen by females. If there was a minimum investment in care required for offspring survival, all males that advertised were predicted to provide similar levels of care and the intensity of the display indicated male quality. When more resource investment was required to produce a display signal than to ensure offspring survival, even though all males invested some of their resources into care, only the males that produced the highest intensity displays were predicted to invest a correspondingly greater proportion of their resources in care (Kelly & Alonzo, 2009).

Although this theory predicted that male advertisement should always be a reliable indicator of parental care in species where offspring survival requires male care, there have been empirical observations of what appears to be unreliable signalling in species with either paternal care (e.g. Gasterosteus aculeatus, threespine stickleback (Candolin, 1999; 2000a)) or obligate bi-parental care (e.g. Carpodacus mexicanus, house-finches (Badyaev & Hill, 2002; Badyaev & Duckworth, 2003; Duckworth et al., 2003)). In both of these species, the production of unreliable signals has been hypothesized to be because of trade-offs between male survival and reproduction. In sticklebacks, low-quality males exhibited terminal investment by investing resources into mate attraction while reducing investment into parental care and their own survival (Candolin, 2000a). In house-finches, brighter males provided less care and survived for longer than duller males (Duckworth et al., 2003).

In this paper, we therefore expand our previous model to explore the optimal allocation of resources between advertisement, parental care and male survival in species where offspring survival requires male care. We investigate theoretically whether the inclusion of a trade-off between reproduction and increased male survival affects the reliability of male advertisement as an indicator of a male’s investment in parental care, or of male phenotypic or genetic quality. Finally, we explore whether the predicted allocation of resources varies among males of different qualities and discuss the life history strategies we may see empirically in species where offspring survival requires male care if the predicted resource allocation varies with male quality.

Model description

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

The predictions of our previous model for male allocation of resources (Kelly & Alonzo, 2009) differed substantially from those of earlier theory which predicted that male signalling can be an unreliable indicator of male parental care (Kokko, 1998) and from empirical observations based on that theory (Candolin, 1999; Duckworth et al., 2003). While some of the differences between the predictions of Kokko (1998) and Kelly & Alonzo (2009) are because of the form of fitness equation used (additive versus multiplicative, respectively), both Kokko’s model and the empirical work suggest that males are trading off investment in parental care for increased investment in their own survival. Here, we investigate whether our predictions for male advertisement as an indicator of male parental care are affected by the presence of an allocation trade-off between current and future reproduction. Our model examines the trade-off in resource allocation between current and future reproduction for males of differing quality. We consider a resource allocation trade-off between current reproduction’s two reproductive traits (male advertisement and parental care), and allocation to increased male survival between reproductive events. As in our previous model (Kelly & Alonzo, 2009) and most other models that examine the expression of condition-dependent traits, we assume that the resources available for allocation to current reproduction and male survival are dependent on a male’s phenotypic quality, Q, and that

  • image(1)

where a is the allocation of resources to the male display trait of advertisement; c is the allocation to male parental care; and x is a male’s allocation to increased male survival between reproductive events. While the model examines male lifetime reproductive success over multiple reproductive bouts, we assume that male allocation does not change over each male’s lifetime. A male’s allocation of resources to increased male survival between reproductive events is also assumed to have no effect on the resources available to him during his next reproductive event and Q is assumed to remain the same for a given male over his lifetime.

As Q is the total amount of resources available for investment, increased investment in any trait is traded off against decreased allocation to the other traits. The fitness gain that a male receives for increased investment in any trait will be dependent on the shape of the gain function for that trait: m(a), f(c) and s(x). m(a) represents the gain in mating success as a function of male allocation to resources to advertisement (a); f(c), the probability of offspring survival with allocation to parental care (c); and s(x), the probability of male survival between reproductive events with allocation to increased male survival (x). Our model assumes that females choose a mate based on the absolute level of advertisement, a, that he displays. Male advertisement will be a reliable indicator of parental care for females if male investment in care, c, increases with increased investment in a (i.e. if females that choose males that advertise more also receive higher levels of care for their offspring), and a reliable indicator of male phenotypic quality, Q, if a is positively correlated with Q. As the model examines the trade-off in available resources between traits, it is possible for a signal to reliably indicate male quality while being negatively correlated or uncorrelated with care (Kokko, 1998).

To facilitate comparison with this earlier theory, we use a fitness optimization approach, rather than a game theoretical model. Thus, we examine optimal male allocation patterns under the assumption that male mating success m(a) does not depend on what other males in the population are doing. This simplifying assumption is biologically consistent with the idea that females choose among males based on their absolute advertisement independent of the frequency distribution of available males (i.e. m(a) is independent of the allocation of other males). While it would certainly be worth investigating the effects of relaxing this assumption in future analyses, this analysis would require making multiple additional assumptions about how female behaviour responds to the distribution of male advertisement and mate availability. Here, our focus is on the trade-off faced by males between multiple components of current reproductive success and survival to future reproduction.

In this model, we assume that a male’s chance of surviving to his first reproductive bout (i.e. from birth/hatching to sexual maturity) is independent of his allocation once reproductive. Using an iterative life history model, we explore the allocation of resources between a male’s current and future reproductive success. As described above, males are assumed to adopt a general chronic allocation, such that the equation for a male’s fitness, w, over his reproductive lifespan takes the form:

  • image(2)

which can be simplified to:

  • image(3)

where m(a), f(c) and s(x) are the functions described earlier. It should be noted that males do not need to allocate any resources to increased male survival to still have reproductive fitness; when s(x) = 0, inline image, which provided that male allocation to both current reproductive traits is nonzero will result in males having some reproductive fitness (as previously modelled in Kelly & Alonzo, 2009). We allow male allocation between current and future reproduction to depend on male quality Q. Consequently, the predicted patterns of male allocation are consistent with male quality being either environmentally or genetically determined. Male quality determines the total amount of resources available for allocation between traits, where males vary in quality between 0 ≤ Q ≤ 1. As eqn (1) implies c = Q –a – x, f(c) can be rewritten as an expression of a: f(Qax) or f(a), allowing us to express current reproductive success in terms of a and x.

We consider two different functional forms for the gain equations involved in current reproductive success, m(a) and f(a), and an additional two functional forms for increased male survival between reproductive events, s(x). The different functional forms represent different biologically relevant scenarios for how mating success, offspring survival and the probability of surviving between reproductive events increase with increased allocation to advertisement, care or male survival, respectively. The effect of male allocation to care on offspring survival is assumed to be independent of the number of offspring for which a male is caring. This is most consistent with forms of male care such as defence from predators and general provisioning of the nest or reproductive site. However, this mathematical form may not fully capture the biology of species where the effect of male care on offspring survival diminishes with the number of offspring receiving care, such as when males individually feed offspring. In total, we examine eight different general cases by considering all possible combinations of the functional forms for m(a), f(a) and s(x) (see Table 1 for complete description of cases).

Table 1.   Summary of gain functions and the full range of parameters explored for each case modelled.
 m(a)f(a)s(x)
Case 1Power function: 0 ≤ βa ≤ 2Power function: 0 ≤ βc ≤ 2Diminishing returns function: 0 ≤ βx ≤ 10; 0 ≤ sb ≤ 0.5
Case 2Power function: 0 ≤ βa ≤ 2Sigmoidal function: 0 ≤ αc ≤ 20; 0 ≤ tc ≤ 1Diminishing returns function: 0 ≤ βx ≤ 10; 0 ≤ sb ≤ 0.5
Case 3Sigmoidal function: 0 ≤ αa ≤ 20; 0 ≤ ta ≤ 1Power function: 0 ≤ βc ≤ 2Diminishing returns function: 0 ≤ βx ≤ 10; 0 ≤ sb ≤ 0.5
Case 4Sigmoidal function: 0 ≤ αa ≤ 20; 0 ≤ ta ≤ 1Sigmoidal function: 0 ≤ αc ≤ 20; 0 ≤ tc ≤ 1Diminishing returns function: 0 ≤ βx ≤ 10; 0 ≤ sb ≤ 0.5
Case 5Power function: 0 ≤ βa ≤ 2Power function: 0 ≤ βc ≤ 2Sigmoidal function: 0 ≤ αx ≤ 30; 0 ≤ tx ≤ 1
Case 6Power function: 0 ≤ βa ≤ 2Sigmoidal function: 0 ≤ αc ≤ 20; 0 ≤ tc ≤ 1Sigmoidal function: 0 ≤ αx ≤ 30; 0 ≤ tx ≤ 1
Case 7Sigmoidal function: 0 ≤ αa ≤ 20; 0 ≤ ta ≤ 1Power function: 0 ≤ βc ≤ 2Sigmoidal function: 0 ≤ αx ≤ 30; 0 ≤ tx ≤ 1
Case 8Sigmoidal function: 0 ≤ αa ≤ 20; 0 ≤ ta ≤ 1Sigmoidal function: 0 ≤ αc ≤ 20; 0 ≤ tc ≤ 1Sigmoidal function: 0 ≤ αx ≤ 30; 0 ≤ tx ≤ 1

Current reproductive success

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

Power function

In species where mating success is closely correlated with male nuptial colouration, or offspring survival is closely correlated with the time spent caring for the offspring, the effect of allocation, a, on male mating success or offspring survival can be represented by power functions such that

  • image(4)

and

  • image(5)

If β < 1 the relationship between resource allocation and mating success or offspring survival has a convex shape and shows diminishing returns. If β = 1, resource allocation has a linear pay-off. Finally, if β > 1, the effect of resource allocation on mating success or offspring survival has a concave shape and shows accelerating returns.

Sigmoidal function

A minimum allocation requirement might exist before individuals obtain high mating success (as in species where females are choosy and only mate with males that advertise above a certain level) or offspring survival (as in species where offspring require a certain level of care to survive) such that

  • image(6)

and

  • image(7)

Here, t is the inflection point of a sigmoidal curve where the gain from allocation switches from accelerating returns to diminishing returns, and α is the slope parameter that determines the steepness of the curve. The higher the value of α the steeper the curve and the higher the minimum requirement for resource allocation required for some mating success or offspring survival.

Male survival between reproductive events

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

Diminishing returns function

We might expect the gain curve for allocation to male survival to show diminishing returns in species where there is a maximum limit on an animal’s ability to increase its probability of surviving to the next mating event. As, for example, in species where there is a limit to the amount of food that an animal is able to sequester to survive a winter. The effect of allocation, x, on a male’s survival between reproductive events can be represented using a modified form of the function for exponentially diminishing returns used by Parker et al. (1989) to describe the effect of parental investment on offspring survival (see also Macnair & Parker, 1979; Jeon, 2008), such that

  • image(8)

where 0 ≤ x ≤ Q. βx is a positive constant determining the rate at which s(x) rises to its asymptotic value (Smax, the maximum probability of surviving to the next mating event), such that the higher the value of βx, the more rapidly the likelihood of surviving to the next reproductive event increases with additional allocation. Smax < 1 so that males cannot have absolute survival between mating events. Sb is a male’s baseline rate of survival where allocation to x increases a male’s survival rate beyond this baseline. Although male allocation to male survival always increases with x, we explored cases where male survival rate is zero (Sb 0) if males do not allocate any additional energy to their own survival (s(x) = 0 when x = 0) as well as cases where males may have a low (Sb > 0) rate of survival even if males do not allocate any additional energy to their own survival (i.e. s(x) >  0 if x = 0).

Sigmoidal function

A minimum allocation requirement might exist before individuals are able to survive to the next reproductive event as, for example, in species where animals have a minimum fat store requirement to survive the winter, such that

  • image(9)

where inline image is the maximum probability of surviving to the next mating event (Smax < 1 to prevent male’s having absolute survival between mating events). tx is the inflection point of a sigmoidal curve where the gain from allocation switches from acceleration returns to diminishing returns. αx is the slope parameter that determines the steepness of the curve. The higher the value of αx, the steeper the curve and the higher the minimum requirement for resource allocation required for some probability of surviving between reproductive events.

Calculating optimal allocation

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

It was not possible to solve analytically for optimal male allocation (because one cannot mathematically solve δw/δx and δw/δa for x* and a* due the exponential portion of the male fitness function). Therefore, we searched numerically for the allocation of resources between traits (i.e. values of a, c, and x such that x) that maximized male expected lifetime reproductive success w as a function of male quality Q (using Matlab©, code available on request). For each value of 0 ≤ Q ≤ 1, we calculated male expected fitness as we increased x incrementally for 0 ≤ x ≤ Q. For each value of x, we allowed allocation to a to increase incrementally for inline image, and set c = Q – a – x. Q, x and a were allowed to increase in increments of 0.001. This allowed us to find the allocation of resources between a, c and x that maximized w for each incremental value of Q. These predictions for the optimal allocation of resources between traits by males of different qualities were then used to plot the relationship between male allocation to advertisement and the benefit received by a female in the form of male quality and allocation to care (Figs 1, 2 and 3 show representative plots of the model’s predictions. Please note that the parameters used in the figures allow the predictions of this model to be compared directly to those of Kelly & Alonzo, 2009).

image

Figure 1.  Optimal male allocations into advertisement (a), parental care (c) and survival (x) when survival between mating events (s(x)) shows diminishing returns: Smax = 0.9, Sb= 0, βx = 5. Case 1: When both mating success (m(a)) and parental care (f(a)) are power functions, βa = 2, βc = 0.5, advertisement is correlated with both parental care and male quality. Case 2: When m(a) is a power function and f(a) is sigmoidal, βa = 0.5, αc = 10, tc 0.5, low allocation to advertisement is correlated with high male quality and allocation to care. Case 3: When m(a) is sigmoidal and f(a) is a power function, αa = 10, ta = 0.5, βc = 0.5, male advertisement is correlated with male quality but, unless a male has high allocation to advertisement, is not a good indicator of his allocation to care. Case 4: When both m(a) and f(a) are sigmoidal, αa = 5, ta = 0.5, αc = 15, tc 0.5, male advertisement is correlated with both male quality and parental care.

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image

Figure 2.  Optimal male allocations into advertisement (a), parental care (c) and survival (x) when the gains from survival (s(x)) have a sigmoidal shape, Smax 0.9, αx = 20, tx = 0.4. All other variables and functions are the same as for Fig. 1. Case 5: When both mating success (m(a)) and parental care (f(a)) are power functions, advertisement is correlated with parental care and male quality. Case 6: When m(a) is a power function and f(a) is sigmoidal, low allocation to advertisement is correlated with high-male quality and allocation to care. Case 7: When m(a) is sigmoidal and f(a) is a power function, male advertisement is correlated with male quality but, unless a male has high allocation to advertisement, is not a good indicator of his allocation to care. Case 8: Even when the minimum allocation to survival (s(x)) required for survival between mating events is low, tx = 0.3, when both m(a) and f(a) are sigmoidal, male advertisement is correlated with both male quality and parental care.

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image

Figure 3.  Optimal male allocations into advertisement (a), parental care (c) and increased survival (x) when the gains from survival (s(x)) have a sigmoidal shape and the minimum threshold for allocation to survival is low: tx = 0.3. All other variables and functions are the same as for Fig. 2. Case 5: When both mating success (m(a)) and parental care (f(a)) are power functions, advertisement is correlated with parental care and with male quality except for high-quality males. Case 6: When m(a) is a power function and f(a) is sigmoidal, low allocation to advertisement is correlated with high male quality and allocation to care. Case 7: When m(a) is sigmoidal and f(a) is a power function, male advertisement is correlated with male quality (with the exception of very high-quality males) but, unless a male has high allocation to advertisement, is not a good indicator of his allocation to care.

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In species with paternal care, the presence of male advertisement is often assumed to act as an indicator of a male’s ability to provide care. We define advertisement to be a reliable indicator of care if males that advertise more never provide less care than males that advertise less. We define advertisement to be a good indicator of the quantity of care a male provides if increases in allocation to advertisement are closely correlated with increases in allocation to care. Our previous theory predicted that male advertisement in species with parental care will always be a reliable indicator of male care when males allocated between current advertisement and care, although the correlation between allocation to advertisement and allocation to care is dependent on whether allocation to care is fitness limiting (Kelly & Alonzo, 2009).

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

We explored a wide range of parameter values for each gain curve to fully explore the range of possible trade-offs that we might expect to see empirically (see Table 1 for complete list of cases, and the parameter values tested for their corresponding gain curves). However, we found that, except for a narrow range of parameter values, the predictions of our model fit two general trends (discussed in detail in the following sections). In general, when we expand the previous model to include male allocation to increased male survival between mating events, males are predicted to (i) allocate all of their resources to current reproduction when both current reproductive traits require a minimum amount of resource investment and/or the function for male survival between mating events has a sigmoidal shape (Fig. 1: case 4 Fig. 2: cases 5–8), and (ii) only higher-quality males are predicted to allocate any resources to increased male survival when s(x) is a power function (Fig. 1: cases 1–3).

Regardless of the shape of the s(x) curve, our model predicts that male advertisement will always be a reliable indicator of care in systems where male care is necessary for offspring survival. Whether male allocation to advertisement is closely correlated with allocation to parental care depends on the shape of the gain curves for current reproduction (see Table 2 for a summary of the model’s general predictions).

Table 2.   Summary of general predictions for allocation and reliable signalling. Exceptions to these predictions found when the threshold for allocation to survival is very low are shown in Fig. 3.
 Cases 1–3: Survival between mating events (s(x)) has diminishing returns Cases 4–8: Survival s(x) has a minimum threshold for allocation or, independent of the shape of survival, both mating success (m(a)) and offspring survival (f(a)) have a minimum allocation threshold
Allocation to current reproductionMales allocate more resources to whichever trait is fitness limiting. Allocation to current reproduction plateaus in high quality males that have maximized their current reproductive fitness Males allocate more resources to whichever trait is more fitness limiting
Allocation to survivalLow quality males allocate nothing to survival. For higher quality males, allocation to survival increases with male quality None
Reliable signal of male quality? Yes 
Reliable signal of male care?Yes. Advertisement is more closely correlated with care when offspring survival is fitness limiting

Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

When male survival (s(x)) has diminishing returns, low-quality males are predicted to allocate all of their resources to current reproduction and none to increased male survival, but males of increasing quality are predicted to allocate an increasing proportion of their resources to their own survival (Fig. 1: cases 1–3). The proportion of their resources that males allocate to increased male survival versus current reproduction will depend on the shapes of the gain curves for mating success and offspring survival and whether m(a) or f(a) is limiting. The value of male quality at which males start to allocate resources to increased male survival is primarily dependent on the relationship between the gain curves for current reproduction. Increasing Sb increases slightly the value of male quality at which males start to allocate resources to male survival but the results remain qualitatively the same. The proportion of males in a population that survive between multiple mating events will depend on the shape of the gain curves and the distribution of male quality in the species.

Generally, when one of the current reproductive traits has diminishing returns, while the other current reproductive trait’s gain curve is limiting, as when it shows accelerating returns (Fig. 1: case 1) or either offspring survival (Fig. 1: case 2) or mating success (Fig. 1: case 3) has a sigmoidal shape, males are predicted to allocate proportionally more of their resources to whichever trait is limiting. For males of higher quality and thus with more abundant resources available, this trait may cease to be limiting. For example, when the increase in gains begins to accelerate in a sigmoidal or accelerating returns curve, resource allocation to the other current reproductive trait is predicted to increase with increasing values of male quality until the gains for allocation to increased male survival are greater than for allocation to current reproduction, the point where the nonlimiting trait’s curve and s(x) intersect (Fig. 1a: cases 1–3). At this point, allocation to current reproduction plateaus and males of high enough quality are predicted to allocate any remaining resources to increased male survival (Fig. 1b: cases 1–3). Males in these species that are likely to survive to the next mating event will all appear to advertise at the same level or provide the same amount of care, whereas males that do not have the resources to allocate to increased male survival will show varying levels of care or advertisement. When one of the current reproductive traits shows accelerating returns and the other has a sigmoidal shape, both reproductive traits will be limiting and the value of male quality at which males start to allocate resources to male survival is predicted to increase as the value of β increases (cases 2 and 3; not shown).

In case 1 (both current reproductive traits are power functions), when both mating success and offspring survival have similar gain curves (i.e. when both show accelerating or diminishing returns; not shown in Fig. 1), then the allocation of resources to current reproduction increases steadily, with similar allocation of resources to each trait, until it begins to plateau and males with enough resources left over begin to allocate their remaining resources to increased male survival. When both m(a) and f(a) show diminishing returns, allocation of resources to current reproduction reaches a plateau rapidly, as the gain for increased allocation begins to decrease, and even low- to medium-quality males are able to allocate resources to increased male survival. In these species, both male advertisement and parental care will require little male investment, and the majority of males (assuming the distribution of male quality is normal) will have multiple mating events. When both m(a) and f(a) show accelerating returns, there is a much greater amount of resources allocated to current reproduction before it starts to plateau, and only high-quality males are able to allocate resources to increased male survival, resulting in species with higher levels of advertisement and care and fewer males surviving over multiple mating events. The different predictions result because males do not stop allocating extra resources to current reproduction until both curves cease to be limiting.

Regardless of the shape of the gain functions for current reproduction, our model predicts that advertisement will be a reliable indicator of a male’s ability to provide care. However, the degree of correlation between advertisement and care will depend on the shape of the gain curves. When mating success is limiting, as in species where females are choosy, advertisement is predicted to be closely correlated with male quality but will either be correlated with relatively low levels of care (Fig. 1c: case 1) or will only be a good indicator of care in high-quality males (Fig. 1c: case 3). When parental care is the fitness-limiting trait, as in species where offspring require a minimum amount of care to survive, males are predicted to produce less exaggerated displays and either any males that are able to advertise will provide the minimum level of care required for offspring survival (Fig. 1c: case 2) or small increases in advertisement will be correlated with larger increases in the amount of care provided (case 1: when f(a) has accelerating returns: not shown). As higher-quality males are predicted to allocate more of their resources to increased male survival than lower-quality males, the increase in male quality with respect to increased advertisement accelerates rapidly at higher levels of advertisement (Fig. 1c: cases 1–3).

Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

When male survival (s(x)) has a sigmoidal shape, males are not predicted to allocate any of their resources to increased male survival (Fig. 2: cases 5–7) unless the minimum threshold for allocation to male survival required to survive between mating events is very low and neither current reproductive trait shows accelerating returns (Fig. 3: cases 5–7). For the majority of possible gain curves, males are predicted to allocate the majority of their resources to whichever current reproductive trait is more fitness limiting. If the limiting trait has a sigmoidal shape (cases 6 and 7), once the trait begins to increase exponentially with increased investment, males should switch to investing increasing resources in the other current reproductive trait (Fig. 2b: cases 6 and 7). Male advertisement is predicted to be a reliable indicator of care and quality when males do not invest resources in increased male survival. There will be a higher level of advertisement in species where mating success is the fitness-limiting function, as for species where females are choosy (Fig. 2c: cases 5 and 7), but care and quality will be more closely correlated in species wher parental care is the fitness-limiting function, as in species where offspring require a minimum amount of care to survive (Fig. 2c: case 6 vs. case 7). The majority of species where male survival between reproductive events requires a minimum resource investment are predicted to allocate all their resources into a single reproductive event by this model.

It is only when the minimum amount of resources required for male survival to the next reproductive event is low and one or more of the current reproductive traits has diminishing returns that males are able to invest any resources into increased male survival, and even then only very high-quality males will be able to invest the required resources into current reproduction and male survival (Fig. 3: cases 5–7). When these conditions are met, there is an abrupt shift in the allocation of the majority of resources between current reproduction and future reproduction from low- to high-quality males. High-quality males are able to allocate the resources required into the current reproductive traits until the gains from increased allocation begin to show diminishing returns and are also able to allocate enough resources to increased male survival to meet the minimum requirement of investment to see any gain in male survival (Fig. 3b). Lower-quality males cannot meet this minimum allocation required for increased male survival and so allocate more of their resources to current reproduction. The lower the minimum resource investment required for male survival between reproductive events the lower the predicted male quality (and if male quality distribution in a population is normal, the greater proportion) of males in these species that are predicted to survive between multiple mating events. Male advertisement is predicted to remain a reliable indicator of the care provided by a male but will not be a reliable indicator of quality (Fig. 3c). High-quality males will advertise at the same level as low- to medium-quality males and at a lower level than some males of lower quality. Females will therefore be unable to distinguish between males of medium and high quality, but will still receive the most direct benefits for choosing males that advertise more.

Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

When both mating success and offspring survival are sigmoidal (Fig. 1: case 4 and Fig. 2: case 8) and therefore require a minimum level of investment in advertisement and care, respectively, before they show increasing gains, males are not predicted to invest any resources in increased male survival. This prediction is independent of the shape of the gain curve for male survival. When both current reproductive traits have a minimum allocation threshold, the gain in fitness is greater for investing additional resources into either current reproductive trait than into future reproduction. Male advertisement is a reliable indicator of both quality and care. Advertisement is a better indicator of quality and care when allocation to advertisement is lower (i.e. when offspring survival is the more limiting function; Fig. 1c: case 4 and Fig. 2c: case 8) than when it is higher (not shown). In species where such gain curves exist, such as those where females are choosy and offspring require a minimum care investment to survive, males are predicted to invest all their resources into a single mating event.

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References

In general, the addition of a trade-off between current and future reproductive investment to our previous model had little effect on the predicted allocation of resources to current reproductive traits or its predictions regarding reliable signalling. Furthermore, our expanded model also finds that resource allocation in species with obligate male care is expected to differ from predictions made assuming female-only or bi-parental care (see Kokko, 1998). It is generally assumed by life history theory that the trade-off between current and future reproduction, usually expressed as survival, will alter the costs and benefits of investment in reproductive traits (Williams, 1966; Stearns, 1992). This trade-off between current and future reproduction has been used to explain empirical observations of decreased male parental care or unreliable male advertisement in species where offspring survival requires male care (Candolin, 2000b; Duckworth et al., 2003). However, our model suggests that the explanation for apparent terminal male investment (Candolin, 2000b) or unreliable male signalling (Duckworth et al., 2003) may be more complicated and may possibly be because of factors other than male investment in survival versus current reproduction. In only 3 of the 8 cases that we examined (Cases 1–3) does our expanded model predict that males would allocate any of their resources to survival between mating events. For the biological circumstances considered here, unless survival between reproductive events falls within a relatively narrow parameter space (when male survival requires little investment of resources), species where offspring survival requires male parental care are expected to concentrate their resources into current reproduction.

The limited parameters under which males are predicted to allocate resources to increased male survival by this model may have implications for species where offspring require paternal care. For biological scenarios consistent with those examined by our model, it may be that the life history trade-off between reproduction and increased male survival to future reproduction is less important than the life history trade-off between reproduction and growth which affects the age of first reproduction. Under the circumstances described by our model, males in the majority of cases, and indeed low-quality males in all cases, are predicted to have semelparous life histories. However, our model does not investigate the allocation of resources into a male’s survival to his first reproductive event, so our prediction that males may delay their age of maturity to when they are large enough or of high enough quality to maximize their current reproductive success is based on an extrapolation of our predictions. It is worth noting that many of the iteroparous species that exhibit male parental care are fish, which continue to grow throughout their lives (Helfman et al., 1997). For such species, it may be difficult to separate allocation to growth from allocation to increased male survival as the probability of an individual surviving between mating events may increase as an individual grows. Indeterminate growth will affect not only male survival but also the resources available to an individual to allocate between current and future reproduction (his Q value in our model). We suggest that a model of chronic allocation may capture the allocation of resources in such species over a short time period, but may not be able to capture long-term allocation in species with indeterminate growth. In addition, the gain curves for mating, parental care and survival may change over the lifetime of the male because of growth or other changes in individual condition, experience and state. Further models should examine whether such state-dependent effects alter the predictions made here and possibly explain observed empirical patterns in iteroparous species with paternal care.

Even under those circumstances where our model predicts that males will allocate resources between both current and future reproduction, male advertisement is predicted to remain a reliable indicator of a male’s ability to provide parental care (Fig. 1c: cases 1–3 and Fig. 3c). Compared to cases where males do not allocate any resources to increased male survival (Fig. 2), males are predicted to reduce investment of resources into both advertisement and care (Fig. 1: cases 1–3). In contrast, for similar biological parameters, previous theory predicted that males would produce unreliable signals of parental care by decreasing investment in one current reproductive trait over the other (Kokko, 1998). Therefore, for species where male care is required for offspring survival under the circumstances considered here, we predict that the overall level of advertisement observed may be decreased, a prediction consistent with empirical observations of lower levels of advertisement than expected in species where females receive direct benefits (Owens & Hartley, 1998; Duckworth et al., 2003).

Why does our model find a relatively small effect on the predicted pattern of advertisement and paternal care when allocation to increased male survival is included? The answer may be that, for species where offspring require male care, male fitness may be primarily dependent on a male’s ability to both attract mates and care for his offspring. As males must allocate some of their resources to both current reproductive traits to have current reproductive success, it is only males that have enough resources to ensure current reproductive success that are able to allocate any additional resources to increased male survival. When males need to allocate a relatively small proportion of their resources to current reproduction (as in Case 1 when both current reproductive traits show diminishing returns; not shown in figures) males of lower quality are able to allocate their additional resources to increased male survival, although higher quality males are able to allocate proportionally more. The lower the allocation of resources required for males to have high current reproductive success in our model’s gain curves, the more likely is that males will be able to survive to multiple mating events.

Given our model’s predictions and our assumption that females rely on male advertisement for information when choosing a mate, it is useful to ask how the reliability of male advertisement might affect female preferences and the direct benefits from paternal care received by females. In most of our biological scenarios, our model predicts that male advertisement is a good indicator of both phenotypic quality and care (Fig. 1: cases 1, 4 and Fig. 2: cases 5–8); in others, advertisement may be a good indicator of care but not phenotypic quality (Fig. 3: cases 5 and 6); or it may be a good indicator of phenotypic quality but not, generally, correlated with care (Fig. 1: case 3 and Fig. 3: case 7). Finally, there may be almost identical levels of care across males but increasing levels of advertisement with quality (Fig. 1: case 2; Figs 2 and 3: case 6). In terms of the behaviours, we might expect to observe empirically, provided that females are choosing males for the direct benefits that they will provide, and these different scenarios give rise to two predictions. (i) If offspring survival does not have a minimum allocation threshold, females are expected to have a preference for males with the greatest level of advertisement, as these males will also provide the most parental care. Depending on how much investment is required for males to produce a signal, female preferences may be more (cases 1 and 5) or less (cases 3 and 7) plastic. (ii) If offspring survival has a minimum care requirement (cases 2 and 6), females need not exhibit strong preferences, provided that they have a minimum advertisement threshold for mate acceptance, as males that are able to produce a signal have already maximized their offspring survival.

That is not to say that our model predicts no dishonesty in species with obligate male care. In cases where high-quality males have the highest fitness from allocating more resources to increased male survival than either care or advertisement (Fig. 3b: cases 5–7), our model predicts that females who prefer high-quality (Q) males will be unable to distinguish between medium- and high-quality males (Fig. 3c: cases 5–7). Females choosing males that produce the highest level of advertisement will not be choosing mates with the highest phenotypic quality. However, they will be choosing males of higher, if not the highest, quality, and these males are predicted to allocate the most resources to parental care. It may be unlikely that females in species where such gain curves occur will be selected to have a preference for the highest quality males; there may in fact be selection against such a preference.

For the scenarios examined, our model also suggests that, while male survival may have an effect on the allocation of resources between advertisement and care, only high-quality males will allocate any of their resources to survival between mating events. It also suggests that for the majority of cases that fit the biological parameters of the model, males will allocate none of their resources to male survival, resulting in semelparous life histories. As many of the empirical observations of unreliable male advertisement are in iteroparous species (Candolin, 1999; Badyaev & Hill, 2002; Badyaev & Duckworth, 2003), our model suggests that the current explanations for unreliable signals in these species may be incomplete and that further investigation into the life history trade-offs in these species may be required. It is possible that the reliability of male advertisement in species where offspring require male parental care may be affected by trade-offs not captured by this model, such as those between current reproduction and growth.

We suggest that there are two main directions that future models should explore in species where offspring require male care. (i) This model, while iterative, assumes that male allocation of resources between current and future reproduction follows a chronic pattern. It also does not examine the presence of age-dependent effects on resource allocation, which may be especially relevant for species with indeterminate growth or where male quality follows a cumulative pattern over additional mating events. Allowing male allocation to increased male survival to contribute cumulatively to the resources available to current reproduction at the next reproductive event may affect the predictions for male allocation to increased male survival. (ii) This model, like many other resource allocation models, is an optimization model. Using a game theoretical model would allow us to examine male allocation of resources when mating success is dependent on the behaviour of other males in the population, i.e. when m(a) is affected by male–male competition. However, before a game theoretical model can be used to fully explore male allocation, we first need to consider how female behaviour might co-evolve with male advertisement in the presence of male-male competition.

While further expansion of current theory is needed, our model shows that, for the biological scenarios examined, previous predictions for male allocation of resources between current and future reproduction do not apply to species where male care is necessary for offspring survival. Our model also shows that because of its effect on the relative importance of current versus future reproduction for male fitness, the expected pattern of male allocation and the reliability of male signalling may depend very strongly on whether paternal care is assumed to be necessary for offspring survival.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model description
  5. Current reproductive success
  6. Male survival between reproductive events
  7. Calculating optimal allocation
  8. Results
  9. Cases 1–3: Resource allocation when increased male survival between mating events shows diminishing returns and one or more of the reproductive trait curves is a power function
  10. Cases 5–7: Resource allocation when male survival between mating events has a sigmoidal shape and one or more of the reproductive trait curves is a power function
  11. Cases 4 and 8: Resource allocation when both mating success and offspring survival have sigmoidal shapes, independent of the shape of the curve for male survival
  12. Discussion
  13. References