Behavioral Ecology
doi:10.1093/beheco/arl080
Advance Access publication 23 November 2006
Capital or income breeding? A theoretical
model of female reproductive strategies
Alasdair I. Houston,a Philip A. Stephens,b Ian L. Boyd,c Karin C. Harding,d and John M. McNamarab
School of Biological Sciences, University of Bristol, Woodland Road, Bristol, BS8 1UG, UK,
b
Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK, cSea Mammal
Research Unit, Gatty Marine Laboratory, University of St Andrews, St Andrews, Fife, KY16 8LB, UK, and
d
Department of Marine Ecology, Göteborg University, Box 461, 405 30 Göteborg, Sweden
a
Energy storage is an important component of life-history variation. A distinction is recognized between species that provision
offspring using energy gained concurrently (income breeders) and those that provision offspring using energy stores accumulated at an earlier time (capital breeders). Although this distinction has been recognized for some time, surprisingly little
attention has been paid to the general adaptive value of the 2 strategies. Here, we present a simple, general framework for
modeling female reproductive strategies. We show that our framework can be applicable either to annual breeders that aim to
maximize the energy delivered to their offspring before independence, or to species with shorter reproductive cycles that aim to
maximize reproductive rate, given that their offspring must build up a given level of reserves before independence. For both
scenarios, we show that the costs of accumulating capital can lead to pure income breeding, pure capital breeding, or a mixture
of the 2 strategies. Our model allows the effects of a variety of parameters to be assessed. Length of gestation, offspring
metabolism, efficiency of energy transfer from mother to offspring, and the relative rates of energy gain by females with and
without offspring are all important factors. The cost associated with accumulated capital is a particularly critical determinant of
the strategy adopted. More detailed approaches to specific systems may provide a greater understanding of the factors promoting
different maternal strategies for offspring provisioning. Key words: blubber, energy storage, lactation, mass-dependent costs,
optimal behavior. [Behav Ecol 18:241–250 (2007)]
ife-history theory is concerned with how an animal should
allocate resources to reproduction over its life in order to
maximize fitness (e.g., Roff 1992; Stearns 1992; Roff 2002).
The storage of energy represents an important component of
life-history variation. One aspect of energy storage that has
generated particular interest is the use of stored energy for
reproduction. A distinction is recognized between income
breeders, in which reproduction is financed using current
energetic income, and capital breeders, in which compensatory feeding takes place in advance of breeding, so that reproduction may be financed from stored energetic capital
(e.g., Thomas 1988; Stearns 1992). In practice, a range of
strategies is possible, from pure income breeding at one extreme to pure capital breeding at the other extreme (Thomas
1988).
The potential costs and benefits of income and capital
breeding have been reviewed from 2 different perspectives
( Jönsson 1997; Bonnet et al. 1998). There is a tendency to
view income breeding as the preferred strategy in endotherms
and capital breeding as a necessary adaptation to adverse conditions. This is because of the costs of maintaining and converting stores, as well as the dangers of error in prior estimates
of required capital (Jönsson 1997). By contrast, many ectotherms are predisposed toward storing energetic capital and
so, for these species, capital breeding strategies may be more
energetically efficient than income breeding strategies (Bonnet
et al. 1998). Although this dichotomy has general support,
there is clearly a great deal of variation among both endotherms and ectotherms. Differing modes of transport and var-
L
Address correspondence to P.A. Stephens. E-mail: philip.stephens@
bristol.ac.uk.
Received 27 March 2006; revised 10 October 2006; accepted 16
October 2006.
The Author 2006. Published by Oxford University Press on behalf of
the International Society for Behavioral Ecology. All rights reserved.
For permissions, please e-mail: [email protected]
iation in the thermal stress of environments can both affect
whether capital or income breeding is more economical in
either type of organism. In low productivity environments,
food availability can also limit the feasibility of income breeding if potential rates of energy delivery to offspring are low.
Although there have been a number of taxon-specific treatments of the behavior and physiology of various capital
breeders (e.g., Festa-Bianchet et al. 1998; Bonnet et al. 1999;
Beck, Bowen, Iverson 2003; Beck, Bowen, McMillan, Iverson
2003), there have been far fewer general attempts to examine
the adaptive value of capital and income breeding strategies.
To date, all such attempts have been limited to the order
Pinnipedia (Boyd 1998, 2000; Schulz and Bowen 2005; Trillmich
and Weissing 2006). Pinniped species show several strikingly different strategies for offspring provisioning, broadly
divided along phylogenetic lines (but see further in Discussion) (Bonner 1984; Oftedal et al. 1987; Boness and Bowen
1996). Among these, female true seals (family Phocidae) tend
to be relatively large, with mothers fasting throughout the
weaning period and relying entirely on stored capital to provision their pups; thus, female phocids are often cited as classic examples of capital breeding. By contrast, female fur seals
and sea lions (family Otariidae) tend to be smaller, and employ a foraging cycles strategy, in which mothers intersperse
bouts of lactation with long foraging trips, thus provisioning
pups from current income; as a result, female otariids are
often cited as classic income breeders. Previous models (Boyd
1998; Trillmich and Weissing 2006) and correlative approaches (Boyd 2000) have shown support for the hypothesis
that body mass is likely to have a strong influence on female
strategies. However, the focus on body mass alone cannot
explain the pinniped dichotomy, owing to overlap in size between capital breeding phocids and income breeding otariids
(Schulz and Bowen 2005; Trillmich and Weissing 2006).
Consequently, it seems likely that a more general approach,
Behavioral Ecology
242
focusing less on body mass, will allow other factors to be
identified for further investigation (see also Schulz and
Bowen 2005).
Here, we present a simple model framework that allows for
different rates of energetic gain depending on whether or not
the female is caring for offspring. Differences may arise because the presence of offspring reduces foraging efficiency or
increases energetic demands or because it is necessary to raise
offspring in an area of relatively low food availability. For clarity of description, our model is framed in terms of mammals,
with a gestation period and a period of lactation. We return to
this point in the Discussion, where we consider its potential
application to, and implications for, other taxa. Previous approaches have used energy delivery rate to offspring as the
sole criterion for optimization (Boyd 1998; Trillmich and
Weissing 2006). Although this is a key aspect of the model
that we present, our framework allows for different implications arising from different ultimate optimization criteria.
Here, we focus on 2 different criteria: maximization of the
rate of offspring production and maximization of the quality
of offspring produced (see further below). Using either criterion, we show that relative rates of maternal energy gain before and after giving birth can predict pure income breeding,
pure capital breeding, or strategies intermediate between the
2. Our formulation also allows the effects of other life-history
parameters to be assessed.
THE MODEL
Basic assumptions and structure
Let t be time in years and t ¼ 0 be the time at which the
previous offspring achieves independence (all model notation
is summarized in Table 1). For simplicity, we consider the case
where a female cannot become pregnant until her previous
offspring is independent and where energy gain rate and
metabolism are independent of t. Beyond this, the model rests
on 5 basic assumptions as follows.
(i) Females will seek to maximize the speed at which offspring energy reserves increase. This will be true regardless of whether the female aims to minimize the
time taken to produce offspring with a given level of
reserves or to maximize the level of her offspring’s
reserves within a given time interval.
(ii) The rate at which a female can accumulate energy
reserves will be affected by her current level of reserves. This will be true where current reserves (e.g.,
fat stores) have an impact on metabolism, behavior, or
foraging ability. Specifically, where current reserves
(above some baseline, viable, nonpregnant level) are
denoted x, we consider the case where the net rate of
energy gain, G(x), is a decreasing function of current
reserves (Figure 1A). It is assumed that G(x) is a decreasing function of x because carrying reserves becomes increasingly costly. We refer to these costs (of
carrying capital reserves) as mass-dependent costs.
Their principal outcome is that G#(x) , 0. We denote
the time taken to raise a given level of reserves, x, as
R x dy
t̂ðxÞ ¼ y¼0 GðyÞ
:
(iii) Immediately after birth, all available capital is transferred to the offspring. Transfer of energy from
mother to offspring need not be completely efficient,
so is modeled as occurring with efficiency a. For simplicity, transfer of capital is assumed to be instantaneous (we return to this point in the Discussion).
(iv) Female net income between birth and offspring independence is not dependent on reserves. This is because all available capital is transferred to the
offspring immediately after birth and, thus, the female
returns to her baseline state of reserves (x ¼ 0) at that
time. She maintains this level of reserves by transferring all income (in excess of her own metabolic needs)
Table 1
Summary of notation used
Notation
Description
t
x
t̂ðxÞ
G(x)
Time in years (t ¼ 0 is assumed to be the time at which a female is free to begin a reproductive attempt).
Female energetic reserves, above a viable, lean, nonpregnant level.
Time taken to raise reserves from zero to x.
A female’s net rate of energy gain during periods when she is not provisioning offspring and can, potentially, accumulate reserves.
This is assumed to be a decreasing function of current reserves (see model assumption [ii]).
A female’s net rate of energetic income when foraging as an income breeder. This is assumed to be independent of current reserves as
the female is assumed to maintain herself at a baseline level (x ¼ 0) while income breeding (see model assumption [iv]).
Efficiency with which female reserves can be transferred to her offspring.
Offspring metabolic rate.
The point at which net rate of energy transfer to offspring that results from capital accumulation (by the mother) before birth is equal
to the rate achieved by income breeding.
The time taken to raise x* reserves (i.e., t * ¼ t̂ ðx *Þ).
The minimum reserves required by the female at the time of birth, comprising the energy content of the offspring at birth, together
with any nonretrievable energy in tissues used to support the pregnancy.
The time taken to raise x0 reserves (i.e., t0 ¼ t̂ ðx0 Þ).
The fixed duration of gestation.
Required offspring energy reserves at independence (in the rate maximization model).
A female’s initial net rate of energy gain during periods when she is not provisioning offspring; cF ¼ G(0).
The proportion by which central place foraging (when caring for offspring) reduces the surplus rate of energy gain in females with
offspring relative to cF.
Parameter describing the strength of mass-dependent costs.
The degree of capital breeding, here defined as K ¼ C/(C 1 I), where C is the increase in offspring energy as a result of transfer from
maternal capital and I is the increase in offspring energy as a result of provisioning from maternal income.
cL
a
c
x*
t*
x0
t0
T
E
cF
q
k
K
Houston et al.
•
Modeling capital and income breeding
Figure 1
Optimal accumulation of reserves in the absence of other
constraints. (A) Increasing reserves lead to decreases in the rate of
energy accumulation, so female reserves, x, show a declining rate of
increase with time (as shown by the black line). If the female gives
birth at any time, t, she will pass her stored capital to her offspring
with efficiency, a. Consequently, the value of capital (in terms of
energy available for the offspring) must be discounted by a (as
indicated by the gray line). We term this discounted value of stored
capital ‘‘effective offspring reserves.’’ (B) If the female pursued an
income breeding strategy from the outset (giving birth at time
t ¼ 0), offspring energy reserves would increase as shown, at the rate
acL c. (C) When female reserves are x*, the rate at which effective
offspring reserves will increase as a result of delaying birth (and
continuing to accumulate capital) is equivalent to the rate at which
offspring reserves will increase as a result of provisioning from
income; that is, aG(x*) ¼ acL c, so the gradients of the effective
capital curve (gray line) and effective income function (black line)
are equal. Ignoring other constraints, this is the point at which the
female should give birth, in order to maximize the speed at which
offspring energy reserves increase. The time taken to raise this
optimal level of reserves is denoted t*.
to her offspring. We denote this net rate of income, cL.
Transfers from income are also assumed to occur with
efficiency a. The rate at which offspring reserves increase between birth and independence is thus acL c,
where c is the offspring’s metabolic rate (Figure 1B).
(v) The female can adjust the time at which she gives
birth, for example, by mating at a given time or by
delaying implantation.
These 5 assumptions dictate that a female will adjust the
timing of her offspring’s birth in order to maximize offspring
energy reserves at any time between birth and independence.
To illustrate this point, consider Figure 1. The gray line in
243
Figure 1A shows the increase in effective offspring reserves
that results from a female accumulating capital until any
time, t. The initial gradient of this line is aG(0), where G(0) is
the initial rate of surplus accumulation for a female with no
capital reserves. Figure 1B shows the rate at which offspring
reserves increase as a result of giving birth and provisioning
from income. This function has gradient acL c [see model
assumption (iv)]. Given that the constraints imposed by caring for offspring are likely to have a negative effect (if any) on
the efficiency with which the female forages, it follows that
cL G(0). This suggests that the initial rate at which energy
accumulated as capital can be translated into offspring
reserves will be higher than the rate at which surplus energetic
income can be translated into offspring reserves, that is,
aG(0) . acL c.
As the female accumulates capital, so her rate of capital
accumulation decreases (i.e., G#(x) , 0). Owing to the asymptotic nature of the capital accumulation curve, G(x) decreases
to zero. It follows, therefore, that there will be some x at which
aG(x) , acL c. The consequence of this is that aG(x), the
function that dictates how energetic capital can be translated
into offspring reserves at birth, has an initial rate of increase
that is greater than, but a subsequent rate of increase that is
less than, the rate at which offspring reserves increase from
income provisioning. The crucial point then is that there will
be an intermediate value of reserves, x*, at which the rate of
increase of effective offspring reserves resulting from the female accumulating further capital before birth is equivalent
to the rate of increase of offspring reserves if the female gives
birth and switches to an income breeding strategy; i.e., aG(x*) ¼
acL c. The time taken to raise x* reserves is t̂ðx *Þ ¼ t *; and this
is the point at which income provisioning becomes more efficient than further accumulation of capital. Ignoring any other
constraints, this will be the optimal point at which the female
should give birth (Figure 1C).
Temporal constraints on optimality
Clearly, the basic model is highly simplified. In particular, it is
extremely unlikely that a female will be free to give birth at
any time from immediately after her previous offspring has
attained independence. In light of this, we introduce 2 further
temporal requirements. First, we assume that a female must
have reserves of at least x0 at term, otherwise her offspring
dies. At the time of birth, the female loses x0 reserves (x0
comprises the energy content of the offspring at birth, together with any nonretrievable energy in tissues used to support the pregnancy). The time taken to raise x0 reserves is
t̂ðx0 Þ ¼ t0 : Second, we assume that there is some fixed gestation time, T, during which the female can accumulate reserves
xT. When T depends on aspects of fetal development, only
weakly related to maternal energy availability (e.g., skeletal
or neuronal development), then T may be unrelated to t0.
Timing of birth, timing of implantation (ti), and maternal
strategy for increasing offspring reserves can all be shown to
depend on the relative magnitudes of the 3 time periods: T, t0,
and t* (Figure 2).
Optimization criteria and maternal strategy
Although maximizing the rate of increase of offspring energy
reserves is a key component of the model, we consider 2 ultimate optimization criteria that both provide a context for that
aim but which have subtly different implications for the degree of capital and income breeding. In the first, which we
term ‘‘rate maximization,’’ offspring must accumulate some
fixed threshold level of reserves, E, prior to gaining independence. We assume that females will seek to minimize the time
Behavioral Ecology
244
taken to raise their offspring to independence, either: 1) because in a seasonal environment, they should make the most
of a limited breeding season and produce offspring as early as
possible during the period of relative food abundance or 2)
because in an aseasonal environment, minimizing the length
of the breeding cycle will have the effect of maximizing their
rate of reproduction and, hence, their fitness. The first of
these would also apply to situations in which multiple breeding attempts were possible within the breeding season. The
alternative criterion that we explore is one of ‘‘offspring
energy maximization.’’ Here, we assume that females are
constrained to annual breeding, with the production of
independent offspring necessarily occurring at a given time
of year. We assume that the value of independent offspring
depends on their condition and that females will seek to maximize the amount of reserves of their offspring at the time of
independence (t ¼ 0 in the following year).
Both the optimization criteria considered are bound by
the constraints introduced in the previous section and, thus,
can be assessed in light of the 3 regions of parameter space
summarized in Figure 2. However, both also introduce an
additional constraint. In the rate maximization model, the
additional constraint is that the offspring’s total reserves at
independence must be E. We assume that E . 0 and is a measure of the increase in offspring reserves after birth (over
and above x0). In the energy maximization model, the additional constraint is that the length of the breeding cycle is
limited to exactly t ¼ 1. Here, we assume that t0, T , 1. We
can assess the consequences for maternal strategies of these
additional constraints by defining a measure of the degree of
capital breeding, K. Within the context of the current framework, a useful metric is K ¼ C/(C 1 I ), where C is the increase in offspring energy as a result of transfer from
maternal capital and I is the increase in offspring energy as
a result of provisioning from maternal income. Using this
metric, the maternal strategies arising under each optimization criterion are summarized in Table 2. Clearly, the outcomes are broadly similar, but the additional constraints
affect the situations under which pure capital or mixed strategies would be expected when t* . T, t0, and also affect the
calculation of K. The reasoning underlying Table 2 is provided in greater detail in the Appendix.
Effects of parameters
In spite of the simplicity of this model, it is clear that a number
of aspects of biology (including metabolic rates, food availability, and efficiency of energy transfers from mother to offspring) can have an important bearing on the model
outcomes. Here, we examine the key parameters of the model
to assess how changes in those parameters will affect maternal
strategies. In particular, we examine the effects of offspring
metabolism, efficiency of energy transfer, surplus energy accumulation rate by a mother with offspring, and different
aspects of the relationship between surplus accumulation with
and without offspring.
Parameters of the basic model may have a variety of effects,
depending on which of the regions of parameter space delineated in Table 2 applies. Here, we consider the unconstrained region of parameter space, where t* . T, t0 (e.g.,
Figure 2A). In this region, we can examine the effects of
parameters using the relationship G(x*) ¼ cL c/a. First,
setting u ¼ c/a allows us to examine the effects of c and a
because x* is a function of this parameter:
Gðx *ðuÞÞ ¼ cL u
ð1Þ
Figure 2
Optimal strategies under different time constraints. In each case, x*
and t* are the same, as are a, cL, and c. Only the relative magnitudes
of t0 and T are varied. Under either optimization criterion (rate
maximization or offspring energy maximization), the timing of birth
(and implantation) is dictated by the larger of the 3 time periods, t*,
t0, and T. (A) t* . T, t0. As in Figure 1, the optimal time to give birth
is t* [except where E , a(x* x0), see further in Table 2 and
Appendix]. Implantation is delayed until ti and capital in excess of
the basic demands of parturition (x* x0) is transferred to the
offspring after birth. (B) t0 . T, t*. The minimum time until birth is
now dictated by the minimum time taken to raise the basic demands
of parturition. Implantation is again delayed but, at birth, there is no
excess of capital and hence no capital transfer after birth. (C) T .
t*, t0. A longer minimum gestation period determines the timing of
birth. Consequently, implantation is immediate, and capital (in
excess of the basic demands of parturition) is available for transfer
immediately after birth. In all scenarios, the heavy black line shows
the trajectory of offspring energy reserves after birth, once the female begins provisioning from income. In scenario B, the female
must pursue a pure income strategy. Only in scenarios A and C will
the female pursue a pure capital or mixed capital and income
breeding strategy. Whether a pure or a mixed strategy is more likely
depends on the ultimate optimization criterion, as shown in Table 2.
Differentiating Equation 1 with respect to u gives
dx *
1
¼
:
du
G#ðx *ðuÞÞ
Because G(x) is a decreasing function, x* increases as u increases. Increasing c for constant a will mean that G(x) will
remain greater than cL c/a for higher x, and offspring
energy gain from capital, C, will increase (Figure 3A).
Houston et al.
•
Modeling capital and income breeding
245
Table 2
Optimal strategies under different optimization criteria
Conditions
Rate maximization
Offspring energy maximization
t* . T, t0
If a(x* x0) , E
Optimal implantation time ¼ t* T; a mixed
strategy results, where K ¼ a(x* x0)/E
If a(x* x0) E
Optimal implantation time ¼ t̂ðx0 1E=aÞ T ;
pure capital breeding results (K ¼ 1).
If t* , 1
Optimal implantation time ¼ t* T; a mixed
strategy results, where K ¼ a(x* x0)/[a(x* x0)1
(1 t*)(acL c)]
If t* 1
Optimal implantation time ¼ 1 T; pure capital
breeding results (K ¼ 1).
t0 . T, t*
Optimal implantation time ¼ t0 T
Pure income breeding results (K ¼ 0).
Optimal implantation time ¼ t0 T
Pure income breeding results (K ¼ 0).
T . t*, t0
Immediate implantation (at t ¼ 0) is optimal.
If a(xT x0) , E
A mixed strategy results with K ¼ a(xT x0)/E
If a(xT x0) E
Pure capital breeding results (K ¼ 1).
Immediate implantation (at t ¼ 0) is optimal.
A mixed strategy results with K ¼ a(xT x0)/[a(xT x0)
1(1 T)(acL c)].
Similarly, the delay between previous offspring independence
and birth, t*, will also increase (Figure 3C). Under both optimization criteria, the probability of the female pursuing a pure
capital breeding strategy will be increased (because increasing
x* will increase the likelihood that a(x* x0) E in the
rate maximization model and that t* 1 in the energy
maximization model). Even where mixed strategies are followed, increasing c will increase the overall degree of capital
breeding (K; see Table 2, Figures 3A,C). Decreasing a will
have similar effects on x* to increasing c. However, decreasing
a will also decrease the relative transfer from capital, reducing
C in either model (Figure 3A). Reducing C in the rate maximization model will also reduce K, but in the energy maximization model, effects on K will depend on the time remaining
for income breeding (which will also reduce because t* increases with decreasing a, Figure 3C), together with the total
amount of energy gained by the offspring during that time.
Differentiating Equation 1 with respect to cL gives
dx *
1
¼
;
dcL G#ðx *ðuÞÞ
so that x* decreases as cL increases. Consequently, higher intake rates during the care period will tend to reduce the proportion of the costs of the breeding effort that are financed by
capital (Figures 3B,D), potentially switching the female to
a pure income breeding strategy.
Following model assumption (ii), we have specified that
G(x) is a decreasing function. It is possible to decompose that
function into 2 components. These are the initial surplus rate
of gain when no offspring are present, G(0), which is affected by
foraging ability and food abundance, and the mass-dependent
costs, which affect how rapidly G(x) declines below G(0) as x
increases. We can consider the importance of these 2 aspects
separately. First, we consider the initial surplus rate of gain;
later, we return to the mass-dependent costs. For ease of notation, we define cF ¼ G(0). To emphasize that accumulation of
capital depends on cF , we use the notation GðxÞ ¼ G̃ðx; cF Þ:
Notice that now, the gain rate may differ in situations where
the mass-dependent costs are identical, solely because the initial rate of gain is different (perhaps because food is less abundant). It follows that
G̃ ð0; cF Þ ¼ cF
and from assumption (ii),
G̃ ðx; cF Þ , cF
for x . 0:
To determine how x* varies with cF , we can follow the same
logic as used above to determine how x* varies with other
parameters. Thus, we now write Equation 1 as
G̃ ðx *ðu; cF Þ; cF Þ ¼ cL u:
ð2Þ
Differentiating Equation 2 with respect to cF gives
dx * @ G̃=@cF
¼
dcF
@ G̃=@x
which is positive. Note that although C increases with increasing cF (Figure 3B), t* may not necessarily increase (Figure
3D). In the offspring energy maximization model, a reduction
in t* will lead to increases in the duration of the income
breeding phase and, potentially, increases in the proportion
of offspring reserves that are derived from income.
In the previous example, cF was assumed to vary independently of cL. However, unless the female is migratory (switching environments between the prebirth and postbirth
periods), it is likely that energy gain rates with and without
offspring will be linked. One way that the surplus rate of
accumulation when caring for offspring could be linked to
the initial surplus rate of accumulation in the absence of offspring is if the need to return to her offspring means that the
mother is a central place forager (Orians and Pearson 1979).
For example, a female without offspring may be able to remain in the vicinity of a food source, but a female that is
caring for offspring must return to her offspring at regular
intervals. In this case, if the amount of food in the environment changes, then both cL and cF will change. For convenience, we assume that this change will be proportional, such
that the surplus rate of accumulation for mothers with offspring is a constant fraction of the initial rate of surplus accumulation for females without offspring, that is, cL ¼ qcF ;
where 0 q 1.
In this situation, we can write
G̃ ðx *ðu; cF Þ; cF Þ ¼ qcF u:
Now, we can examine the consequences of varying food availability, given that changing cF will have a proportional impact
Behavioral Ecology
246
Figure 3
Effects of parameters on model outcomes. (A, B) Effects of changes in parameters on the amount of capital transferred after birth: (A) effects of
changing intrinsic factors as indicated by the legend in (C) (including k, the parameter controlling the magnitude of costs associated with
accumulated reserves; c, offspring metabolic rate; and a, the efficiency of energy transfer from mother to offspring); (B) effects of changing
extrinsic factors as indicated by the legend in (D) (including cL, the surplus accumulation rate for mothers with offspring; cF , the initial surplus
accumulation rate for mothers without offspring; cF1, both accumulation rates when mass-dependent costs are an additive function of
accumulated capital; and cF3, both accumulation rates when mass-dependent costs are a multiplicative function of accumulated capital; see text
for further details). (C, D) Effects of changes in parameter values on t*, the time taken to raise x* reserves: (C) effects of changing intrinsic
factors, as before; (D) effects of changing extrinsic factors, as before. Note that all comparisons are relative to the situation shown in Figure 2A.
Note, also, that changes leading to an increase in transfer from capital do not necessarily lead to an increase in t* (e.g., effects of changing a
in [A] and [C]). In the rate maximization model, increases in a(x* x0) will always lead to increases in the degree of capital breeding.
In the energy maximization model, this is less clear and depends on the time remaining within the annual cycle. In this case, if a(x* x0)
increases but t* decreases (e.g., cF1 in [B] and [D]), it is possible that the degree of capital breeding might also decrease.
on cL. In this case, optimal strategies depend on the interaction between intake rates and foraging cost due to increased
reserves. The consequences of this interaction may depend
critically on the form of the function that specifies how surplus accumulation declines with increasing x. For example,
accumulated reserves may have a straightforward, additive effect on accumulation rate. This may happen, for instance, if
accumulated reserves affect only one aspect of energy balance,
such as the metabolic rate (or cost of foraging). In this case,
G̃ ðx; cF Þ ¼ cF hðxÞ; where h#ðxÞ.0: For example, let
G̃ ðx; cF Þ ¼ cF kx 2 : Then,
h#ðx *Þ
dx *
¼ 1 q;
dcF
so that
dx *
. 0:
dcF
As previously, x* will increase with cF (Figure 3B), but t*
may decrease (Figure 3D), leading to uncertain impacts
on K.
It is also possible to envisage situations where the accumulation of reserves affects several components of energy balance simultaneously, potentially leading to multiplicative
impacts on the rate of surplus accumulation. Here, it may
be that G̃ ðx; cF Þ ¼ cF H ðxÞ; where H #ðxÞ,0: For example,
G̃ ðx; cF Þ ¼ cF =ð11kx 2 Þ: Then,
H ðx *Þ ¼ q u=cF ;
and hence x* decreases as cF increases (Figure 3B). This will
tend to reduce K as cF increases and may thus have the opposite effect to the case where the costs of accumulated reserves
were additive. This demonstrates the critical importance of
understanding the functional form of the costs of carrying
capital.
Finally, we note that in either of the cases above, increasing k
will increase the strength of mass-dependent costs, causing net
energy gain rate to decline more rapidly with increasing accumulation of capital. In the absence of changes in other parameters, this will always have the effect of decreasing both x* and t*
(e.g., see Figures 3A,C for the additive case), making a pure
income breeding strategy more likely and reducing K in both
the rate maximization and energy maximization models.
DISCUSSION
The simple model illustrated above demonstrates that females
may adopt a range of breeding strategies, including pure income breeding, pure capital breeding, and a mixture of the
2 strategies. The model can be applied to annual breeders
and to species with shorter reproductive cycles. By maintaining
a general structure, we have shown how different aspects of
ecology and physiology might be expected to affect the selected strategy. Previous models (e.g., Boyd 1998; Trillmich
Houston et al.
•
Modeling capital and income breeding
and Weissing 2006) have tended to focus on body mass as
the major determinant of the adaptive value of capital and
income strategies. By contrast, our model enables us to pose
a number of general hypotheses regarding other differences
that may have led to divergent strategies among related species. These include that capital breeding would be favored
among species: 1) with relatively low neonate mass or relatively long fixed terms of gestation; 2) with offspring that
have relatively high metabolic rates; 3) where female foraging
is relatively constrained while caring for offspring; and 4) where
relatively low costs are incurred by carrying stored capital. The
model also indicates ways in which efficiency of energy transfer from mother to offspring and environmental richness
(or food availability) might be expected to influence maternal
strategies. These impacts are less clear cut, however, and will
depend on specific circumstances, including the values of
other model parameters. Finally, our model also identifies
the complex relationships between rates of energy gain for
females with and without offspring and the specific form of
mass-dependent cost functions. Clearly, the costs of carrying
high-energy reserves (which might be additive or multiplicative and can be affected by a range of physiological and
locomotory factors, as well as absolute body size) must be
understood if predictions about maternal strategies are to
be made regarding the consequences of variation in food
supply.
Our findings can be interpreted in light of other general
assessments of the adaptive value of maternal provisioning
strategies. For example, it is argued that large-bodied pinnipeds are more likely to be capital breeders because they seldom have access to rich local food supplies at the breeding
site, whereas small bodied pinnipeds tend to be income
breeders because they cannot support large reserves (Boyd
1998). Within our framework, these findings are equivalent
to relatively high costs of capital storage among smaller pinnipeds and, potentially, a higher difference between cF and cL in
larger species (perhaps because of the constraints imposed by
the need for central place foraging). By identifying the importance of the costs of accumulated reserves, we have to some
extent formalized arguments regarding which taxa might be
predisposed toward capital breeding due to lower costs of
carrying capital (Bonnet et al. 1998).
Our model goes beyond previous efforts in 3 ways. First, it is
more general and therefore applicable to a greater range of
species than previous models that have been constructed and
parameterized specifically for pinnipeds (Boyd 1998; Trillmich
and Weissing 2006). Indeed, in contrast to these previous models, our framework has no requirement for central place foraging, showing that capital breeding can still arise under a variety
of conditions. Second, our model is novel in that it allows for
a mixture of strategies, focusing attention on the degree of
capital or income breeding, rather than viewing these as dichotomous strategies. Smaller phocids are known to employ mixed
strategies (e.g., Bowen et al. 1999) and our model might suggest
explanations for these intermediate situations. Third, rather
than focusing on body mass as the sole potential driver of different strategies, our framework allows an assessment of the
effects of a variety of other parameters. Although there is a general tendency for capital breeding phocids to be larger than
income breeding otariids (Costa 1991), exceptions do exist
(e.g., the Steller sea lion, Eumetopias jubatus) (Pitcher and
Calkins 1981), confounding attempts to explain maternal strategies solely on the basis of maternal body size (Trillmich and
Weissing 2006). Although many of the parameters in our model
are likely to show some relationship with lean body mass, some
parameters may clearly be affected by other broad differences
in life history. In particular, differing modes of insulation, foraging, and locomotion among phocids and otariids might be
247
expected to lead to different mass-dependent costs in the 2
families, whereas physiological constraints (such as mammary
gland function, e.g., Lang et al. 2005) might also affect the
choice of strategy.
We have deliberately maintained a general model structure,
and so it is unlikely that our model will apply precisely to any
particular system. Nevertheless, it is worth considering the
realism of the major underlying assumptions. Perhaps most
critical to the model is the assumption of a negative relationship between the quantity of stored capital and the rate of
capital accumulation (i.e., the assumption of an asymptotic
maximum level of stores). Although the relative costs of reserve accumulation among small and large-bodied animals
has been implicit in previous models of capital and income
breeding, our model draws explicit attention to the critical
importance of these costs in dictating maternal breeding strategies. Loss of mobility (Kullberg et al. 2005) and resultant
vulnerability to predation are widely cited as costs of stored
reserves in birds (e.g., Hedenström 1992; Witter and Cuthill
1993; Gosler et al. 1995), but these and other costs and benefits in other taxa are less well researched. Among air-breathing
divers (many of which are constrained while tending by the
need to raise offspring on land but forage at sea), emerging
evidence suggests that efficiency of locomotion (Hind and
Gurney 1997; Fish 2000), diving behavior (Lovvorn and Jones
1991; Beck et al. 2000; Sato et al. 2003; Miller et al. 2004), and
thermoregulation (Kvadsheim et al. 1997; Costa and Gales
2003; Harding et al. 2005) are all affected by fat reserves.
However, these relationships are rarely demonstrated in terms
of their effects on optimal fat loading or rates of energetic
gain. Similarly, the longer term health costs of increased work
rates owing to increased body mass are, as yet, poorly understood (e.g., Nilsson 2002). Overall, it seems reasonable to
assume that all motile organisms will have a maximum energy
storage capacity and that most will show some decline in the
rate of energy accumulation before they reach that maximum.
Even among sedentary organisms, energy stores will require
supporting tissues, the manufacture and maintenance of
which will be associated with costs. Large size (resulting from
large stores of capital) is also likely to be associated with
higher threats from predation and parasitism, both of which
may incur costly avoidance strategies. Consequently, our assumption that net energy gain will be a declining function
of stored capital is likely to be valid, at least at high levels of
stored energy.
One way in which many species might be expected to differ
from the types of energy accumulation functions that we
present is where low reserves also carry a cost. For example,
if limited energy stores are associated with poor insulation,
rapid energy loss and, thus, high metabolism, then a sigmoid
gain function might be more reasonable than the monotonic
functions that we have discussed. These might be expected to
favor income breeding if they lead to a long delay in accumulating the capital required for birth. However, the outcome of
such situations is not intuitively obvious as low initial rates of
capital accumulation also imply low rates of surplus energy
gain for income breeding females. Furthermore, given the
dangers (in a stochastic environment) of maintaining a lean
body when capital can only be accumulated slowly, we might
expect individuals in such a scenario to maintain an intermediate level of energy stores as their baseline (thus, omitting
the lower part of the sigmoid curve and conforming to the
monotonic functions that we have discussed).
A second assumption of our model is that of the instantaneous transfer of capital from mother to offspring after birth.
Although some capital breeders achieve very high-speed transfers of energy to their young after birth (e.g., the hooded seal,
Cystophora cristata) (Bowen et al. 1985), instantaneous transfers
248
are not an accurate reflection of the average capital
breeding system. However, the purpose of this model is merely
to explore the circumstances that favor capital breeding or
income breeding. Lower metabolic overheads during the
provisioning phase are often cited as an advantage of capital
breeding (e.g., Schulz and Bowen 2004, 2005), a factor exaggerated by our assumption of an instantaneous capital phase.
Were this assumption relaxed to allow for longer capital
transfer phases, we would still expect the same qualitative conclusions. Similar arguments may be advanced for 2 other assumptions, including female maintenance of reserves during
the income breeding phase and female ability to manipulate
timing of birth. Although neither of these are likely to be exactly accurate (e.g., see Boyd et al. 1997 for an example of where
the former assumption is violated), both are plausible and sufficiently accurate for the purposes of a heuristic model.
Finally, our model also assumes that whether they breed
annually or within shorter cycles, females will seek to maximize the speed at which they transfer energy to their offspring. This is not necessarily the case. Decisions regarding
offspring condition and the speed at which offspring reserves
should be increased can only properly be viewed in light of
optimization over a full annual routine (e.g., Houston and
McNamara 1999). Trillmich and Weissing (2006) are concerned that annual routine models which consider longer
term aspects of fitness would require such species-specific
parameterization that they would be vulnerable to accusations
of adaptive storytelling (cf. Gould and Lewontin 1979). We disagree that this is necessarily the case. For example, models of
generalized pinniped life histories can examine the influence
of one or a few parameters on reproductive strategies, without
the requirement to tailor these to specific species. Such approaches would also permit an examination of other aspects
of food supply, such as seasonality and predictability (to which
our analytical model is not well suited).
There is some debate about how the costs of reproduction
should be measured in the context of capital breeding (Bonnet
et al. 1999; Lewis and Kappeler 2005), and it is often far
from clear which components of these costs are financed
from stored reserves and which are financed from concurrent intake (Casas et al. 2005). Consequently, the best metric to indicate the degree of capital or income breeding is
also unclear. We have suggested one metric here (total energy from capital as a proportion of total offspring reserves
at independence) because it is easily envisaged and applied
within the framework of our model. However, we do not
suggest that this is the only metric or, indeed, the best.
Other measures may be more easily quantified in empirical
studies or more appropriate to other taxa. For situations
where our model predicts mixed strategies, the influence
of parameters on the degree of capital breeding is likely
to depend on the metric used.
Our model has been formulated and presented in terms
most relevant to mammalian life history. However, there is
no reason why it should not be adaptable to other taxa. Although it is framed in terms of a female producing a single
offspring, it applies equally well to multiple offspring. Similarly, we describe a female that expends substantial reserves at
the time of birth and, thereafter, transfers energy to her offspring, for example, by lactation. Again, however, there is no
reason why the initial expense should not apply to the production of eggs, while subsequent transfers of resources may
represent a combination of heat transfers (during incubation)
and direct provisioning thereafter. Such adaptations would
require careful thought regarding the meaning of capital
and income breeding. For example, some authors have
suggested that for birds, the source of energy used for egg
production (rather than energy used for subsequent pro-
Behavioral Ecology
visioning) defines the capital-income split (Klaassen et al.
2001). Such a view would be hard to reconcile with our framework. Even in cases where the provisioning of offspring is
clearly the focus of strategic differences, the model that we
present would require substantial modifications before it
could be applied to species in which males play an active role
in offspring provisioning.
There are clear parallels between the offspring provisioning
strategies that we consider and maternal reproductive strategies that have been the focus of recent interest in several other
taxa. For example, Trexler and DeAngelis (2003) used modeling approaches to consider how matrotrophy, the nourishment of developing embryos by a source other than yolk, has
evolved from lecithotrophy (in which developing embryos are
provisioned only from resources stored in the egg prior to
fertilization) in fish. In lecithotrophic systems, all the resources required for development to independence must be supplied at the outset (analogous to capital breeding), whereas
matrotrophic systems allow for continued maternal inputs as
the embryo develops (akin to income breeding). Johnson
(2006) used an experimental approach to assess the factors
underlying foraging by ant queens between founding a colony
and producing their first brood. Claustral queens store reproductive energy prior to mating flights and rear their first
broods from stored reserves (i.e., they are capital breeders),
whereas semi-claustral queens must forage prior to producing
their first brood (and so are viewed as income breeders)
( Johnson 2006). In both of these studies, the authors found
compelling evidence to suggest that environmental variability
and predictability were important factors that would be difficult to incorporate into our structure (requiring, instead, a dynamic programming or simulation approach). Nevertheless,
our simple model might have implications for other factors
that could influence the transition between lecithotrophic
and matrotrophic fish or claustral and semi-claustral ants. In
particular, neither Trexler and DeAngelis (2003) nor Johnson
(2006) considered explicitly the initial costs of raising capital
that must be invested at the outset, which our model suggests
could have important implications for strategic optima.
A final limitation of our model is that it focuses on energy,
making no allowance for other types of stores accumulated in
advance of reproduction. For birds, there is evidence that, for
egg development, protein may be a more important factor
than energy (Meijer and Drent 1999). Similarly, in gray seals
(Halichoerus grypus), maternal protein stores after parturition
may be more important than fat stores in determining pup
weaning weight (Mellish et al. 1999). Storage and catabolism
of protein and fat may be associated with very different costs
and benefits, but a model considering different types of stores
would, necessarily, be considerably more complex than the
one that we have presented (e.g., Noren and Mangel 2004).
Indeed, the costs of mobilizing long-term and short-term stores
of different types clearly vary greatly among taxa (Jönsson 1997;
Bonnet et al. 1998), and our single parameter for controlling
transfer of efficiency is very crude in this regard.
To keep our model simple and thereby maximize its generality and heuristic value, we have deliberately ignored many
behaviors that might reasonably be incorporated. The model
could be extended to include more explicit costs of travel, in
utero investment and the costs of pregnancy, spatial variation
in food availability, the demands of mate encounters, and risks
of predation. In spite of these and other limitations discussed
above, we believe that our model should stimulate further
consideration of the factors underlying maternal strategies
for offspring provisioning. More detailed annual routine models of specific systems and, in particular, more attention to the
form of mass-dependent cost functions could reveal much
about the adaptive significance of income and capital
Houston et al.
•
Modeling capital and income breeding
breeding and the relative frequency of these and intermediate
strategies in different taxa.
APPENDIX
For both optimization criteria, we consider 3 regions of parameter space. Here we clarify the optimal decisions in each
case, showing how the degree of capital breeding, K (as given
in Table 2), is derived in each case. Our reasoning rests on the
fact that, without additional constraints, the female will favor
giving birth (and switching to income provisioning) at t*. This
is the point at which income provisioning becomes more efficient than further accumulation of capital. If t* is longer
than t0 and T, then the female will usually be free to follow
this strategy. If either t0 or T is the longer period, however,
the female will not be free to give birth at t*, and other strategies will be favored.
249
capital after birth) will be a(xT x0). This presents 2 possibilities as follows:
1. If a(xT x0) , E, then the female cannot raise sufficient capital in T for her offspring to reach independence through the transfer of capital alone.
Remaining reserves required must come from income
provisioning, so the proportion of offspring reserves
at independence that come from maternal capital will
be K ¼ a(xT x0)/E.
2. If a(xT x0) E, then capital to supply all the energy
that the offspring requires can be raised in t T. Either
the female will limit her accumulation of capital or she
will only transfer a proportion of her reserves as capital
following birth. In either case, she will transfer E/a reserves as capital, so that all offspring reserves prior to
independence come from capital, that is, K ¼ 1.
The offspring energy-maximization model
The rate-maximization model
t* . T, t0
The female is free to delay giving birth until t*. Female capital
reserves at t* are, by definition, x*. Thus, energy available to
her offspring (transferred as capital after birth at t*) would be
a(x* x0). This presents 2 possibilities as follows.
1. If a(x* x0) , E, then the female cannot raise sufficient
capital in t* for her offspring to reach independence
through the transfer of capital alone. She will delay giving birth until t ¼ t* and will delay implantation until ti ¼
t* T. Offspring energy reserves will increase (following
birth) by a(x* x0) as a result of transfer from capital.
Remaining reserves required must come from income
breeding, so the proportion of offspring reserves at independence that come from maternal capital will be K ¼
a(x* x0)/E.
2. If a(x* x0) E, then capital to supply all the energy
that the offspring requires can be raised in t t*. Specifically, the offspring will require E reserves, so the female must raise x0 1 E/a reserves before giving birth.
The time taken to raise this quantity of reserves is
t̂ðx0 1E=aÞ: In order to give birth at this time, the female
must delay implantation until ti ¼ t̂ðx0 1E=aÞ T : Birth
will occur at ti ¼ t̂ðx0 1E=aÞ: In birth, the female will lose
x0 reserves and so will transfer E/a reserves to her offspring with efficiency a. The offspring requires no additional reserves for independence, and so all offspring
reserves prior to independence come from capital, that
is, K ¼ 1.
t0 . T, t*
Here, the time taken to raise the minimum reserves needed
for birth is the largest of the 3 time periods. Consequently, the
female will give birth at t0 (implanting at ti ¼ t0 T). Notice
that this is after the point at which income provisioning becomes more efficient than further accumulation of capital. By
the time of birth, therefore, the female will only have accumulated x0 reserves, all of which will be lost in birth. As a
result, she will have no additional reserves to pass to her
offspring. All offspring reserves must therefore come from
income provisioning, so the degree of capital breeding is
K ¼ 0.
T . t *, t 0
In this case, the gestation period dictates the time until birth,
so there will be immediate implantation. By definition, the
total reserves accumulated by the female by T will be xT and
the amount of energy available to her offspring (transferred as
t* . T, t0
As in the rate-maximization model, if the female delays giving
birth until t*, the amount of energy available to her offspring
(transferred as capital after birth) will be a(x* x0). Again,
this presents 2 possibilities as follows:
1. If t* , 1, then the female will have time remaining after
giving birth (at t*) in which to provision from income.
The amount of time remaining will be 1 t*, during
which time she will be able to supply her offspring with
(1 t*)(acL c) from income provisioning. She will
therefore follow a mixed strategy with K ¼ a(x* x0)/
[ a(x* x0) 1 (1 t*)(acL c)].
2. If t* 1, then offspring energy reserves will be maximized when the female spends the whole year accumulating capital, giving birth (and transferring all available
capital to her offspring) at t ¼ 1. She will not benefit
from giving birth before this, as (by definition) the point
at which income provisioning becomes the more efficient mode of maximizing offspring energy reserves
(t*) will not be reached within the annual cycle. Consequently, pure capital breeding results (K ¼ 1) and implantation occurs at ti ¼ 1 T.
t0 . T, t*
Here, the time taken to raise the minimum reserves needed
for birth dictates the timing of birth. As in the rate-maximization model, the female will give birth at t0 (implanting at ti ¼
t0 T). By the time of birth, the female will only have accumulated x0 reserves, all of which will be lost in birth. As a result, she will have no additional reserves to pass to her
offspring. All offspring reserves must come from income provisioning, so the degree of capital breeding, K ¼ 0.
T . t*, t0
In this case, the gestation period dictates the time until birth,
so there will be immediate implantation. By definition, the
total reserves accumulated by the female by T will be xT and
the amount of energy available to her offspring (transferred as
capital after birth) will be a(xT x0). By definition, T , 1 and,
consequently, some time will remain for income provisioning.
The female will follow a mixed strategy, where K ¼ a(xT x0)/
[a(xT x0) 1 (1 T)(acL c)].
This work was funded by the Natural Environment Research Council
(grant number 2003/00616). A.I.H. and J.M.M. were supported by
Leverhulme Trust fellowships. We thank the editorial team and 2
anonymous referees for helpful comments on an earlier draft.
250
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