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vol. 159, no. 6 the american naturalist june 2002

Sexual Conflict about Parental Care: The Role of Reserves

Zoltán Barta,1,* Alasdair I. Houston,2,† John M. McNamara,1,‡ and Tamás Székely 3,§

1. Centre for Behavioural Biology, School of Mathematics, and thus, parents should have sufficient reserves to raise
University of Bristol, University Walk, Bristol BS8 1TW, United their young to independence. Parents in poor condition
Kingdom;
may terminate care and abandon their nest or young (Sny-
2. Centre for Behavioural Biology, School of Biological Sciences,
University of Bristol, Woodland Road, Bristol BS8 1UG, United der et al. 1989; Olsson 1997; reviewed by Clutton-Brock
Kingdom; 1991; Székely et al. 1996). Given that energy reserves play
3. Department of Biology and Biochemistry, University of Bath, an important role in determining patterns of care, an evo-
Bath BA2 7AY, United Kingdom lutionary account of care should adopt a state-dependent
approach (Houston et al. 1988; Houston and McNamara
Submitted February 5, 2001; Accepted November 30, 2001
1999; Clark and Mangel 2000). State-dependent models
have been developed to investigate the decision of a single
sex by Kelly and Kennedy (1993) and Webb et al. (in press).
For instance, Kelly and Kennedy (1993) showed that fe-
abstract: Parental care often increases the survival of offspring, male Cooper’s hawks Accipiter cooperi should desert their
but it is costly to parents. Because of this trade-off, a sexual conflict young when their reserves fall below a critical level. In
over care arises. The solution to this conflict depends on the inter- their model, however, the behavior of deserting females
actions between the male and female parents, the behavior of other
was constrained by their not allowing the females to re-
animals in the population, and the individual differences within a
sex. We take an integrated approach and develop a state-dependent mate. In a more general model by Webb et al. (in press),
dynamic game model of parental care. The model investigates a single a deserting female can remate and renest within the same
breeding season in which the animals can breed several times. Each breeding season. Interestingly, their model found that a
parent’s decision about whether to care for the brood or desert female deserts not only when her reserves are low (and
depends on its own energy reserves, its mate’s reserves, and the time thus she is threatened by starvation) but also when her
in the season. We develop a fully consistent solution in which the
reserves are high. In the latter case, the deserting female
behavior of an animal is the best given the behavior of its mate and
of all other animals in the population. The model predicts that fe-
immediately starts to search for a new mate and thus in-
males may strategically reduce their own reserves so as to “force” creases her reproductive success by remating.
their mate to provide care. We investigate how the energy costs of In many birds, fish, and mammals, however, the payoffs
caring and searching for a mate, values of care (how the probability from caring and deserting not only depend on the envi-
of offspring survival depends on the pattern of care), and population ronment and the energy reserves of the parents but also
sex ratio influence the pattern of care over the breeding season. on the behavior of other animals in the population. First,
the success of the current breeding attempt depends on
Keywords: sexual conflict, parental care, offspring desertion, dynamic whether the focal animal’s mate cares or deserts. Second,
game, reserves, body mass regulation.
the operational sex ratio (Emlen and Oring 1977) can
determine the time to remate and hence influence the
payoff from desertion. There is evidence that animals re-
Parental care is an energetically demanding behavior
spond to the operational sex ratio. For example, male fish
(Ricklefs 1974; Golet and Irons 1999; Hõrak et al. 1999),
more often terminate care if the sex ratio is biased toward
* Corresponding author. Present address: Behavioural Ecology Research females (Keenleyside 1983; Balshine-Earn and Earn 1998),
Group, Department of Evolutionary Zoology, University of Debrecen, Deb- and female rock sparrows Petronia petronia desert their
recen H-4010, Hungary; e-mail: zbarta@delfin.klte.hu. brood in years when many unmated males are in the pop-
†
E-mail: a.i.houston@bristol.ac.uk. ulation (Pilastro et al. 2001). These aspects of parental care
‡
E-mail: john.mcnamara@bristol.ac.uk. require a game-theoretic analysis (Maynard-Smith 1982;
§
E-mail: t.szekely@bath.ac.uk. Hammerstein and Parker 1987) in which the behavior of
Am. Nat. 2002. Vol. 159, pp. 687–705. 䉷 2002 by The University of Chicago. the focal pair as well as other members of the population
0003-0147/2002/15906/0008$15.00. All rights reserved. is considered. The importance of including the behavior
688 The American Naturalist

of other members of the population is illustrated by Webb (i.e., the net energy gained in a day). Searching for a mate
et al. (1999). They show that modeling the behavior of a decreases the reserves of the male, but he may find a mate.
single pair in isolation from the rest of the population The mate-finding probability of a male depends on the
results in different evolutionarily stable strategies (ESSs) number of mate-searching individuals of both sexes (Webb
from ones in which future expectations, including remat- et al. 1999). To illustrate this, let us assume that only a
ing probabilities, depend on the behavior of all members few females are searching for a mate. If only the focal male
of the population. Taken together, a full understanding of searches for females, he will find one with high probability
parental care requires an analysis that is both state de- despite their low numbers. However, if many males search,
pendent and game theoretic. For a related discussion in the focal male’s probability of finding a mate will be low.
the context of alternative reproductive behaviors, see The probability of finding a mate also depends on the
Alonzo and Warner (2000). mate-search efficiency k, which may reflect population
Here, we develop a state-dependent game model in density (McNamara et al. 2000). If the focal male finds a
which the parents’ decisions depend on their own energy mate, he becomes mated. We assume that mate-finding is
reserves, their mate’s reserves, and time in the season. The random (i.e., there is no mate choice). As a consequence,
payoffs for performing each action are not specified in the level of energetic reserves of the mate is drawn from
advance, since they depend on future expectations, which, the distribution of the reserves of the unmated mate-
in turn, depend on the future behavior of the focal animal searching females (i.e., at the time of pair formation, one
and the behavior (both past and future) of other popu- member of the pair has no information about the energy
lation members. We use the model to show that state- reserves of its partner).
dependent models can produce very different predictions Once a pair has formed, they then produce some off-
from state-independent ones because they allow the ani- spring after a fixed period. For example, in birds, this
mals to optimize simultaneously their behavior and re- period may represent the time required to prepare a nest
serves. We then investigate how the energetic costs of var- or to produce eggs. Producing offspring reduces the re-
ious activities (e.g., parental care, reproduction, mate serves. This could represent the energetic cost of nest
search), care parameters, and the population sex ratio in- building, territory defense, or egg laying. We assume that
fluence parental decisions. if the reserves of the male’s mate are not high enough to
cover this cost, then she dies and no offspring are pro-
duced. The male then becomes unmated after the time
The Model
period needed for offspring production.
We consider the behavior of a focal male and a focal female Having produced the offspring, both members of the
in a large population of NM males and NF females (N p pair decide whether to care for the offspring until they
NM ⫹ NF) during a breeding season of T d. For simplicity, become independent or to desert (i.e., we assume that the
we only describe the behavior of a focal male. If we do animals decide once for each breeding attempt). We as-
not explicitly specify the female’s behavior, then similar sume that by this time, each parent has been able to ob-
reasoning applies to her too. For a technical account of serve the energy reserves of its partner. Thus, when making
the model, see appendices A and B. decisions, both parents know each others’ reserves exactly.
The state of a focal male is represented by his marital The male decides first, and the female then decides on the
status (unmated or mated), his energy reserves, and, if he basis of her partner’s decision. This is a reasonable as-
is mated, the energy reserves of his mate. The male dies sumption in internally fertilizing animals such as birds,
of starvation if his reserves fall to zero. However, his re- mammals, and some fish because the male can leave the
serves cannot increase above some maximal level. All family immediately after fertilization, whereas the female
changes in reserves are stochastic. has to wait to finish offspring production (e.g., laying the
All animals are unmated when they arrive at the breed- eggs or giving birth). If the male deserts, his reserves re-
ing ground. Members of the population arrive gradually main unchanged and he will be unmated at the beginning
(i.e., on a given day, only a proportion of the whole pop- of the next day (i.e., he has again the choice of resting,
ulation arrives). The expected reserves of an arriving in- foraging, or searching for a mate). If the male cares, he
dividual may depend on its date of arrival. remains with the offspring until they become independent,
The available behavioral actions of a male depend on or he dies of starvation. Care is costly in terms of both
his marital status. If a male is unmated at the beginning time (since the caring parent is not available for remating)
of a given day, then he can rest, forage, or search for a and energy. The energetic cost to a parent is, however,
mate during that day. Resting does not change the reserves typically less if both parents care for the offspring (cost
of the male, whereas foraging increases his reserves. The of biparental care) than if the parent cares alone (cost of
extent of this increase depends on the efficiency of foraging uniparental care). If a caring parent dies of starvation, we
 Sexual Conflicts and Reserves 689

consider the consequences for the brood to be as if that Table 1: Parameters of the model and their baseline values
parent had not cared at all. For instance, if both parents Parameter Symbol Value
decide to care but the female dies during care, then the
Length of the breeding season (d) T 100
brood receives male-only care. In this case, the male’s
Duration of offspring production (d) tl 4
reserves decrease by the cost of uniparental care instead Age of offspring at independence (d) tc 10
of by the cost of biparental care. Although the conse- Maximum value of reserves L 25
quences of starvation have to be taken into account, we Efficiency of mate search k .3
emphasize that for the parameters used, starvation almost Value of no care VDD 0
never occurs. Value of male-only care VCD .7
When the offspring become independent, their contri- Value of female-only care VDC .7
bution to the male’s reproductive value is RAB(x, y, t). We Value of biparental care VCC 1
assume that this reproductive value of offspring depends Net energy gain while foraginga xf, M 2
on the pattern of care they receive AB, the time in the Energetic cost of restinga xr, M 0
breeding season t, and in some cases, the parents’ reserves Energetic cost of mate searchinga xs, M 1
Energetic cost of offspring productiona x l, M 3
x and y. There are four possible patterns of care: no care
Energetic cost of desertiona xd 0
(i.e., both parents desert, DD); male-only care (i.e., the Energetic cost of biparental carea xc, M(C) 10
male cares and the female deserts, CD); female-only care Energetic cost of uniparental carea xc, M(D) 12
(i.e., the male deserts and the female cares, DC); and bi- Dependency of offspring value on reserve h 0
parental care (i.e., both parents care, CC). We are inter- Number of males NM 5,000
ested in species in which at least some care is essential for Number of females NF 5,000
the offspring to reach independence; therefore, we assume Median arrival time 6
that the offspring die if both parents desert (i.e., Note: The sexes are not different under the baseline case, so we give only
R DD[x, y, t] p 0). Furthermore, the young typically have the values for the male.
their highest reproductive value when they are cared for a
Variable is set to zero in the reserve-independent scenario.
by both parents so that R CC ≥ R CD, R DC ≥ R DD. The effect
of uniparental care on the value of offspring is character- breeding season. We call this optimal policy the “best re-
ized by the care parameters (baseline values given in table sponse” B(p) to the resident population strategy p (Mc-
1). The offspring’s reproductive value is assumed to de- Namara et al. 1997, 2000). A necessary condition for a
crease with time in the season to account for the general strategy p ∗ to be an evolutionarily stable strategy (ESS) is
trend in birds that offspring produced late in the season that it is a best response to itself (i.e., B[p ∗] p p ∗).
have lower reproductive success than offspring produced To calculate the ESS, we use a technique commonly
early (Daan et al. 1989; Székely and Cuthill 1999). In the called the “iteration of the best response map.” This starts
baseline case, the reserves of the parents have no effect on with an arbitrary strategy p0 and finds the sequence (p0,
the reproductive value of the young. We also investigate p1, p2, …) of strategies where each strategy is the best
cases in which the reproductive value of the offspring in- response to the previous strategy in the sequence (i.e.,
creases with the reserves of the caring parent(s). B[pi] p pi⫹1). The sequence converges to a limiting strat-
If the male survives until the end of the breeding season, egy p ∗ in some cases, but it fails to converge in others.
he receives an additional reward that represents his re- The problem arises because the best response B(p) does
productive success in future breeding seasons. This ter- not change continuously with p. To overcome this diffi-
minal reward is taken to be independent of his reserves. culty, we use damping and errors in decision making (for
further details, see McNamara et al. 1997). The error is
controlled by the parameter d. When d p 0, the animals
The Desertion Game
do not make any errors, whereas with increasing d, the
We consider a population whose members follow a policy animals are increasingly unable to differentiate between
p. We refer to this policy as the “resident policy” (Mc- actions with similar consequences. We use a small error
Namara et al. 1997, 2000). Given this policy and some (d p 0.015) that produced the evolutionarily stable so-
initial distribution of reserves, we calculate the number of lutions in all cases. These stable solutions give the optimal
mate-searching males and females, the distribution of their policies of males and females over the breeding season.
reserves, and, consequently, the probability of finding a We consider a population in which these policies are used.
mate with given reserves for all days in the breeding season. Following the population forward in time gives the dis-
On the basis of these distributions, we can determine the tribution of reserves of population members and the pro-
optimal policy for a “mutant” by solving the dynamic portion of population members taking each action at each
programming equations backward from the end of the time in the breeding season. (Note this is a forward cal-
690 The American Naturalist

Figure 1: The pattern of parental care over the season. A, Reserve-independent scenario. B, Reserve-dependent scenario: the costs of uni- and
biparental care are identical (both p 12 ). C, Reserve-dependent scenario: the costs of care are different (uniparental p 12 ; biparental p 5). Solid
lines p biparental care; dashed lines p female-only care; dotted lines p male-only care. Other parameters are as in table 1.

culation rather than simulation; see section 3.3 in Houston because it is more beneficial for an animal to desert than
and McNamara 1999.) The results presented in the figures to care, given that its mate will care.
are based on this computational procedure. For a detailed Then we move on to investigate a reserve-dependent
technical account, see appendix B. scenario by setting the reserve parameters to the values
given by table 1, except that the costs of both uniparental
and biparental care are 12 units each. Comparison of this
Results case with the reserve-independent case establishes which
First, we investigate a reserve-independent scenario. In this effects in the reserve-dependent scenario are specifically
scenario, all parameters influencing the animals’ reserves due to the inclusion of reserves. Finally, we investigate
(e.g., net energy gain or cost of biparental care; see table biologically important questions that have been discussed
1) are set to zero, and other parameters are kept at their in studies of parental care (Clutton-Brock 1991; Székely
baseline values (table 1). We choose the baseline values et al. 1996) by exploring the effects of changing model
such that they allow us to concentrate on the details of parameters.
the game between the parents. For instance, the sexes are
not different from each other, and one parent on its own Reserve-Independent Scenario
is reasonably successful at rearing the offspring. Conse- In the reserve-independent case, an oscillation occurs be-
quently, a conflict of interest arises between the parents tween female-only and biparental care (fig. 1A). This pat-
 Sexual Conflicts and Reserves 691

tern of care is similar to, and arises for the same reason
as, the one predicted by the non-state-dependent dynamic
game model of McNamara et al. (2000). We only note
here that when female-only care occurs, it is because the
male, who decides first, has “forced” the female to care
by deserting her. For a full analysis, see McNamara et al.
(2000).

Reserve-Dependent Scenario
Restoring the effects of reserves has a major effect on the
pattern of care (fig. 1B). Given that the costs of uni- and
biparental care are the same, male-only care is the most
common form of care even though the male decides first.
Biparental care is frequent only near the end of the breed-
ing season (fig. 1B).
The reason for the difference between the reserve-
independent and the reserve-dependent scenarios is as fol-
lows. In the reserve-independent scenario, the male can
force the female to care by deserting her because he decides
first (McNamara et al. 2000). In the reserve-dependent
scenario, an animal can only raise the young and survive
the period of care if it starts to care with reserves higher
than a critical level (the sum of the cost of parental care
and a safety margin serving as an insurance against the
stochasticity in reserve dynamics). Consequently, in this
case, the male can only force his mate to care by deserting
if the female has reserves higher than her critical level;
otherwise, she also deserts in order to avoid starvation.
Therefore, the inclusion of reserves creates an opportunity
for the female to circumvent the male’s intended behavior
by keeping her reserves below the critical level. If the fe-
Figure 2: The mean reserves of mated males (solid lines) and females
male does this, she will have to desert whatever the male (dashed lines) over the breeding season. The costs of uni- and biparental
does. Thus, if the young are not to die, the male must care are (A) identical (both p 12) and (B) different (uniparental p 12;
ensure that he has sufficiently high reserves to care. Con- biparental p 5). The dotted lines mark the critical levels of reserves.
versely, assume that the male has high reserves. Then, if
the female were to have high reserves, the male would have already seen, however, that if males have high re-
desert, thus forcing the female to care. It follows that the serves, then it is best for females to have low reserves. It
female’s best strategy is to have low reserves, thus forcing follows that low reserves for females and high reserves for
the male to care. From this argument, it can be seen that males is the unique stable pattern of reserves. The outcome
the pattern of high reserves for the male and low reserves of this pattern is uniparental care by the male.
for the female (see fig. 2A) is evolutionarily stable.
Is it possible that there are other stable patterns; in
Difference between Energetic Costs of
particular, could it be stable for a male to have low reserves
Uni- and Biparental Care
and hence force the female to care? If females have high
reserves, then it is optimal for males to exploit this fact Until now, we have investigated the case in which the
by deserting, so forcing the female to care. (Note the asym- energetic costs of uni- and biparental care are the same.
metry here; the female could not force a male with high This may be a reasonable assumption for species with less
reserves to care because such a male would have deserted demanding young (e.g., birds with precocial chicks). In
by the time she made her care decision.) But if a male is contrast, the costs of uni- and biparental care may sig-
going to desert, it is better if he has high reserves. This is nificantly differ in species that have more demanding
primarily because high reserves mean that he can devote young (e.g., birds with altricial chicks). We investigate the
his time to searching for a mate rather than foraging. We effects of the difference between these costs by keeping the
692 The American Naturalist

cost of uniparental care at its baseline value and decreasing termediate levels of reserves. Even if the female attempts
the cost of biparental care. to keep her reserves in this region, she cannot reliably do
When the difference between these costs is small (!5), so when the critical levels of uniparental and biparental
then the female can use her strategy of reserve regulation care are close because of stochasticity in the dynamics of
to force her mate to care; thus, the result is male-only care reserves. This explains why male-only care occurs at small
during most of the season (results are similar to those differences between the costs of uni- and biparental care.
shown in fig. 1B). In contrast, when the difference between In particular, for the parameter values given in table 1,
these costs is large, the male can circumvent the female the difference in costs is small, and there is male-only care
strategy by keeping his reserves at an intermediate level, for much of the season. Since results for these parameters
and this results in biparental care (figs. 1C, 2B). are very similar to those given in figures 1B and 2A, they
The occurrence of biparental care when the difference are not shown. When the difference is large, however, the
between costs is large can be understood as follows. We male can safely reduce his reserves and biparental care is
first consider the evolutionarily stable level of reserves for the result.
the female. There are three possibilities. She can have high
reserves (above the level required for uniparental care),
intermediate reserves (below the level for uniparental care Energetic Cost of Reproduction
but above the level for biparental care), or low reserves
We manipulated the energetic cost of reproduction (the
(below the level for biparental care). Note that a female
sum of energetic costs of offspring production and parental
with high reserves can care on her own, with intermediate
care) by varying either the cost of offspring production or
reserves she can only care together with her mate, whereas
the cost of parental care in such a way that the difference
with low reserves she cannot care. Suppose that the female
between the costs of uni- and biparental care remains as
has high reserves. Then, the best action for the male is to
in table 1. If the energetic cost of reproduction is signif-
desert, leaving the female to care. Given that he is going
icantly less than the maximal reserves, the female forces
to desert, it is best for the male to have high reserves. But
the male to care in the same way as seen previously. If the
then the female should reduce her reserves below the level
cost of reproduction for one sex approaches the maximal
for uniparental care, forcing the male to care. Thus, there
level of reserves, then that sex deserts more often than the
can be no ESS in which the female has high reserves. Now,
other sex. If an animal’s cost of reproduction exceeds its
suppose that the female has low reserves. She then always
maximal level of reserves, then it always deserts.
deserts whatever the decision of the male. The male is
forced to care in these circumstances, and given this, it is
best if he has high reserves. But the best action of a male Energetic Cost of Mate Search
with high reserves is to care not only if the female has low
reserves but also if she has intermediate reserves. So given As the energetic cost of mate search increases, a single
this behavior of the male, it is best for the female to have individual must spend a greater proportion of time on
reserves as high as possible while still forcing the male to foraging. As a consequence, the time needed to find a new
care. Thus, she should have intermediate reserves. It fol- mate increases, so the number of broods produced over
lows that there is no ESS in which the female has low the breeding season decreases (fig. 3A).
reserves; if an ESS exists, the female must have interme- We separately investigate symmetric cases (the search
diate reserves. cost is the same for both sexes) and asymmetric cases (the
What, then, is the stable level of reserves for the male? search cost is different for the sexes). In symmetric cases,
In line with the above reasoning, let the female have in- if the search cost is low, the females desert more often
termediate reserves. The male then has to care. Given this, than the males (fig. 3B). This is because the females keep
he can have high reserves, in which case the female will their reserves low (fig. 3C) to force the males to care. If
desert and he will care on his own, or intermediate re- the search cost is high, the sexes do not exploit each other
serves, in which case the female will be forced to care as (fig. 3B). Large search cost means that it is more difficult
well. Since in the baseline case biparental care is signifi- to maintain high reserves. Hence, the males’ reserves de-
cantly better than uniparental care, and he must care what- crease (fig. 3C ), which in turn makes them less vulnerable
ever happens, the male’s best strategy is to have inter- to exploitation by the females. Another consequence of
mediate reserves. The resulting ESS is for both the male the low level of reserves is that the frequency of biparental
and the female to have intermediate reserves and for both desertion increases with search cost. The animals do not
to care (figs. 1C, 2B). know the reserves of their mates in advance, and if the
The male, however, should only reduce his reserves to average level of reserves in the population is low, both
intermediate levels when his mate is expected to have in- members of the pair are more likely to find themselves
 Sexual Conflicts and Reserves 693

Figure 3: The effects of mate-search cost. A, The expected number of broods produced during the breeding season plotted against the mate-search
cost of females at different values of mate-search cost of males (circle p 0 ; triangle p 1 ; plus p 2 ; cross p 3 ; diamond p 4; inverted triangle p 5).
B, The pattern of care plotted against the symmetric (same for both sexes) cost of mate search. Proportion of broods produced over the season
receiving biparental care (circle), female-only care (triangle), and male-only care (plus) is given. C, The mean reserves of mated males (circle) and
females (triangle) plotted against the symmetric mate-search cost. Reserves are averaged over the season. D, The proportion of caring males to all
caring individuals over the season plotted against the difference between the mate-search costs of males and females (average cost of the pair, ca, is
given; circle: ca ≤ 1; triangle: 1 ! ca ≤ 2; plus: 2 ! ca ≤ 3; cross: 3 ! ca ≤ 4; diamond: 4 ! ca).

below the critical levels of parental care, which results in sequently, when an animal does mate, its reserves will be
biparental desertion. low, and it will be unable to care.
In asymmetric cases, the sex with the higher search cost
deserts more often than the sex with lower cost (fig. 3D).
Foraging Efficiency
This result is counterintuitive because high search costs
decrease the proportion of time a single animal searches When the net energy gain while foraging is increased sym-
for a mate and hence increases the time to find a mate. metrically for each sex, the baseline pattern of care does
Thus, high search costs decrease the advantage of deser- not change its qualitative form; male-only care remains
tion. However, in order to be able to care, an animal has the most common type of care, and biparental care occurs
first to raise its reserves above the critical level of parental close to the end of the season. The details of the pattern
care, and a high cost of mate search makes it difficult to of care, however, are changed. Animals start to care earlier
raise reserves and find a new mate at the same time. Con- in the season. The frequency of biparental care at the end
694 The American Naturalist

of the season increases since it is easier to raise reserves in symmetric cases, they found female-only care when each
when there is more food, and, hence, less females are below sex can raise the young effectively on its own.
the critical level of biparental care at the end of the season.
By changing the net energy gain asymmetrically, the sex
that finds more food always cares. This is because the sex
that is less efficient at foraging takes longer to replenish Search Efficiency and Population Sex Ratio
the energy spent during care, and, hence, if it was going
Increasing search efficiency, k, has a major effect on the
to care, it would pay a greater cost in terms of future
pattern of care (fig. 5A). When search efficiency is low
matings.
(i.e., it is very difficult to find a new mate), the most
common form of care is biparental. When the search ef-
Effects of Care Parameters ficiency increases, the pattern of care changes dramatically;
the frequency of deserting females increases, and male-
When the value of uniparental care by both sexes is low only care is observed. The reason for this change is the
(VCD , VDC ≤ 0.5), neither parent can effectively raise the strategic reserve regulation by females; when the payoff
young on its own so both parents care (fig. 4). Increasing from desertion increases, the females reduce their reserves
the value of uniparental care leads to increased temptation and force the males to care (fig. 5B).
for desertion and increased frequency of uniparental care. The effects of population sex ratio NM : NF depend on
When the increase is symmetric, the female can exploit the search efficiency. When the search efficiency is low,
the male by keeping her reserves low, and male-only care the sex ratio has no effect on the pattern of care (fig. 6A)
is the result (fig. 4). When the increase in care parameters since desertion is not beneficial even at very extreme sex
is asymmetric, the sex that raises the young less effectively ratios (NM : NF p 3 : 1 or 1 : 3). When the search efficiency
on its own will desert (fig. 4). These results are in accor- is high, the payoff from desertion increases. The propor-
dance with those of McNamara et al. (2000), except that tion of male-only care increases, whereas the proportion

Figure 4: The pattern of care plotted against the values of uniparental care of each sex. Each pie chart shows the proportion of broods that receive
female-only (white), male-only (black), or biparental (grey) care during the breeding season. The frequency of biparental desertion was negligible
in all cases.
 Sexual Conflicts and Reserves 695

Figure 5: The effects of search efficiency k. A, The proportion of broods receiving biparental (circle), female-only (triangle), and male-only (plus)
care. B, The mean reserves of males (circle) and females (triangle).

of female-only and biparental care decreases as the sex Value of Brood Depends on Parental Reserves
ratio shifts from female biased to male biased (fig. 6B).
This result agrees with previous models (e.g., Maynard So far, we have assumed that the value of the brood is
Smith 1977; McNamara et al. 2000). Unlike these models, independent of parental reserves. One can argue, however,
however, in our model, even in a strongly female-biased that parents with high reserves can provide better care
(NM : NF p 1 : 3) population, male-only care is more com- than parents with low reserves. We investigate this pos-
mon than female-only care. This result arises because the sibility by letting brood reproductive values increase with
strategic reserve regulation by females overcomes the effect parental reserves (see app. B). If the reproductive value of
of sex ratio. the offspring depends strongly on the reserves of the caring
696 The American Naturalist

of the sex deciding first (labeled as males) force the mem-

bers of the other sex (labeled as females) to care by de-
serting. In contrast, when the reserves have effects and the
difference between the costs of uni- and biparental care is
small, the female circumvents the intended behavior of
the male by keeping her energy reserves low. Thus, even
though the male decides first, he cares instead of deserting.
Furthermore, when the difference between the costs is
large, each sex avoids being exploited by keeping its re-
serves between the level required for biparental care and
the level required for uniparental care. This is a novel form
of strategic body mass regulation that arises only in a
game-theoretic context (i.e., animals reduce their reserves
in order to manipulate their mate’s behavior). Previous
studies of strategic body mass regulation focused on single

Figure 6: The effects of population sex ratio NM : NF when the searching

efficiency is (A) low (k p 0.1) and (B) high (k p 0.7). For symbols, see
figure 5A.

parent(s) (h 1 0.2), biparental care is the most common

form of care during the season (fig. 7A). The reason for
this result is that strong reserve dependence leads to the
elimination of strategic reserve regulation, as shown by
the increasing reserves of females with reserve dependence
(fig. 7B), since by strategically reducing the reserves, the
parents pay a cost in terms of reduced brood value. If the
dependence is weak (h ≤ 0.2), females still use regulation
of reserves to force the males to care.

Discussion Figure 7: The effects of the dependency h of offspring value on the

Our state-dependent dynamic game model shows that the reserves of the parent(s). If h p 0, the brood reproductive value does
not depend on the parental reserves. As h increases, the value of having
strategic regulation of body mass can have important ef- high reserves increases. A, The proportion of broods receiving biparental
fects on the solution of sexual conflict over parental care. (circle), female-only (triangle), and male-only (plus) care. B, The mean
When reserves are not included in our model, members reserves of males (circle) and females (triangle).
 Sexual Conflicts and Reserves 697

individuals and investigated the various costs (e.g., in- lish the implications of using mass loss to generate cred-
creased predation hazard) and benefits of carrying fat (re- ible threats in a realistic context.
viewed by Witter and Cuthill 1993; Cuthill and Houston It is widely accepted that the ability of one parent to
1997). raise the young limits the possible patterns of care (Lack
A threat must be credible in order to settle a conflict 1968; Temrin and Tullberg 1995). For instance, if a sin-
between animals. For example, imagine a male and female gle parent is able to provide appropriate care for the
deciding about whether to care or desert as in our model. offspring, then uniparental care is expected. It has been
Suppose that the energetic costs of care can be ignored, found, however, that in many species with biparental
and it is more beneficial for an animal to desert than to care, one parent can raise the young nearly as success-
care, given that its mate will care. Then a pair of mutual fully as both parents (Bart and Tornes 1989; Wolf et al.
best responses, that is, a Nash equilibrium, is for the male 1990; Liker 1995). Interestingly, our model predicts
(who decides first) to care and the female (who decides that, despite the fact that one of the parents can raise
second) to desert whatever the male decides. In this case, the young efficiently, the pattern of care can still be
the female is forcing the male to care by means of the biparental if the difference between the costs of uni-
threat of desertion even if the male deserts. Although these and biparental care is large. Then both sexes can avoid
strategies are a Nash equilibrium, they are not evolution- being deserted by keeping their reserves at an inter-
arily stable (Houston and McNamara 1999). Consider the mediate level.
alternative female strategy: desert if the male cares and Our model reveals that, as a consequence of the ef-
care if the male deserts. Given that the male always cares, fects of reserves, the parameters that influence reserves
this strategy for the female has exactly the same payoff as before mating (e.g., foraging efficiency, cost of mate
always deserting. It follows that this alternative strategy search) can have a large effect on the pattern of parental
can increase by random drift (cf. Selten 1980). Further- care. In general, the sex that can improve its reserves
more, if males make mistakes and desert (or do not care more efficiently or pay less energetic cost will care. This
because, unknown to the female, they have been killed), emphasizes that one cannot investigate the behavior of
then the alternative strategy is strictly better for the female. unmated or mated individuals in isolation from each
For these reasons, the alternative strategy can be expected other (Webb et al., in press).
to increase in the population up to a level at which it It could be argued that three of our assumptions may
becomes better for males to exploit this strategy by de- limit the generality of the results on strategic regulation
serting (Houston and McNamara 1999). Thus, in evolu- of body mass. First, we assume that each animal knows
tionary terms, the threat of desertion by the female even exactly the reserves of its mate. This expectation is prob-
if the male deserts is not credible. Once reserves are in- ably unrealistic, especially in cases when the male and
cluded, the above argument no longer holds. If a female female interact only briefly. Note, however, that the level
keeps her reserves below the level required for uniparental of reserves is set while the individuals are unmated (i.e.,
care, then her best action when deserted is also to desert. before they mate). Therefore, the male’s behavior can
It follows that the threat of desertion by a female with low be an evolutionary response to the female’s expected
reserves is now credible. As a result, there is an evolu- low reserves (and vice versa), in which case it is un-
tionarily stable solution in which the female forces the necessary to assume that the pair members should be
male to care by keeping her reserves low. aware of each other’s reserves. Consequently, the as-
Many birds often lose mass during breeding (re- sumption that mated animals know each other’s re-
viewed by Moreno 1989). This change is traditionally serves exactly in our model may not have a crucial effect
interpreted as either a sign of stress (Ricklefs 1974; Nur on the results. Our second assumption is that an animal
1984) or an adaptation to increase flight efficiency dur- must have reserves higher than a certain limit before
ing a demanding period of brood rearing (Freed 1981; breeding in order to breed successfully. Stearns (1992)
Norberg 1981; Houston 1993; Hillström 1995). Our defines a capital breeder to be an organism that uses
model raises the possibility of a novel explanation for stored energy for reproduction and an income breeder
this mass loss; the parents lose mass to manipulate their to be an organism that uses energy that it obtains during
mate’s behavior. If this explanation is correct, we might the reproductive period rather than stored energy for
expect that a female would increase her level of reserves reproduction. Our model applies to capital breeders. It
when her mate is removed. In our current model, there does not apply to a pure income breeder in which en-
is only a single decision per breeding attempt, so such ergy reserves at the start of reproduction make no con-
an effect cannot be predicted. In reality, birds are likely tribution. This pure form of income breeding may be
to make a sequence of state- and time-dependent de- rare. There is a continuum of breeding patterns from
cisions. Further theoretical work is necessary to estab- pure capital breeding to pure income breeding (Thomas
698 The American Naturalist

1989; Stearns 1992). Our model applies to forms of lactation in mammals). For these asymmetric cases, the
breeding along this continuum provided that for low model predicts female-only care. In addition, our model
reserves, income is not enough to cover costs, so that has no scope for males to monopolize access to groups
individuals whose reserves are sufficiently low at the of females.
beginning of care are likely to starve during care. We Details of the decision process may also influence our
emphasize that the existence of a critical level of reserves results. First, we assume that the male decides first. In
does not mean that offspring desertion due to low re- nature, the opposite can also hold (i.e., the female de-
serves can be observed frequently in nature since ani- cides first). For instance, in externally fertilizing animals
mals follow policies that make such desertion unlikely. such as many fish and amphibians, the female releases
The third assumption is that mate choice is random. It the gametes first (Dawkins and Carlisle 1976). Second,
might be suggested that a female can use strategic body we assume that the pair decides about care only once
mass regulation if the male does not know the reserves during a breeding attempt. In reality, the decision pro-
of his prospective mate at the time of pair formation. cess is probably more complicated; it may involve a
If, however, the male knew the reserves of his future series of interactions between the male and female.
partner, he could potentially avoid mating with a female Therefore, the final outcome (i.e., the observed pattern
with low reserves. Therefore, in a population of choosy of care) can be the result of this prolonged negotiation
males, females should have high reserves. If all females process. For instance, in the Kentish plover Charadrius
have high reserves, however, it is no longer worth alexandrinus, the parents alternate care of the brood
choosing among females especially if choosiness incurs until one parent deserts. These shifts may be part of
a nonzero cost. Consequently, nonchoosy males can the negotiation process in which the parents test their
spread into the population, creating an opportunity for own abilities to attract a new mate and renest as well
the females to use strategic body mass regulation. as testing the ability of their mate to care for the brood
Therefore, the population of choosy males and females unassisted (T. Székely, personal observation). The ef-
with high reserves cannot be stable, so mate choice may fects of these repeated interactions on the pattern of
not prevent the use of strategic body mass regulation. care, however, are not known. It is possible that when
Further theoretical work is needed, however, to explore the caring and deserting decisions are the results of a
the detailed effects of the role of information and the long process (e.g., one that lasts for the whole breeding
inclusion of mate choice in our model. attempt), the caring female can use the strategic body
A consequence of random mate choice is that bipar- mass regulation to ensure that her mate does not desert.
ental desertion occurs, especially when the average level These arguments emphasize that the decision process
of reserves in the population is low. In this case, the itself requires detailed future theoretical and empirical
individuals do not know the reserves of their prospec- studies.
tive partner, and hence, it may turn out that neither To conclude, our model predicts that strategic reg-
parent can cover the cost of care, which in turn leads ulation of body mass has an important role in the so-
to biparental desertion. Similarly, Webb et al. (1999) lution of sexual conflict in parental care. A fundamental
find that uncertainty about the quality of a partner can effect of reserves is that they allow animals to make
result in biparental desertion. credible threats.
Strategic regulation of reserves has far-reaching con-
sequences in our model. For instance, there is more
desertion by females than by males even in a population
with a strongly female-biased sex ratio. Consequently,
Acknowledgments
our model predicts male-only care for a large parameter
space. This prediction is different from the empirical We thank two anonymous referees for their helpful
findings that female-only care is more common than comments. Z.B. was supported by a Leverhulme Trust
male-only care in most animals (Clutton-Brock 1991). Linked Fellowship and a fellowship from the Soros
We suggest a number of reasons for this discrepancy. Foundation (Budapest). T.S. was supported by a Natural
First, the sexes are typically more asymmetric in nature Environment Research Council grant (GR3/10957) to
than we assume during the computations. For instance, A.I.H., Innes C. Cuthill, and J.M.McN. The study was
males often have higher costs of mate search than fe- also supported by grants from the Hungarian Ministry
males. Also, care by the female may be much more of Education (FKFP-0470/2000) and the Hungarian Sci-
beneficial to the young than care by the male (e.g., entific Research Fund (T030434 and T031706).
 Sexual Conflicts and Reserves 699

APPENDIX A

Basics

State Variables and Their Dynamics

The state of a focal male at time t is represented by his marital status mM(t), the level of his energy reserves XM(t),
and, if he is mated, the energy reserves of his mate YM(t). A male can be either unmated, m M(t) p 0, or mated,
m M(t) p 1. The male’s energy reserves XM(t) take integer values between 0 and L. The male dies of starvation if its
reserves fall to zero. Any energy that would raise reserves above L is lost. The same constraints apply to the energy
level of the male’s mate YM(t). In defining the dynamics of reserves, we use the truncation function chop(x) p
max [0, min (L, x)] (Mangel and Clark 1988).
An unmated male chooses between resting, foraging, or searching for mate. A mated male must either care or desert
once the young have been produced. If a male with reserves x on day t rests, then his reserves become X M(t ⫹ 1) p
chop(x ⫺ x r, M ⫺ z i), x r, M ≥ 0, with probability Pi, where Pi p {0.25, 0.5, 0.25}, and z i p {⫺1, 0, 1}. If the male forages,
then his energy reserves increase; X M(t ⫹ 1) p chop(x ⫹ x f, M ⫹ z i) with probability Pi, where z i and Pi are defined as
above. One can consider xf, M the net energy gain during a day; this gain can depend on either the foraging efficiency
of the male or the food abundance in the environment. Resting and foraging animals remain unmated (i.e.,
m M[t ⫹ 1] p m M[t] p 0).
If a male decides to search for a mate on day t, then his marital status will be m M(t ⫹ 1) p 1 (mate found), with
probability PM(t) (defined by eq. [A4]), and m M(t ⫹ 1) p 0 (mate not found), with probability 1 ⫺ PM(t). If a male
does not find a mate, then his reserves are X M(t ⫹ 1) p chop(x ⫺ x s, M ⫺ z i) , with probability Pi. If he does find a mate,
then the pair produce offspring after tl d, and his reserves at this time will be X M(t ⫹ t l ⫹ 1) p chop(x ⫺ x l, M ⫺ z i),
with probability Pi. Under the assumption of random mating, the reserves of the male’s mate is given by the probability
distribution of reserves of unmated, mate-searching females: YM(t ⫹ t l ⫹ 1) p chop(x ⫺ x l, F ⫺ z i), with probability
Pi Psearch, F(x, t), where Psearch, F(x, t) gives the probability that a mate-searching female has reserves x at time t:

PF(x, 0, t)psearch, F(x, t)

Psearch, F(x, t) p , (A1)
冘 P (x, 0, t)p
L

F (x, t)
search, F
xp1

where PF(x, 0, t) gives the probability that a female is unmated and has reserves x on day t while psearch, F(x, t) is the
probability that an unmated female with reserves x on day t searches for a mate. This latter probability is derived
from the females’ optimal policy pF (see below). If the male’s mate died because of starvation during the laying period
(i.e., YM[t ⫹ t l ⫹ 1] p 0), then no brood is produced, and the male becomes unmated at the end of the laying period
(i.e., m M[t ⫹ t l ⫹ 1] p 0).
After finishing offspring production on day t, both members of the pair decide whether to desert or care for the
offspring until their independence. An individual who deserts becomes independent (m[t ⫹ 1] p 0 ), and its reserves
are decreased by x d ⫹ z i, x d ≥ 0, with probability Pi.
If a male decides to care for his offspring, then he does so until they become independent after tc d. Care of the
young is energetically costly, and the male’s reserves at the end of care are X M(t ⫹ tc ⫹ 1) p chop[x ⫺ x c, M(A) ⫺ z i],
with probability Pi. The extent of the cost of care depends on whether the mate of the focal male cares (A p C,
biparental care) or deserts (A p D, uniparental care). Uniparental care may cost more than biparental care (i.e.,
x c, M[D] ≥ x c, M[C]). Caring for young leads to the production of independent offspring after time t ⫹ tc ⫹ 1 when the
caring male becomes unmated; m M(t ⫹ tc ⫹ 1) p 0.

Arrival
All males (as well as females) are unmated when they arrive at the breeding ground. The proportion of males arriving
on day t, G(t), is given by

G(t) p (t ⫹ 1)a M⫺1e⫺lMt. (A2)

700 The American Naturalist

This distribution is normalized so that 冘T⫺1

tp0 G(t) p 1 . The approximate mean of G(t) is aM/lM, while its approximate
variance is aM/l2M.
The reserves of males that arrive on day t are distributed approximately as a normal distribution with mean, mM(t),
and standard deviation, jM. The arriving males’ mean reserves depend on t as follows:

m M(t) p A M ⫹ B Mt. (A3)

A similar equation applies to the arriving females. In all of our computations, a M , a F p 2; l M , l F p 0.25; jM , jF p
5; A M , A F p 5; and B M , B F p 0.

Probability of Finding a Mate

We assume that the probability that a male finds a mate is given by the following function

PM(t) p k 冑 NF(t) NF(t)

NM(t) ⫹ NF(t) N
, (A4)

where NM(t) is the number of males searching for females, NF(t) is the number of searching females, N is the population
size, and k is the efficiency of searching, 0 ! k ≤ 1 (McNamara et al. 2000). The values of NM(t) and NF(t) are calculated
from the distributions of state variables (reserves and marital status) and the optimal strategy of each sex: NM(t) p
NM 冘Lxp1 [PM(x, 0, t)psearch, M(x, t)], where NM is the number of males currently in the breeding area, PM(x, m, t) is the
probability that a male following the optimal strategy has energetic level x and marital status m at time t, and
psearch, M(x, t) is the probability of searching for a female. Similar equations apply to the females.

APPENDIX B

The Game

Reproductive Value of the Brood

We consider the reproductive value of the brood RAB(x M , x F , t) to have two components, the first of which represents
the effects of care and parental reserves rAB(x M , x F), while the second one corresponds to the seasonal effects S(t):

RAB(x M , x F , t) p rAB(x M , x F)S(t), (B1)

where AB specifies the pattern of care (see “The Model”), while x M and x F give the reserves of the caring male and
female, respectively, at the end of the caring period. If a parent does not care, we set its reserves to zero in the above
equation. This notation also allows us to incorporate easily the effect of the death of one or both parents occurring
during care. The effect of care and parental reserves is modeled as follows.
If both parents desert, then the young certainly die:

rDD(0, 0) p 0. (B2)

When only the male cares, the value of the offspring is

hVCD
rCD(x M , 0) p (1 ⫺ h)VCD ⫹ x M. (B3)
LM

Here, h controls the effect of reserves on the young’s value. If h p 0 , then the male’s reserves do not affect the young’s
value, while if h p 1, his reserves fully influence the offspring’s value (hence offspring cared for only by a male with
reserves close to zero have close to zero survival expectation). The maximal reserves with which a male can finish care
are given by L M p L ⫺ x c, M(D). The value of care by a male with reserves L M caring alone is given by VCD. The value
 Sexual Conflicts and Reserves 701

of female-only care is similarly defined. If both parents care and both are alive at the end of the period of care, then
the offspring’s value is

1
rCC(x M , x F) p [a M(x F) ⫹ b M(x F)x M ⫹ a F(x M) ⫹ b F(x M)x F] , (B4)
2

where

xF
a M(x F) p hVDC ⫹ (1 ⫺ h)VCC , (B5)
LF

and

b M(x F) p
h
[ xF
V ⫹ (VCC ⫺ VCD ⫺ VDC ) .
L M CD LF ] (B6)

The values of a F(x M) and b F(x M) are defined by similar equations. In these equations, h also controls the effect of
reserves on the young’s value. If h p 0 , then the parents’ reserves do not affect the young’s value, while if h p 1, the
parents’ reserves fully influence the offspring’s value. The maximal reserves with which a female can finish the care
are given by L F p L ⫺ x c, F(C). The value of care by parents with maximum reserves L M and L F is VCD, VDC, and VCC
in the cases of male-only, female-only, and biparental care, respectively.
The function defined by equation (B4) has the following properties. It gives higher offspring value for biparental
care unless VCC p VCD p VDC and the reserves of the parent with higher V has stronger effect on the young’s value
than those of with lower V.
The effect of time of the season on the offspring value is represented by the following function:

S(t) p {(
0
1⫹ 1⫺( t
T
) v

)( (T ⫺ t)v
(T ⫺ t)v ⫹ (T ⫺ t)v)when 0 ≤ t ! T

when t ≥ T.
(B7)

The shape of this time-dependent function is controlled by two parameters: v gives the abruptness of the transition
from high brood value to low brood value (the larger v is, the more sudden the decrease), and t is the time at which
this transition occurs (Webb et al., in press). We use v p 2 and t p 80 in all of our computations.

Solution of the Game

To solve the dynamic parental care game, we use the “errors in decision making” approach (McNamara et al. 1997).
Let UM(x, t) be the reproductive value of an unmated male with reserves x on day t. Let WM(x, y, t) be the reproductive
value of a mated male whose own reserves are x and whose mate’s reserves are y on day t, just after finishing offspring
production and about to decide on care. We assume that UM(0, t), WM(0, y, t) p 0 (i.e., an animal with zero reserves
dies) and that WM(x, 0, t) p UM(x, t) (i.e., if the male’s mate dies during the offspring production, then no surviving
offspring are produced, and the male becomes unmated). We use

exp (x/d)
E(x) p (B8)
exp (x/d) ⫹ 1

as an error function with d p 0.015 as the extent of error.

Decision of an Unmated Animal. In this section, we give the equations for an unmated male; analogous notation is
used for unmated females. Let
702 The American Naturalist

H rest, M(x, t) p 冘n

ip0
i M{chop(x ⫺ x r, M ⫺ z i), t ⫹ 1},
PU (B9)

H forage, M(x, t) p 冘n

ip0
i M{chop(x ⫹ x f ⫹ z i), t ⫹ 1},
PU (B10)

and

H search, M(x, t) p [1 ⫺ PM(t)] 冘n

ip0
i M{x , t ⫹ 1} ⫹ PM(t)
PU 
冘冘冘
L n n

yp1 ip0 jp0

Psearch, F(y, t)Pi PW
j M{x , y , t ⫹ t l ⫹ 1},
 
(B11)

where Psearch, F(y, t) is defined by equation (A1), x  p chop(x ⫺ x s, M ⫺ z i), x  p chop(x ⫺ x l, M ⫺ z i), and y  p
chop(y ⫺ x l, F ⫺ z j ).
Let H max , M(x, t) p max a Ha, M(x, t), where a is either “rest,” “forage,” or “search,” and qa, M(x, t) p E[Ha, M(x, t) ⫺
H max , M(x, t)], where E is given by equation (B8). Then

qa, M(x, t)
pa, M(x, t) p
冘q
a
a, M(x, t)
(B12)

gives the probability that the male with reserves x on day t performs action a. The value of pa, M(x, t) is also the best
response with error (cf. McNamara et al. 1997) of an unmated male. Then the unmated male’s reproductive value is

UM(x, t) p 冘a
pa, M(x, t)Ha, M(x, t). (B13)

Decisions of a Mated Female. Assume that a pair has finished the production of offspring at time t and that the male
has decided whether or not to desert. Let HAB, M(x, y, t) be the payoff of the desertion game for the male making a
decision at time t, while HAB, F(x, y, t) is the payoff for the female. The variables x and y denote the energy reserves
of the focal animal and its partner, respectively. The caring decision of the pair is given by AB (see above). Let
t  p t ⫹ tc ⫹ 1 throughout. If the male deserts, then the expected reproductive value of the female if she cares is

冘
n

DC(0, x , t ) ⫹ UF(x , t )},

   
HDC, F(x, y, t) p P{R
i (B14)
ip0

where x  p chop[x ⫺ xc, F(D) ⫺ z i]. The reproductive value of a deserting female is

冘
n

HDD, F(x, y, t) p i F{chop(x ⫺ x d ⫺ z i), t ⫹ 1}

PU (B15)
ip0

because R DD(x, y, t) p 0. Using the error function, we get that the probability of female desertion given that her male
deserts is

pdesert, F(D)(x, y, t) p E[HDD, F(x, y, t) ⫺ HDC, F(x, y, t)]. (B16)

Similarly, if the male cares,

H CC, F(x, y, t) p 冘冘
n

ip0 jp0
n

Pi P{R
j CC(y , x , t ) ⫹ UF(x , t )},
    
(B17)

where x  p chop[x ⫺ x c, F(C) ⫺ z i], and y  p chop[y ⫺ x c, M(C) ⫺ z j]. If the male cares but the female deserts,
 Sexual Conflicts and Reserves 703

H CD, F(x, y, t) p 冘冘
n

ip0 jp0
n

Pi P{R
j CD(y , 0, t ) ⫹ UF(x , t ⫹ 1)},
  
(B18)

where x  p chop(x ⫺ x d ⫺ z i), and y  p chop[y ⫺ x c, M(D) ⫺ z j]. The probability that the female deserts, given her
mate cares, is then

pdesert, F(C)(x, y, t) p E[H CD, F(x, y, t) ⫺ H CC, F(x, y, t)]. (B19)

Similarly to the case of unmated individuals, pdesert, F(A)(x, y, t) also gives the best response of a mated female.

Decision of a Mated Male. Let the male decide on desertion at time t. As before, let t  p t ⫹ tc ⫹ 1. Then

冘冘
n n

CC(x , y , t ) ⫹ UM(x , t )},

    
H CC, M(x, y, t) p Pi P{R
j (B20)
ip0 jp0

where x  p chop[x ⫺ x c, M(C) ⫺ z i], y  p chop[y ⫺ x c, F(C) ⫺ z j];

冘
n

CD(x , 0, t ) ⫹ UM(x , t )},

   
H CD, M(x, y, t) p P{R
i (B21)
ip0

where x  p chop[x ⫺ x c, M(D) ⫺ z i];

冘冘
n n

DC(0, y , t ) ⫹ UM(x , t ⫹ 1)},

  
HDC, M(x, y, t) p Pi P{R
j (B22)
ip0 jp0

where x  p chop(x ⫺ x d ⫺ z i), and y  p chop[y ⫺ x c, F(D) ⫺ z j]; and

冘
n

HDD, M(x, y, t) p i M{chop(x ⫺ x d ⫺ z i), t ⫹ 1}.

PU (B23)
ip0

Let b F(C) p pdesert, F(C)(x, y, t) and b F(D) p pdesert, F(D)(x, y, t), then the expected reproductive value for the male if he
cares is

WM(C) p [1 ⫺ b F(C)]H CC, M(x, y, t) ⫹ b F(C)H CD, M(x, y, t), (B24)

and the expected reproductive value if he deserts is

WM(D) p [1 ⫺ b F(D)]HDC, M(x, y, t) ⫹ b F(D)HDD, M(x, y, t). (B25)

Thus, the male deserts with probability

b M p pdesert, M(x, y, t) p E[WM(D) ⫺ WM(C)], (B26)

which also gives his best response with error.

Payoff of the Desertion Game. Given the above reproductive values and probabilities, the expected reproductive value
of a paired male on finishing offspring production at time t is

WM(x, y, t) p (1 ⫺ b M)WM(C) ⫹ b MWM(D), (B27)

whereas for a female on the same occasion it is

704 The American Naturalist

WF(x, y, t) p (1 ⫺ b M){[1 ⫺ b F(C)]H CC, F(x, y, t) ⫹ b F(C)H CD, F(x, y, t)}

⫹ b M{[1 ⫺ b F(D)]HDC, F(x, y, t) ⫹ b F(D)HDD, F(x, y, t)}. (B28)

The above equations are solved by working backward from the final time when

UM(x, T) p RT(x), (B29)

where RT(x) may represent the dependence of overwinter survival on energy reserves at the beginning of the winter.
We assume RT(x) will be the same for both sexes. In the computations, RT(x) was represented by a threshold function

RT(x) p {10 ifif xx ≤ xx ,

1 L

L
(B30)

where x L p 0.

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