Survival costs of reproduction predict age-dependent variation in maternal investment

Authors


Holly K. Kindsvater, Department of Ecology and Evolutionary Biology, Yale University, P.O. Box 208106, New Haven, CT 06520, USA. Tel.: +1 203 4324077; fax: +1 203 4322374; e-mail: holly.kindsvater@yale.edu

Abstract

Life-history theory predicts that older females will increase reproductive effort through increased fecundity. Unless offspring survival is density dependent or female size constrains offspring size, theory does not predict variation in offspring size. However, empirical data suggest that females of differing age or condition produce offspring of different sizes. We used a dynamic state-variable model to determine when variable offspring sizes can be explained by an interaction between female age, female state and survival costs of reproduction. We found that when costs depend on fecundity, young females with surplus state increase offspring size and reduce number to minimize fitness penalties. When costs depend on total reproductive effort, only older females increase offspring size. Young females produce small offspring, because decreasing offspring size is less expensive than number, as fitness from offspring investment is nonlinear. Finally, allocation patterns are relatively stable when older females are better at acquiring food and are therefore in better condition. Our approach revealed an interaction between female state, age and survival costs, providing a novel explanation for observed variation in reproductive traits.

Introduction

Life-history theory predicts that females face a complex trade-off balancing future reproduction with current reproductive effort, which includes a trade-off between offspring number and effort-per-offspring (Williams, 1966; Stearns, 1992; Sinervo, 1999). Although larger offspring generally have greater fitness than smaller offspring, a rate-maximization model of offspring size suggests that maternal fitness is optimized at an intermediate level of investment (Smith & Fretwell, 1974). Therefore, although a female is predicted to vary her fecundity according to her expectation of future fitness, she is expected to make only optimal-sized offspring. In addition, if an older female is larger than a younger female, we might expect her to produce more offspring in each clutch.

Although the Smith & Fretwell (1974) prediction of a single offspring size is appealingly simple, empirical data suggest that offspring size and number may be correlated with maternal traits such as size, age or condition. In some species, females in better condition or older females are observed to produce larger offspring, often in greater numbers (Berkeley et al., 2004; Beckerman et al., 2006; Fischer et al., 2006; Gagliano & McCormick, 2007; Plaistow et al., 2007; Marshall et al., 2010). These empirical patterns have motivated the development of more sophisticated theory on female investment in offspring size and number (Parker & Begon, 1986; Winkler & Wallin, 1987; Sakai & Harada, 2001; Kindsvater et al., 2010; Marshall et al., 2010). One prediction to come out of this work is that density-dependent competition among siblings can favour increased investment in offspring size, accompanied by a reduction in fecundity. Females are predicted to make larger offspring than they would otherwise in order to minimize density effects (Parker & Begon, 1986). This mechanism could therefore explain positive correlations between maternal age, offspring size and fecundity (Parker & Begon, 1986; Kindsvater et al., 2010), but the assumption of sibling competition may not be generally applicable.

Another important factor highlighted by this work is that maternal traits may be confounded with maternal age. Specifically, maternal physiology and size have been proposed as mechanisms contributing to the production of larger offspring by older females. In a plant model, Sakai & Harada (2001) suggested that larger females could be more efficient in transferring resources to offspring. Therefore, if developmental time is a constraint, larger mothers can produce larger offspring in the same amount of time. Through this mechanism, maternal size alone could explain the differences in offspring size among females in the same environment. In a more recent model of fish reproduction, Marshall et al. (2010) proposed that size constraints may prevent small females from producing offspring at the size predicted by an analysis based on the Smith and Fretwell model. If this is the case, larger females will produce larger offspring. If maternal size and age are correlated, older females will be observed to produce larger offspring. However, Marshall et al. (2010) noted that there is little evidence for such constraints in highly fecund organisms.

Although these theoretical advances indicate that sibling competition or size constraints could explain why older or larger females produce larger offspring in some species, these factors do not explain the entirety of observed variation. Empiricists have suggested that considering more biological detail in life histories may lead to departures from the Smith and Fretwell prediction (Brown & Shine, 2009; Marshall et al., 2010). Natural and experimental observations suggest that females in better condition or females with lower expected future fitness have more reserves to spend on reproduction and therefore make larger offspring and larger clutches. To investigate whether these mechanisms can lead to departures from the Smith and Fretwell prediction, we developed a model of maternal allocation as a function of state and age. Our model includes both variation in maternal condition and the trade-off between current and future reproduction. We use the model to understand how survival costs associated with reproduction affect maternal allocation (Shine & Schwarzkopf, 1992). We also explore how age-dependent changes in resource availability, which lead to age-dependent variation in female condition, are predicted to affect maternal allocation patterns.

In a previous paper motivated by this problem (Kindsvater et al., 2010), we developed a model of the three-way trade-off among future reproduction, offspring size and offspring number (where size and number make up current reproduction). We found that variation in offspring size is predicted among females with low clutch sizes, especially when the number of offspring produced negatively affects offspring fitness (i.e. density-dependent sibling competition). However, our model predictions were limited to the simplified case of an organism that matures with all its resources available for reproduction (i.e. a capital breeder).

Here, we extended our framework to include income breeding (the case where mature females acquire resources and use them for reproduction repeatedly over their lifespan). Our results address the effects of maternal age and state on allocation patterns in species with survival costs associated with reproduction. Specifically, we address the cases where maternal survival is negatively affected by fecundity and by her total reproductive effort. In each case, reproduction reduces her expected future reproductive success. Our model also examines how food availability and these survival costs interact to affect female state and consequently allocation to reproduction. We found that the interaction between state and age can lead to departures from the Smith and Fretwell prediction when reproduction decreases female survival. Our findings therefore suggest a novel explanation for observed variation in offspring size and number.

We used dynamic programming to find the allocation strategy that maximizes expected lifetime fitness for females in differing ages and states (see Kindsvater et al., 2010). We first used a simple version of the model that accounts for female age and state, but does not include costs associated with reproduction or age-dependence in food availability. This simple model provides a baseline prediction for comparison when we add more biological complexity. In order to understand how the female’s expectation of future reproductive success affects her current allocation strategy, we investigated two types of reproductive costs and also the effects of age-dependence on state. We first asked how survival costs associated with fecundity affect the predicted offspring size and number. We then explored the effects of costs that depend on total reproductive effort. Finally, we asked whether allocation patterns change when older females are more likely to be in good condition because of age-dependence in food encounter and quantity. In general, we aim to clarify when age- and state-dependent variation in offspring size and number is predicted and whether these patterns can better explain observed empirical patterns.

Model description

State and age dynamics

We considered a model in which a female matures with a given quantity of energy for reproduction, which we refer to as her reproductive state g. We modelled her reproductive life history as a discrete sequence of reproductive events, or time steps. After a given number of time steps, the female reaches a point beyond which further fitness is not possible. This point, which we denoted as maximum age A, could represent either maximum lifespan or the age of reproductive senescence. Before she reaches A, the female may die from extrinsic mortality or starvation (explained below). Assuming that the female has a fixed period of reproduction allowed us to calculate expected fitness from the current age, a, to the end of life. We then determined the investment strategy that maximizes expected fitness F(g,a) as a function of current state, g, and age, a.

A female’s state increases when she encounters food and decreases with reproduction (Fig. 1). The female encounters food with probability λ in each time step. If food is encountered, her state increases by a fixed quantity y. In each time step, if the female survives, she can save her state or spend any amount of it on reproduction. For simplicity, we did not explicitly consider the trade-off with metabolic costs of somatic maintenance. If she reproduces, the female can vary clutch size and offspring size (Fig. 1). Her state decreases by the amount of the total reproductive effort, which is the product of offspring size, s, and the number of offspring, n, in that reproductive bout. We assumed that there is no variation in offspring size within a clutch, such that all offspring produced in each time step are the same size. We also assumed that there is no parental care; once a female invests in an offspring, she does not need to invest in it further, and her remaining energy reserves are available for reproduction in the next time step. Therefore, if the female encounters food, her future state is

image(1)
Figure 1.

 Schematic representation of our model of reproductive allocation. The size of the grey boxes represents female resources and investment. Items in bold italics (reproduce, allocate to ns) are outcomes predicted by the model. Food encounter rate and survival are input parameters (specified in Table 1).

(where g is the state at age a, g′ is the state at + 1 and y is the quantity of food gained). If her state is less than a critical value (g′ gcrit), then the female starves, and there is no possibility of current or future fitness [F(gcrit, a) = 0]. If the female does not find food, her future state is

image(2)

where g is state at age a and g′ is the state at a + 1. As mentioned above, if g′ < gcrit the female starves, and no future reproduction is possible. We also assumed that there is a maximum amount of energy gmax that a female can store. Any energy that the female gains through encountering food exceeding this maximum cannot be saved or spent on reproduction.

The probability a female survives from age a to + 1 is α. If the female reaches maximum age A, we assume there is no opportunity for current or future reproductive success. Therefore, the fitness at age A is defined as

image(3)

Assuming that the female has a fixed maximum age in eqn 3 allows us to optimize lifetime fitness using dynamic programming (Mangel & Clark, 1988).

Offspring fitness

Following Smith & Fretwell (1974), we modelled offspring fitness as a sigmoid function of offspring size, s. The female’s current reproductive success P is the product of the number of offspring produced, n, and the fitness gained from each offspring size of size s:

image(4)

where K, p, m and χ determine the shape of the investment function (Fig. 2, Table 1). The sigmoid function is flexible, but is always bounded and therefore will always yield a single optimal offspring size in a Smith and Fretwell analysis. In the absence of a trade-off between current and future reproduction and state-dependent effects, we expect the optimum will be at the level where the rate of return on maternal investment is maximized (Fig. 2). Therefore, the functional form of eqn 4 provides a prediction of offspring size. We refer to this optimum as inline image; it provides a natural basis for comparison with our model predictions of offspring size.

Figure 2.

 Offspring fitness as a function of size. Smith & Fretwell (1974) predicted that maternal investment is maximized at the intersection between the curve and the tangent passing through the origin. Here, the optimum for female investment in offspring size is inline image = 8.

Table 1.   Description of model parameters.
ParameterValueDescription
Kmpχ105521.5Shape parameters in the offspring fitness function (Fig. 2). Together, they determine the steepness of the curve, the inflection point and the asymptote
inline image8Optimal offspring size predicted by a Smith–Fretwell analysis
α0–1Female survival from one age to the next
μ0.01–0.1Constant determining mortality risk in the baseline scenario
b0–0.5Shape parameter determining mortality risk with survival costs of reproduction
c0.0001–0.001Constant determining mortality risk with survival costs of reproduction
λ0.3–1Probability of encountering food at each age
y100–500, or depends on ageQuantity of food obtained (added to state) if food is encountered
F60–100Scaling factor determining the abundance of food at each age
G, granged from 1 : 500Energetic reserves, equivalent to the female’s state at age
A, aranged from 1 : 10Female age; A is the maximum length of the reproductive period, after which there is no opportunity for reproduction
nranged from 1 : 80The number of progeny the female can make in each reproductive bout; the optimal number is n*
sranged from 2 : 18Offspring size or per-offspring effort; the optimal size is s*
PParental fitness associated with a reproductive event at a given age and state

Optimization methods

We used a dynamic programming approach to find the allocation strategy that maximizes expected fitness as a function of state and age. This method allows us to examine explicit age-dependent allocation strategies in the context of lifetime fitness (Mangel & Clark, 1988; Houston & McNamara, 1999; Clark & Mangel, 2000; Kindsvater et al., 2010).

With the offspring fitness function, the probability of maternal survival, and the probability of food availability (defined in the previous section), female fitness from age a to the end of life A is

image(5)

where g′ and g″ are given by eqns 1 and 2, and P is given by eqn 4. Equation 5 is a dynamic programming equation (DPE). The solution of this DPE is an allocation matrix that specifies the optimal offspring size and number at every combination of state and age (Mangel & Clark, 1988; Kindsvater et al., 2010). Although we modelled state discretely, we used linear interpolation to smooth out discontinuities in the fitness landscape (Clark & Mangel, 2000). We also modelled reproduction as an integer even though one component of reproductive investment, offspring size, is continuous. Therefore, to minimize spurious results caused by rounding, we subdivide maternal investment into finer units, approximating continuous investment in offspring size. We refer to the investment that maximizes expected lifetime reproductive success for a given state and age as s* (size) and n* (number).

The baseline model

Initially, we assumed that female survival at each age α was a constant [α = exp(−μ), where μ is a shape parameter (Table 1)]. Therefore, extrinsic mortality risk (e.g. predation) is 1 − α. We also assumed that food encounter rate λ and food quantity y were constant.

We also explored the effect of an age-dependent increase in maximum fecundity. This scenario applies to taxa that face a physical size constraint on fecundity or reproductive effort when they are young [reviewed by Beekey & Hornbach (2004)]. For simplicity, we assumed that the maximum clutch size possible for each female increased linearly with female age. This assumption reflects the pattern that variation in offspring size is usually relatively small relative to variation in fecundity in organisms with a physical size constraint on fecundity or reproductive effort (i.e. those with indeterminate growth).

Survival costs associated with reproduction

We expanded the baseline model to include costs of reproduction that negatively affected female survival and thereby decreased her expected future fitness. We first assumed that the costs scale nonlinearly with female fecundity so that survival was

image(6)

Equation 6 defines the nonlinear increase in mortality associated with greater clutch sizes. As before, α is the probability of surviving to the next age; c is a constant, and b is a shape parameter that determines the degree of nonlinearity in the costs. In this case, we assume that fecundity alone negatively affects female survival (Pianka & Parker, 1975). The strength of these costs is determined by the shape parameter b. Fecundity can negatively affect maternal survival if egg density affects predation probability (e.g. burst-speed in livebearing fish) or if the energetic requirements of manipulating offspring size are less than costs of investing in additional offspring. For example, in some marine invertebrates, it has been demonstrated that the egg jelly coat associated with each offspring is costly and can influence the trade-off between egg size and number (Podolsky, 2004).

We next examined the case where the survival costs of reproduction depend on both offspring size and number. We assumed that survival was

image(7)

In this case, we assume that reproductive costs result in a nonlinear increase in female mortality. Gravid females may be less effective at escaping predators (Ghalambor et al., 2004) or suffer reduced locomotor performance (Cox & Calsbeek, 2010). All other parameters are the same as in eqn 6. In both cases, costs of reproduction decrease the female’s chance of surviving to reproduce again and are independent of her energetic state.

Effects of age-dependent food

We also addressed the possibility that older females are more successful in obtaining food y in each time step because of an advantage in experience or size. We added age-dependence in y to both the baseline model (The baseline model) and both survival cost scenarios (Survival costs associated with reproduction) to determine whether a correlation between age and condition changes the predicted allocation pattern in either case. We assumed a linear relationship between female age and food such that

image(8)

F is a constant that determines the amount by which her state increases if a female encounters food.

As a consequence of this age-dependence, in this scenario older females that survive are more likely to be in good condition than younger females. This could affect allocation patterns in several ways. A young female that faces restricted food quantities may chose to invest less in offspring size or number in order to avoid starvation. Conversely, an older female in better condition may invest more in offspring size and number, because she has a lower expectation of future reproductive success and is likely to replenish her state if she finds food (Creighton et al., 2009).

Finally, we explored how state-dependence in food availability affected our model predictions. In this case, we assumed that females in higher state were more likely to encounter food. The relationship between food encounter and state was linear.

Sensitivity analyses

To ensure that our results were robust, we explored a range of values for all parameters to determine how they affected the predicted offspring size and number (Table 1). We focussed on the key parameters that determine the conditions under which variation in offspring size and number were predicted. We asked whether survival costs (α negatively affected by n) led to a different outcome than the baseline scenario (constant α). We varied the probability of encountering food, λ, and quantity of food, y, to explore how resource availability affected allocation. We also assessed the effect of including age-dependence in food quantity by varying F.

Results

The baseline model

We found that when survival was constant, our model did not predict variation in response to maternal state or age. Fecundity, however, was predicted to increase with female state. The predicted offspring size was close to the Smith and Fretwell optimum inline image (where = 8), although there was some minor variation due to integer effects. Integer effects arise because all offspring within a clutch must be of the same size; as a result, there may be some fraction (remainder) of energy that is distributed among all offspring in the clutch (Charnov & Downhower, 1995; Ford & Seigel, 2010). We found that integer effects were most evident at low clutch sizes and disappeared as clutch size increased. Figure 3a shows the predicted offspring size and number combinations predicted at every state and age by the allocation matrix when food is of high quality (y = 300) and abundant (λ = 0.9). We plot this relationship to demonstrate that a sample drawn from a population of females of heterogeneous age and state may not exhibit a negative relationship between offspring size and clutch size (i.e. a trade-off). However, the range of offspring sizes decreased with increasing clutch size. We note that because small clutches are produced by low-state females, low-state females show the most variation due to integer effects on offspring size in this scenario.

Figure 3.

 The relationship between offspring size and clutch size for each mortality scenario when food is constant. Data are drawn from the optimal allocation matrix, which includes the predicted allocation pattern for every combination of age a and state g. In each case, λ = 0.9 and y = 500. (a) Offspring size s* predicted for every clutch size in the baseline scenario. Deviation from inline image is due to integer effects, which are most prominent at small clutch sizes (n < 10). (b) The range of predicted offspring sizes with survival costs of fecundity (eqn 6; c = 0.001, = 0.3). The variation represents the range of combinations of offspring size and clutch size that are predicted for every state g and age a in the allocation matrix. Note that the range of offspring size at intermediate clutch sizes (e.g. n = 30) exceeds the variation due to integer effects (see panel a). Offspring size decreases at the largest clutch sizes (which are the terminal investment of females in high state). (c) Offspring size at each clutch size for the scenario with costs of total reproductive effort (eqn 7; c = 0.0005, = 0.3).

Our model did not predict that offspring size or number varied with food availability λ or y (Fig. S1a). Regardless of her chances of replenishing state, a female is predicted to spend her available energy on current reproduction, because survival to the next age is uncertain, and there is a maximum amount of energy she can store. In Fig. 4, we plot the data in Fig. 3 as a surface plot of the optimal offspring size and number for a given state and age. When food is readily available, the pressure to spend, not save, current resources amplifies the integer effects seen in low-state females. Figure 4a shows that with the exception of these integer effects, our baseline model confirmed the Smith–Fretwell prediction of minimal variation in offspring size at every combination of state and age.

Figure 4.

 The predicted offspring size (left column) and number (right column) for all combination of state and age in three mortality scenarios. Food abundance is constant; as in Fig. 3, λ = 0.9 and y = 500. (a) The baseline scenario. (b) Survival costs of fecundity: young females in high state are predicted to produce the largest offspring. (c) Survival costs of total reproductive effort: older females produce larger offspring.

Age-dependent fecundity constraints

We also explored the case where older females are larger and therefore capable of producing larger clutches. We found that when the maximum clutch size increased linearly with maternal age, young females in high state increased offspring size (Fig. S2), whereas older females, capable of producing large clutches, allocate inline image to each offspring. Therefore, when fecundity is limited by female size but resources are abundant, females may increase current reproductive effort by increasing offspring size.

Survival costs associated with reproduction

Survival costs of fecundity

To clarify how survival costs of reproduction shape trade-offs between offspring size and number, we examined the scenario in which fecundity has a negative effect on female survival (α decreased with the number of offspring produced n). The severity of the costs depends on the shape of the survival function, determined by b (in eqn 6). We found that including these costs of reproduction increased the predicted offspring size for females in high state. The effect is most pronounced in young females that matured in good condition. These females were predicted to make fewer, larger offspring than older females in the same state in order to increase survival. In contrast, older females have lower expected future fitness and make more offspring of size inline image.

We evaluated the values of b ranging from weak (= 0.1) to strong (= 0.5) mortality. In Fig. 3b, we focus on the scenario where = 0.3 because it clearly demonstrates the effect of weak to intermediate survival costs on allocation patterns. We discuss the effects of stronger survival costs (> 0.3) later. We assume abundant food in Figs 3b and 4b (λ = 0.9, y = 500) as when food is scarce, young females and females in low state reduce their reproductive effort by decreasing fecundity, or skipping reproduction altogether. Variation in offspring size is not affected (Fig. S1b).

Figure 3b shows that when survival costs are present, many combinations of offspring sizes and clutch sizes are predicted from the allocation matrix. In comparison with the baseline predictions (Fig. 3a), we found greater variation in offspring size. The range of predicted offspring sizes is greatest at intermediate clutch sizes, as we see in the hump-shaped relationship between offspring size and number in Fig. 3b. To understand this relationship, we plot the optimal offspring size and number for each combination of state and age in Fig. 4b. Young females initially produce fewer, larger offspring. As they grow older, females adopt a more risky strategy by making larger clutch sizes of optimum-sized offspring. These patterns explain why we did not see females with large clutches producing large offspring (i.e. why Fig. 3b shows a hump-shaped relationship between offspring size and number).

Although intermediate survival costs (b = 0.3) shift the balance between number and size of offspring, they do not affect the age at first reproduction. In contrast, with greater survival costs (b > 0.3), the concavity of the survival cost function increases (eqn 6), and costs become so severe that young females are predicted to delay reproduction until later in life. Eventually, current reproduction is costly enough that a female delays reproduction until the last time step (i.e. semelparity evolves). In that case, the model did not predict variation in offspring size. Therefore, in this scenario, we expect the greatest age-dependent variation in offspring size and number will occur in species with intermediate or weak survival costs associated with reproduction.

Survival costs of total reproductive effort

We then evaluated the effects of survival costs of reproduction that are based on a female’s total reproductive effort nS. As mentioned previously, the severity of these costs depends on the survival function (eqn 7). We found that when costs are intermediate and very slightly nonlinear (b = 0.3, c = 0.0005) and food is abundant (λ = 0.9, y = 500), our model predicts minor variation in offspring size (Fig. 3c). However, this variation in size and number is correlated with female age (Fig. 4c). Young females reduce total reproductive effort by decreasing both offspring size and number. By reducing offspring size, females increase their chances of surviving; the small reduction in per-offspring fitness is compensated for by the increase in survival.

We found that age-dependent increases in offspring size were only present at intermediate costs of reproductive effort. With weaker costs, females did not change their investment patterns from the baseline scenario (Fig. S3a) With stronger costs, young females in poor condition skip reproduction until they are older or have excess state (Fig. S3b). Furthermore, we found the age-dependence in offspring size arose only when food was relatively available (λ > 0.5 and y > 300). Similarly, with scarce food, females in low state skipped reproduction entirely when young, as they were unlikely to replenish their state (Fig. S1c). By skipping reproduction, females increase their changes of surviving to encounter food at a later age.

Age- and state-dependent food availability

To understand how age-dependence in female state affects the predicted variation in offspring size and number, we also considered age-dependence in the amount of food a female acquired, although the rate of food encounter was constant. We assumed that older females are able to acquire more food through an advantage in experience or size and thus are more likely to be in good condition than young females. We considered several functional forms of age-dependence in food, but finally chose to use a linear relationship for simplicity. We varied the scaling factor F, but it did not strongly affect the model predictions. We explored how this age-dependence affected allocation in all three mortality scenarios (Fig. 5, for all three scenarios F = 80 and λ = 0.9).

Figure 5.

 The predicted offspring size (left column) and number (right column) for all combinations of state and age in three mortality scenarios with age-dependence in food availability. In each case, λ = 0.9. (a) The constant mortality (baseline) case with age-dependent food does not change female allocation to offspring size, but affects predicted fecundity at age. (b) Survival costs of fecundity interact with age-dependent food to favour reduced fecundity and increased offspring size among young females. (c) Survival costs of total reproductive effort favour skipped reproduction among young females, especially those in low state.

We also considered how adding state-dependence in food availability might affect model outcomes. We assumed that food encounter was a linear function of state, and therefore females in low state were less likely to encounter food. However, this assumption reduced total reproductive effort in low-state females and decreased variation in offspring size and overall number.

We found that with constant (baseline) mortality and age-dependent food, females did not change their allocation to size and number and therefore produced offspring of size inline image (Fig. 5a). In the scenario with survival costs associated with reproduction, the model predicts a shift in the reproductive effort of young females in low state (Fig. 5b). These females produce few offspring or even skip reproduction in order to minimize the risk of starvation until they are old enough to replenish their state. Unlike the low-state females, young females that mature in high state are predicted to increase offspring size and decrease fecundity in order to minimize the survival costs associated with reproduction. This is the same mechanism and pattern as that discussed in the analysis of survival costs of reproduction alone (Survival costs associated with reproduction). Therefore, the shift in offspring sizes seen in Fig. 5b is a result of the interaction between survival costs of reproduction and age-dependence in food availability.

When food depends on female age, and the costs of reproduction depend on total reproductive effort, young females forego reproduction altogether (Fig. 5c) in order to increase their survival. Although we do see some age-dependence in offspring size, this variation is also present in the constant food case (Fig. 4c). Furthermore, the model predicts that a smaller proportion of females will reproduce. Therefore, when female state and age are correlated, and survival negatively affected by reproductive effort, age-dependence in offspring size might be difficult to detect.

Discussion

We found that costs of reproduction can predict both negative and positive correlations between offspring size and female age when we consider a heterogeneous group of females that vary in age and state. Our model predicted that the optimal investment in offspring size and number will be greater or less than the Smith–Fretwell optimum depending on whether costs are a function of fecundity or of total reproductive effort. When reproductive costs are based on offspring number, offspring size is predicted to increase with female state, and fecundity is predicted to increase with female age. In contrast, when reproductive costs depend on total reproductive effort, we found that young females reduce offspring size by a small amount; old females make the largest offspring. In this case, our model predicted little state-dependent variation in offspring size, but strong state-dependent variation in offspring number. Our results are a novel explanation for observed departures from the Smith & Fretwell (1974) prediction.

We found that older females were predicted to produce larger offspring when females incurred survival costs of total reproductive effort, and food was of intermediate abundance. In this case, young females incrementally reduced offspring size instead of sacrificing clutch size. Females reduced offspring size in order to reduce total reproductive effort and thereby increase survival. In this scenario, the benefit of reducing offspring size or clutch size has an equivalent effect on survival, but the accompanying cost to current fitness (which also depends on size and number) is smaller if the female decreases offspring size than if she decreases number. This is because offspring fitness is a nonlinear function of offspring size. Therefore, young females are predicted to reduce offspring size in order to maximize survival; older females who are approaching the end of life maximize the fitness from reproduction. Although the magnitude of this effect will vary depending on the offspring fitness function and the shape of the cost function, it is generally applicable to species that experience survival costs of reproduction, including highly fecund organisms, such as fish (e.g. Berkeley et al., 2004).

We found that fecundity-dependent survival costs favoured increased offspring size in younger, high-condition females relative to older females. Females minimized fecundity early in life in order to delay paying survival costs. This pattern is similar to the pattern of increased brooding in smaller females described for some marine invertebrate taxa (Strathmann & Strathmann, 1982), although the pattern Strathmann and Strathmann describe is evident when comparing pairs of closely related taxa, not conspecifics. Furthermore, fecundity-dependent survival costs and physical constraints on fecundity in younger females both predict the same qualitative pattern of larger offspring size in young females and increased clutch sizes in older females. Therefore, a negative correlation between maternal age or size and offspring size can be explained by two mechanisms, which are not mutually exclusive. This finding motivates work exploring the relative importance of fecundity costs and size constraints in species with variable offspring sizes.

The baseline version of our model predicted minimal variation in offspring size when food was abundant. This baseline model also predicted that females in better condition will produce more offspring, but that age-dependence in fecundity arises only when food is scarce. This pattern suggests that the trade-off between current and future reproductive effort may be detected only when resources are limiting, except in species with an age-dependent constraint on fecundity (i.e. due to female body size), or when age and state are strongly correlated.

Although this pattern is consistent with classical theory, our naïve expectation was that adding age-dependence in the food quantity a female obtained through foraging would lead to a prediction of increased variation in offspring size as well as in total reproductive effort. Contrary to expectations, our model predicted little variation in offspring size in populations where only older females are in good condition. Specifically, we found that if females are likely to be in better condition as they grow older (due to a correlation between age and food availability), females adopt a risk-averse strategy and reduce reproduction early in life. As a result, only older females reproduce and allocate according to the Smith and Fretwell prediction. Furthermore, the largest amount of variation in offspring size occurred when increased fecundity negatively affected female survival, and young females were likely to be in good condition. This result suggests that variable offspring sizes may be most prevalent in species that are not resource limited. Additionally, our prediction of variable offspring sizes is most relevant to iteroparous species with age-structured populations. In a high-mortality scenario, in which females have little expectation of future reproductive success, our model predictions were consistent with the Smith and Fretwell prediction.

Previous theory addressing age effects on allocation (Marshall et al., 2010) found that maternal size constraints were necessary to predict age-dependent variation in offspring size. Our model evaluated the effects of age on offspring size independently of maternal size. Although we implicitly assumed that maternal size and age were correlated in our analysis of age-dependence in food availability (which might result from increased gape width or another factor related to female size), we did not find that this mechanism affected our prediction of variable offspring sizes. Therefore, our findings underscore the importance of considering the effects of maternal size and age independently in empirical studies of reproductive investment.

Our results suggest that maternal survival costs of total reproductive effort may be driving the empirical pattern where older females allocate more to offspring size because they have a lower expectation of future fitness. However, our findings that fecundity costs may lead to variable offspring sizes are relevant to empirical studies of condition-dependence in offspring size. In a study of reef fish, Gagliano & McCormick (2007) found that females with extra energy increase egg provisioning, not fecundity. Gagliano & McCormick (2007) provided extra food to elevate the condition of female reef fish during gametogenesis. They measured elevated lipids in the offspring of food-supplemented mothers, but did not find a difference in clutch size. Our results motivate further research into the costs associated with reproduction to clarify their role in shaping female investment patterns, as we found that the two types of reproductive costs have different predictions.

Our current model is a significant extension of our previous work (Kindsvater et al., 2010), as we now are able to distinguish variation due to integer effects (e.g. Fig. 3a) from variation due to state and age (e.g. Fig. 3b). The current model incorporates income breeding and therefore predicts a greater maximum clutch size (n* > 50) than our previous model (Kindsvater et al., 2010). We find that variation in offspring size due to integer effects is minimized at these larger clutch sizes. However, we still see the pattern of increased offspring size when there are costs of reproduction. We therefore infer that the mechanisms contributing to variation in offspring size are relevant across a larger range of clutch sizes.

The Smith and Fretwell model of offspring size has greatly influenced studies of the evolution of life-history traits. Although Smith & Fretwell (1974) originally developed their optimality model to understand maternal allocation in birds, the model has been largely successful when applied to other taxa. However, in some cases, empiricists are confronted with variation in offspring size that is not predicted by the Smith and Fretwell model, but may be associated with physiological or ecological factors (Sakai & Harada, 2001; Brown & Shine, 2009; Marshall et al., 2010). Empiricists are then faced with competing explanations for this variation, including physiological constraints, genetic constraints and neutral variation. Our results demonstrate the utility of dynamic state-dependent models in understanding when departures from the Smith and Fretwell prediction can be optimal. Incorporating the interaction between mortality and age into our state-dependent framework reveals that maternal survival costs associated with reproduction can influence the relationship between offspring size and number, suggesting a novel mechanism for observed complexity in maternal investment strategies.

Acknowledgments

This work was supported by the US EPA STAR Fellowship (H.K.K.), Yale University (S.H.A. and H.K.K.), NSF (S.H.A. and H.K.K) and the Royal Society (M.B.B.). We thank Michael Jennions and an anonymous reviewer for comments that greatly improved this manuscript.

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