J. theor. Biol. (1990) 146, 513-522
On the Fitness Functions Relating Parental Care to
Reproductive Value
GuY
BEAUCHAMPt
AND ALEJANDRO
KACELNIK:~
t Sub-Department of Animal Behaviour, University of Cambridge, Madingley,
Cambridge CB3 8AA and ~; King's College Research Centre, King's College,
Cambridge CB2 1ST, U.K.
(Received on 13 December 1989, Accepted in revised form on 14 May 1990)
Models of maximization of reproductive value are based on functions relating the
intensity of parental care to two main fitness components: current offspring viability
and future offspring production. A trade-off between the two components may be
highly dependent on the shape of the underlying functions. Most theoretical models
of parental care, however, have only presented an intuitive analysis of their properties. We present an analytical study of some properties of the offspring viability
curve in birds. Reproductive value may be maximized in a region of the offspring
viability curve that is decelerating. However, when future offspring production
decreases at an increasing rate with the intensity of parental care, reproductive value
may also be maximized in an accelerating portion of the curve. It is shown that the
quantitative and qualitative details of parental care models depend to a large extent
on the shape of the underlying fitness functions.
Introduction
M a n y theoretical discussions on parental care have focused on how parents ought
to allocate resources to the young. This question has been investigated under the
conceptual framework of optimal life-history theory. Such an approach assumes
that the resources that the parent allocates to the young are withheld from parental
self-maintenance and thus from future reproduction, and because of this, they are
a part of parental investment. The intensity of parental behaviour is thus viewed as
a c o m p r o m i s e between survival and reproduction (Charnov & Krebs, 1974; Schaffer,
1974).
In this context, the relationship between current offspring production and the
intensity o f parental behaviour is a key element to the understanding of reproductive
strategies. Most authors agree that the nature of the trade-off between current and
future reproduction is highly dependent upon the function relating offspring production to parental care. In most theoretical models, however, the nature and location
of the trade-off on the parental effort continuum have not been analysed. Moreover,
consequences of the shape of this function on the details of parental care models
have been examined intuitively rather than analytically. For instance, it is assumed
that current offspring production must increase relatively slowly (Brockelman, 1975;
Sibly & Calow, 1983; Winkler, 1987; Winkler & Wallin, 1987) or not at all (Smith
& Fretwell, 1974; Parker & Stuart, 1976; Nur, 1984; Houston & Davies, 1985;
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0022-5193/90/020513 + 10 $03.00/0
(~ 1990 Academic Press Limited
514
G.
BEAUCHAMP
AND
A.
KACELNIK
Kacelnik, 1988) at low intensity of parental behaviour. Then, rapid gains must
gradually diminish until reaching a plateau at high level of intensity.
An exponential family of curves is often used to model these relationships on the
basis of mathematical tractability. It is not known to which extent the predictions
of these models depend on the exponentiality assumption. Hence, it is worth
examining the possible consequences of using differently shaped curves. The purpose
of this paper is thus to present a deductive analysis of some properties of the fitness
curve relating current offspring production to parental care.
The Model
Let B, represent fecundity at age x and P, the probability of survival of a parent
bird between age x and age x + 1. For an iteroparous bird reproducing at age x in
a stable population, the reproductive value V, is given by
V. = Bx + p~B,+, + P~Px+,B~+2+... + P,P~+, • • • Pz-,B~,
(1)
where x = 1, 2, . . . , z, and z tends to infinity. Fecundity during a breeding episode
is defined as the expected fledgling production and can be written as follows
Bx = nOx(E, n),
where n is brood size and O the probability of survival of any one offspring in a
brood where effort, E, is divided equally among nest members. Parental effort, E,
is defined as the total amount of time and energy invested while raising young.
When current intensity of parental behaviour affects fecundity and survival in
future breeding episodes, reproductive value may be decreased. This effect represents
a cost of reproduction. Evidence for a cost of reproduction in birds is inconsistent
thus far. Parental survival rate and fecundity appear to be related to the intensity
o f parental behaviour in some reports (see reviews by Partridge & Harvey, 1985;
Reznick, 1985; Nur, 1988a). However, some recent studies have failed to demonstrate
a cost o f reproduction (Gustaffson & Sutherland, 1988; Korpimaki, 1988; Pettifor
et al., 1988). Notice that investigations on the costs of reproduction in birds are
hampered by the difficulty of linking indirect measures of survival to reproductive
effort (Jones, 1987; Nur, 1988b). To illustrate the effect of a cost of reproduction
on properties of the offspring production curve, the decrease in reproductive value
through reproductive effort is assumed to be brought by reduced parental survival
to the next breeding season. Several forms of survival vs. parental effort will be
considered here. Under the above assumption, the general expression of V can be
written as
V. = nO,(E, n) +/3P~(E),
(2)
where 13 is the expected future fledgling production./3 is usually considered to be
a function o f the mean population fecundity and adult survivorship (Kacelnik,
1988). It is assumed that future brood sizes are more accurately predicted by the
population mean than by the size of the current brood. Considerations about brood
size heritability are thus left aside. Hence, the model is more realistic for male parent
FITNESS
FUNCTIONS
AND
PARENTAL
CARE
515
birds. Because/3 is a constant in this formulation, eqn (2) represents the reproductive
value of a parent bird tending a brood of size n during a given breeding season,
and then reverting to the common pattern of reproduction for this population in
subsequent years. From the equation, a high level of effort leads to an increase in
current offspring production nO, but reduces parental survival rate and thus future
offspring production.
Let us define two parental phenotypes, 1 and 2, that use E1 and E2 in a given
brood size. In terms of reproductive value, the advantage of phenotype 2 over
phenotype 1 is:
V 2 - V1 = fl[ P ( E 2 ) - P(E1)]+ n[ O( E2, n ) - O ( E1, n)].
(3)
One can investigate the behaviour of eqn (3) with respect to the allocation of
phenotype 2 in a given brood by setting
a( V 2 - V1)/aE2 = fl OP(E2)/aE2 + n OO(E2, n)/aE2,
and finding the strategy E* that has no advantage over different strategies when
0 ( I / 2 - V1)/OE2=O;
i.e. - ( f l / n ) O P ( E * ) / O E * = a O ( E * , n)/aE*.
(4)
Equation (4) states that under strategy E* the slope of O vs. E is a function of two
factors: (i) the slope of the future offspring production curve at the optimal value
E*, and (ii) the ratio fl/n. The increment in fitness resulting from an increase in
parental effort in a given brood must be equal to the proportional reduction in
future reproduction, with the proportionality factor being fl/n. This ratio can be
thought of as a scaling factor of the expected future reproductive success. A value
of the ratio smaller than one suggests that a parent bird expects less from future
reproduction than from the current attempt, while a value o f the ratio greater than
one indicates that future reproduction counts more than current productivity. When
fl = n the slopes of P and O at E* are equal in absolute value. In general, eqn (4)
indicates that for a given value of n and of OP/OE* in a narrow region about E*,
the higher the future expectation the lower the expected offspring production in the
current breeding attempt.
When OP/OE equals 0, adult survival prospects are independent of reproductive
effort, that is reproduction is not costly. In this case the optimal allocation of effort
during one breeding attempt can be found by solving O0/cgE = 0 for E, given that
the optimal value is a maximum of O. No inferences can be drawn for O if
reproduction is not costly, as eqn (4) now only depends on O. It is clear that E*
should be very close to the maximum intensity of parental effort in any one breeding
attempt. Nevertheless, stochasticity in the environment between breeding episodes
may influence offspring production nO. The bad-year effect is usually thought to
favour intermediate brood sizes which maximize long-term recruitment (Boyce &
Perrins, 1987). Consequently n becomes a variable and in these conditions current
productivity may sometimes be submaximal although E* would be close to the
maximum intensity o f parental effort. The following section deals with cases where
OP/OE # O.
516
G.
BEAUCHAMP
AND
A.
KACELNIK
If the strategy E* specified by eqn (4) is to be stable then E* must be a maximum
of ( V 2 - V1) with respect to E2. For this to occur, the second derivative of eqn (2)
must be negative when E2 = E*. We thus have:
fl 82P( E*)/ O(E*)2 + n O20(E *, n )/ O(E*) 2< O;
which reduces to
O20(E *, n)/O(E*)2 < -(fl/n) O2P(E*)/O(E*) :.
(5)
Equation (5) allows one to investigate some properties of O vs. E in a narrow region
about E* given some assumptions regarding the shape of P vs. E. It is assumed
now that OP/oE* is negative and O0/OE* positive. The value of the second derivative
o f P about E* might thus take three different forms depending on the shape of P
(Fig. 1).
The first form of P represents an exponential decay where parental survival rate
decreases at an increasing rate with parental effort. Mortality rate is thus increasing
rapidly at high level of investment. In contrast, the third form represents an exponential decay in which mortality rate decreases at a decreasing rate at higher levels of
parental care. The second form depicts a linear decrease of parental survival rate
where the slope is independent o f parental effort. The first type of function has been
used by Houston & Davies (1985), Winkler (1987) and Kacelnik (1988), while the
second type by Charnov & Krebs (1974). Mortality patterns that follow the third
type of function may not be very common. We included this type of function for
the completeness of the analysis. Notice that Parker (1984) has used a function o f
the third type to model offspring survivorship as a function of begging rate. Using
the value o f the second derivative of P about E* and substituting in eqn (5) for
the three types of function give the sign of the second derivative of O about E*:
020(E *, n)/O(E*)2 <-(fl/n)O2P(E*)/O(E*) 2.
(6)
(ii)
O20(E *, n)/O(E*)2<O
(7)
(iii)
O20(E *, n)/O(E*)2 <O
(8)
(i)
The strategy E* is stable in a region of O that is decelerating when O2P(E*)/O(E*) 2
0. However, this region may also be accelerating when 02P(E*)/O(E*)2< 0 (i.e. the
first type of function) because the r.h.s, of eqn (6) would become a positive value.
Therefore, the region of O where E* is stable depends upon the sign of the second
derivative of P in a narrow region about E*.
QUALITATIVE
PREDICTIONS
OF THE
MODEL
In models of parental care where it is assumed that offspring do not contribute
to parental fitness until they receive a minimum amount o f care, and where subsequent gains show diminishing returns (i.e. Houston & Davies, 1985; Kacelnik,
1989) the second derivative of O is negative in the region of effective parental care.
The above analysis showed that reproductive value may be maximized in a region
o f diminishing returns regardless of the actual relationship between adult survival
FITNESS
FUNCTIONS
AND
PARENTAL
CARE
517
(a)
{b)
i
Parental effort
FIG. 1. Three patterns of adult survivorship vs. the intensity of parental effort are illustrated. (a),
Survival rate follows an exponential decay and decreases at an increase rate with the intensity of parental
care. (b), Mortality rate per unit of parental effort is independent of parental care such that survival rate
is a linear function of effort. (c), Parental survival also follows an exponential decay but this time the
rate of mortality decreases at a decreasing rate with effort.
518
G.
BEAUCHAMP
AND
A. KACELNIK
probability and effort, as eqns (6), (7), and (8) allow for a negative solution. However,
when parental survival rate decreases at an increasing rate with effort, reproductive
value may be maximized in a region of O that is accelerating, as eqn (6) allows for
a positive solution. Reproductive value cannot be maximized in an accelerating
region of O if the curve linking survival rate to parental effort has a different shape.
If, for instance, future offspring production decreases linearly with effort, then
reproductive value has no maxima in the interior region of O. In this case, the
optimal strategy o f a parent bird is either to forego reproduction or to become
semelparous, and provide care at a maximum rate to the young. The optimal strategy
depends on the relative value o f fl and n. When fl is greater than n, parents should
forego reproduction because future offspring production counts more than current
production. When n is greater than/3, the reverse argument applies. When the two
values are equal, parent birds should be indifferent.
The previous analysis has demonstrated a complex pattern of interaction between
the two fitness components. Moreover, various shapes of the offspring production
curve allow for the maximization of reproductive value in the interior region. This
goes against the traditional assumption that the stable pattern of allocation of effort
can only be found in a decelerating region of the curve. In conclusion, the qualitative
details o f a parental care model depend on the shape of the underlying fitness curves.
QUANTITATIVE
PREDICTIONS
OF THE
MODEL
It is of interest to know whether the exact location o f the stable strategy E* on
the curve relating offspring production to parental effort depends also on the shape
of P. Such a knowledge would be helpful as the pattern o f adult survivorship in
relation to the intensity of parental behaviour may be one variable involved in
between-species differences in investment patterns.
Let us consider two forms of P discussed previously (see Fig. 1). In the linear
model (P2), mortality rate per unit of parental effort is independent of parental
care. In the other model (P1), adult survival prospects decrease at an increasing
rate with parental effort, and drop most sharply at high intensity of effort. Equation
(4) states that in following the optimal policy E*, the increment in fitness resulting
from an increase in parental effort in a given brood must be equal to the proportional
reduction in future reproduction, with the proportionality factor being/3/n. Let us
assume that expected future reproduction equals current productivity. This would
be so for starlings and other passerines in the particular case where mortality between
breeding attempts is constant and close to one half (Kacelnik, 1988). The slopes of
P and O at E* must then be equal in absolute value in a narrow region about E*.
As the stable strategy can be found in a decelerating region of O, is it possible to
determine where will E * I and E*2, which represent the optimal allocation of effort
under P1 and P2 respectively, be located?
Figure 2(a) shows the two resulting patterns of future offspring production (/3P1
and tiP2) and the current offspring production curve (nO). In Fig. 2(b), the slope
of each of these hypothetical functions is plotted against parental effort. E* can be
found at the intersection o f the two types o f curves. At this point the slopes are
FITNESS
FUNCTIONS
AND
PARENTAL
~~...
CARE
519
Ca)
~
~
-
OP
Q.
r"
a.
m
/
\
v(,
0
(b}
o
o
_=
o
.a
\\\\
\-
"=",,,,.=/
E2
J
BP2
Parental effort
FIG. 2. Effects of the shape of the future offspring production curve on optimal allocation of effort
to the chicks. (a), Two patterns o f future offspring production are illustrated (tiP1 and tiP2, based on
mortality functions described in Fig. 1). OP represents the offspring production curve. (b), The absolute
value of the slope of each of these curves is illustrated. The value of two slopes is equal in absolute
value at the point of intersection. The location of the optimal allocation of effort clearly depends on the
shape of the future offspring production curve.
520
G.
BEAUCHAMP
AND
A. K A C E L N I K
equal in absolute value. A linear reduction in adult survival prospects gives rise to
a lower allocation of effort. The location of the stable strategy on the parental effort
continuum is then dependent on the shape of the function linking adult survival
prospects to the intensity of parental care. In conclusion, quantitative details derived
from models of parental care are also conditional upon the shape of the two fitness
functions. It is suggested that the empirical rule used to determine the location of
E * would be a useful tool to investigate the effects o f a change in the shape o f the
two fitness functions on quantitative predictions of parental care models.
The location of E* can also be influenced by the ratio/3/n. From eqn (4), birds
that expect more from future reproduction than from the current breeding attempt
(i.e./3/n > 1) should invest less in each offspring. E* would be shifted toward low
levels of investment. On the other hand, higher expectation from the current breeding
attempt (i.e. fl/n < 1) should lead to a shift o f the optimal strategy toward high
levels of investment.
Discussion
Current models of parental care in birds assume specific forms for the offspring
production curve. Relatively little is known to support these assumptions. This paper
has presented a deductive analysis of some properties of the offspring fitness curve
and located the stable allocation strategy on the parental effort continuum. The
curve relating offspring production to parental effort is non-linear and characterized
by a region where gains show diminishing returns. The stable strategy of allocation
can be found in such a region regardless of the shape of the curve linking adult
survival rate to parental effort. If survival decreases at an increasing rate with
parental effort, then the stable strategy can also be found in an accelerating region
of the offspring curve. The shape of the offspring curve is therefore in agreement
with the forms suggested by various authors (e.g. Sibly & Calow, 1983; Winkler,
1987). However, there exists a complex pattern of interaction between the two fitness
components involved in the maximization of reproductive value. Parental care
patterns that emerge from maximization models can vary on a quantitative and
qualitative basis depending on the shape of the functions used to model the two
fitness components. Predictions from a model of parental care must therefore be
reached after a detailed consideration of the shape of both the offspring and future
production curves.
The present analysis has examined the location of the optimal pattern of allocation
from a parental point o f view. From the point o f view o f an offspring, the relative
balance between current production and future reproductive potential is different.
In this case, current productivity is valorized to a greater extent because one chick
in the brood benefits more from an increase in allocation now than from an increase
in future production of siblings (/3), to which it is only half related. Parent-offspring
conflicts in terms of allocation of effort can thus arise (Trivers, 1974). In general,
the optimal allocation o f effort from the point o f view o f an offspring will be greater
than that from the parental point of view. Changes in the shape of the curve linking
adult survivorship to intensity of parental care (P) have been shown to shift the
FITNESS FUNCTIONS
AND PARENTAL CARE
521
position of E*, the optimal allocation strategy from a parental point of view. If
parent-offspring conflicts are viewed as a result of the discrepancy between parentoffspring optima, then changes in the shape of P may also affect conflict intensity.
The conclusion is valid only if/3 is unaffected by changes in the shape of P, so that
everything else remains equal. At the moment, it is only possible to speculate on
the validity of this assumption.
Few studies have documented the shape o f the functions relating the intensity o f
parental care to current and future offspring production. For instance, Nur (1988b)
has presented evidence that parental survival rate in blue tits (Parus caeruleus)
follows a linear decrease with manipulated brood size. Reid (1987) has shown an
almost linear pattern in glaucous-winged gulls (Laurus glaucescens). Some recent
field experiments have provided little support for the notion of reproductive costs
in two small passerine birds (Gustaffson & Sutherland, 1988; Pettifor et al., 1988)
and in Tengmalm's owls (Korpimaki, 1988). Nur (1988b) in blue tits, Krementz et
al. (1989) in starlings and Smith et al. (1989) in great tits have shown that offspring
survival is negatively related to manipulated brood size. Hence, offspring survival
prospects are directly linked to the amount of care provided by the parents.
Future research would be needed to document the shape o f the two fitness curves
in other species. Most of the above relationships have only documented the sign of
the first derivative. What is needed now is some indication regarding the sign of
the second derivative, or the degree of curvature of these fitness functions. This
would further our understanding of the trade-off between survival and reproduction
in birds and clarify the links between parental survival rate, offspring production
and parental effort.
We would like to thank A. Desrochers and an anonymous referee for comments on the
manuscript, and L. Barden for drawing the figures. The first author was supported by a FCAR
(Canada) post-graduate scholarship.
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