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vol. 155, no. 4 the american naturalist april 2000
Reproductive Effort and Reproductive Values in
Periodic Environments
Jon Brommer,
1,
*
Hanna Kokko,
2,†
and Hannu Pietia¨inen
1,‡
1. Department of Ecology and Systematics, Division of Population
Biology, P.O. Box 17 (Arkadiankatu 7), FIN-00014, University of
Helsinki, Finland;
2. Department of Zoology, University of Cambridge, Downing
Street, Cambridge CB2 3EJ, United Kingdom
Submitted December 21, 1998; Accepted November 13, 1999
abstract: Life-history theory concerns the optimal spread of re-
production over an organism’s life span. In variable environments,
there may be extrinsic differences between breeding periods within
an organism’s life, affecting both offspring and parent and giving
rise to intergenerational trade-offs. Such trade-offs are often dis-
cussed in terms of reproductive value for parent and offspring. Here,
we consider parental life-history optimization in response to varying
offspring values of a population regulated by territoriality, where the
quality of the environment varies periodically. Periods are interpreted
as either within-year (seasonality) or between-years variation (cyclic-
ity). The evolutionarily stable strategy in a general model with two-
phased periodicity in the environment can generate either higher or
lower effort in the more favorable of the two phases; hence knowing
survival prospects of offspring does not suffice for predicting repro-
ductive effort—the future of all descendants and the parent must be
tracked. We also apply our method to data on the Ural owl Strix
uralensis, a species preying on cyclically fluctuating voles. The ob-
served dynamics are best predicted by assuming delayed reproductive
costs and Type II functional response. Accounting for varying off-
spring values can lead to cases where both reproductive effort and
recruitment of offspring are higher in the phase when voles are not
maximally abundant, a pattern supported by our data.
Keywords: life history, reproductive effort, reproductive value, inter-
generational trade-off, Ural owl, evolutionarily stable strategy.
*
To whom correspondence should be addressed; e-mail: jon.brommer@
helsinki.fi.
†
E-mail: h.kokko@zoo.cam.ac.uk.
‡
E-mail: hannu.pietiainen@helsinki.fi.
Am. Nat. 2000. Vol. 155, pp. 454–472. q2000 by The University ofChicago.
0003-0147/2000/15504-0003$03.00. All rights reserved.
The life history of an organism can be viewed as an
adaptation to environmental conditions (Bradshaw 1965;
Southwood 1977), which presents a solution to the prob-
lem of optimizing reproductive schedules under con-
straints. Proximately, different sets of trade-offs are
thought to shape the evolution of particular life histories
(Stearns 1989); these trade-offs must exist since otherwise
the existence of “Darwinian demons” (Law 1979) with
infinite longevity and infinite reproductive rates would
be permitted. However, deciding when what kind of
trade-off is operational is not a straightforward matter
(Maynard Smith 1993). Broadly, two classes of trade-offs
are distinguished: those operating within one individual
(intraindividual trade-offs) and those operating between
the parent and its offspring (intergenerational trade-offs;
Stearns 1989, 1992). Several authors (e.g., Williams 1966;
Schaffer 1974a, 1974b; Hirschfield and Tinkle 1975) have
used the Fisherian reproductive value (Fisher 1930) to
visualize trade-offs as a dilemma of energy allocation. By
allocating more of the available energy into reproduction,
the current component of Fisherian reproductive value,
a parent diminishes the energy it can put into the future
components of reproductive value: survival and future
reproductive output. Indeed, several experimental studies
have shown that when a higher proportion of the energy
available to an organism has to be allocated to repro-
duction, costs are paid by a reduction in survival to the
next season (e.g., Daan et al. 1990, 1996) or by a re-
duction in reproductive output in the next season (e.g.,
Gustafsson and Sutherland 1988).
Hirschfield and Tinkle (1975) postulated that for or-
ganisms living in an environment with juvenile survivor-
ship varying between seasons, parents should increase their
reproductive effort in favorable years (see also Carlisle
1982). This is an intergenerational trade-off where exter-
nally caused variation in a life-history trait of offspring
affects the parental life history. In discussing intergener-
ational trade-offs in variable environments, empirical
workers often argue in terms of a trade-off between the
reproductive value of the parent and its offspring (e.g.,
Daan et al. 1990; Hakkarainen and Korpima¨ki 1994a;
Reproductive Values under Periodicity 455
Amat 1995; Dale et al. 1996; Korpima¨ki and Rita 1996;
Wiebe 1996; Bruun et al. 1997; Po¨ysa¨ et al. 1997). Fisher’s
(1930) classical formulation, nevertheless, gives all off-
spring the reproductive value 1. More general definitions,
however, allow for differences in reproductive values for
individuals in various states (e.g., Taylor 1990; Kokko and
Ranta 1996; Leimar 1996; Marrow et al. 1996; McNamara
and Houston 1996). Here, we use a method in which
reproductive values of parents are expressed through those
of their offspring, which allows us to follow reproductive
values of adults and their offspring in different phases of
a periodic environment. Specifically, we apply this method
to periodic environments where density dependence reg-
ulates the population. Periodicity of the environment can
be interpreted as recurring within (seasonality) or between
years (cyclicity).
Possibly the best example of species living in an envi-
ronment that is varying predictably between years is pro-
vided by northern forest owls preying on voles, which show
a 3-yr cycle with low, increase, and peak phases (Korpima¨ki
1988a; Pietia¨inen 1989; Brommer et al. 1998). For the Ural
owl Strix uralensis, the juvenile survivorship varies dras-
tically in a predictable manner between years, such that
the recruitment probability for a hatchling in the increase
phase is twice as high as for a hatchling in the peak phase
(Brommer et al. 1998). A similar difference in recruitment
between phases has also been found in Tengmalm’s owls
Aegolius funereus (Korpima¨ki 1988a), Eurasian kestrels
Falco tinnunculus breeding in southwestern Finland (Kor-
pima¨ki and Rita 1996), and great horned owls Bubo vir-
gianus in Alaska (Rohner and Hunter 1996). Differences
in offspring reproductive value are hence expected, and in
the phase with a higher recruitment probability, “high re-
productive effort, possibly associated with high costs,
should be observed” (Hirschfield and Tinkle 1975, p.
2229). Indeed, there is evidence that male Tengmalm’s owls
increase their feeding effort in increase vole years above
that of peak years, despite higher food abundance in the
latter. This strategy is interpreted as a response to the lower
recruitment probability of peak-year hatched offspring
that face a crash in prey numbers early in life (Hakkarainen
and Korpima¨ki 1994b).
However, parents cannot be expected to respond only
to intergenerational fitness benefits without considering
intraindividual constraints. Clearly, parents jeopardize
their own residual reproductive value by increasing their
reproductive effort in favorable years, and solving the
whole allocation dilemma requires following the future
success of both parents and their offspring. Optimizing
reproduction in owls may also require taking into account
within-year variation in food density because parents pro-
vide care for their offspring over an extensive period of
time (about 6 mo) and because the timing of costs during
this time period may be important. For example, feeding
hatchlings may require less effort from adults in peak than
in increase vole years, but subsequent survival is facilitated
in increase years, due to high food abundance in the fol-
lowing winter, whereas voles crash during the summer of
a peak phase. We show that our method allows tackling
problems of such complexity.
The goal of this article is thus twofold. First, we explore
Hirschfield and Tinkle’s (1975) idea of adjusting repro-
ductive effort to the survival probability of offspring in a
two-phased period. Special emphasis is put on the com-
bination of intergenerational and intraindividual trade-off
in determining optimal reproductive effort. Furthermore,
we consider a more detailed scenario of life-history op-
timization for predators living on cyclically fluctuating
prey, taking the Ural owl as an example.
Reproductive Values in Cyclic Environments
Linking Reproductive Values to the Future:
The Chain Procedure
The growth of a population structured in several stages
can be described using matrix theory (Caswell 1989) as
N(t11) = A(t)N(t), (1)
where N(t) is a column vector whose elements present the
number of individuals in each stage of the population and
Ais a matrix describing the transition probabilities be-
tween the stages. In age-structured populations, elements
in N(t) correspond to ages and all stages contribute to the
first class, the stage of newborns (see Caswell 1989). How-
ever, this need not be the case in a periodic environment;
instead, we can categorize each individual by the phase in
which it was born and the phase in which it is now (Gour-
ley and Lawrence 1977). A periodic environment can either
refer to a cyclic environment in which there are between-
year variations in the environment or to a seasonal within-
year variation that recurs year after year: for example, a
two-phase period could represent two breeding attempts
within a single summer. Finding optimal life histories in
periodic environments requires taking into account den-
sity dependence (McNamara 1995, 1997; Mylius and Diek-
mann 1995; Pasztor et al. 1996). To provide a simple ex-
ample of density dependence, we assume that the
population is limited by the scarcity of nesting sites (ter-
ritories) and, hence, that individuals are additionally cat-
egorized into territorial breeders and nonterritorial floaters
(fig. 1). Floaters are assumed to become territorials as soon
as vacancies occur and, once territorial, are assumed to
remain so until death.
Furthermore, to provide the simplest example of en-
456 The American Naturalist
Figure 1: Life cycle graph of a population living in an environment with
a two-phased cycle (phases aand b). Stages and transitions between stages
(vectors) are characterized by the year in which they were born (first
letter) and the year in which they are now (second letter). Uppercase first
letter, territorials. Both letters lowercase, nonterritorial floaters. Newborns
are counted in the year following the year in which they were born
(prebreeding census; Caswell 1989). Solid line, survival (P) and territory
acquisition (T). Dashed line, reproduction (M). See text for further
details.
vironmental variation, our first model comprises two en-
vironmentally determined phases, aand b. These represent
reproductive periods of different quality, regarding, for
example, the abundance of food. The total energy available
to an individual is thus different in the two phases; also,
the trade-offs between survival and reproduction may be
phase dependent. For example, phases could represent two
breeding attempts during a single summer, in which case
within-summer survival from ato bis certainly different
than overwintering survival from bto a.
In appendix A, we describe how the calculation of the
population dynamics proceeds in the case of such a di-
vision in phases. The population vector N(t) can be written
as
N(t)
Aa
N(t)
aa
N(t)
Ba
N(t)
ba
N(t)=.(2)
N(t)
Ab
N(t)
ab
N(t)
Bb
N(t)
bb
Here, tdenotes the period, where each period consists of
two phases, aand b. Stages are characterized by a double
coding. The first index refers to the individual’s status
(territorial or floater) and to the phase in which an in-
dividual was born. The second refers to the current phase
of the environment, where, for example, N
Ab
(t) describes
the number of territorial individuals in period t, which
were born in an “a” phase that are present in the popu-
lation in a “b” phase; uppercase “A” is used to denote
territoriality, such that the corresponding number of float-
ers is denoted N
ab
(t). Transitions M(fecundity), P(sur-
vival), and T(territory acquisition; app. A; fig. 1) are char-
acterized similarly by two indices. Thus, M
Ab
equals
fecundity in phase bof a territorial individual born in
phase a;P
bb
, the chance for a b-born floating individual
who currently lives in phase bto survive to the next phase
(i.e., from bto the next a); and, T
ab
, the probability that
an a-born individual who still floats at phase bgains a
territory in the next transition from bto a(provided that
she survives as well).
In calculating T
ij
, the probability that an i-born floater
in phase jbecomes territorial in the next phase, we con-
sider all individuals (a- and b-born) competitively equal:
, and ; thus T
ij
is denoted as T
j
.T =T =T T =T =T
aabaa abbbb
With phase kfollowing phase j,
T=no. of free territories(k)/N(k)
jfloater
=[no. of all territories 2N(k)]/N(k).
territorial floater
Once a floater becomes territorial, she will be so until
death. Thus, with two phases, aand b, we have
N(k)=N 1N=PN1PN
territorial Ak Bk Aj Aj Bj Bj
and
N(k)=N 1N
floater ak bk
=PN 1PN 1MN1MN.
aj aj bj b j Aj Aj Bj Bj
The territories thus “fill up” instantaneously, and as long
as there are more floaters present than free territories, all
territories will be occupied. Every floater has a chance,
P
ij
T
j
, to survive and to become a territorial the next year.
Likewise, the chance to survive to the next season but to
remain a floater is given by P(1 2T).
ij j
As individuals born in the a- and the b-phase can later
Reproductive Values under Periodicity 457
produce offspring in both phases of the cycle (app. A; fig.
1), they do not just contribute to a single class of offspring.
Furthermore, the population matrix assumes that adults
can reproduce repeatedly in the same stage. Thus, in this
simple case, it is assumed that survival is independent of
age and depends only on current phase, phase at birth,
and current reproductive effort. An individual in the ab
stage can be either a juvenile born the previous (a) phase
or an adult of age ( ). Note that N(t)
112qq=1, 2, ...
contains information on the population sizes of two suc-
cessive phases (app. A). Thus, the time unit in tequals
the length of the period. This is especially convenient as
it allows us to study deterministically cycling populations:
fluctuations in population size can be viewed as occurring
between corresponding elements within N(e.g., from N
Aa
to N
Ab
), while stability properties still apply when N(t)is
viewed as a whole. Hence, by stating that Nis stable, we
do not exclude population fluctuations but merely imply
that the population dynamics repeat the same sequence
of population sizes after one time unit (equal to the length
of the cycle, e.g., 2 yr) and returns to this sequence if
disturbed.
Assuming a fixed number of territories implies negative
density dependence. With higher population sizes, the
chance to obtain a territory, T
j
, decreases, and a larger
fraction of the population will become nonreproductive.
This will stabilize the population (in terms of having an
unchanging N: note again that Nmay nevertheless incor-
porate variations within the period t). Thus, population
dynamics will converge to a stable N. When Nis stable,
a transition matrix Bcan be built to contain all the tran-
sitions that occur during one period, as outlined in ap-
pendix A. The elements of the left eigenvector of the tran-
sition matrix Bare a measure of the weight of each stage
for population growth. They form the matrix equivalent
of Fisherian reproductive value (Caswell 1989). Taking the
left eigenvector of B( , where lis the dominant
vl=vB
eigenvalue of the matrix B) gives
vl=M[Tv1(1–T)v]1Pv,
Aa a a Aa
Aa Ab ab Ab
vl=PTv1P(1–T)v,
aa a aa a
aa Ab ab
vl=M[Tv1(1–T)v]1Pv,
Ba a a Ba
Ba Ab ab Bb
vl=PTv1P(1–T)v,
ba a ba a
ba Bb bb
vl=M[Tv1(1–T)v]1Pv,(3)
Ab b b Ab
Ab Ba b a Aa
vl=PTv1P(1–T)v,
ab b ab b
ab Aa aa
vl=M[Tv1(1–T)v]1Pv,
Bb b b Bb
Bb Ba ba Ba
vl=PTv1P(1–T)v.
bb b bb b
bb Ba ba
Here, represents the reproductive value of an i-bornv
ij
floater in phase j; and, , the reproductive value of an i-v
Ij
born territorial in phase j.
The reproductive value of a floater is a weighted sum
of the reproductive values of territorials and floaters in
the next phase. Weighting is done according to transition
probabilities to these two stages; these weights naturally
include survival prospects. For territorials, the value of
their breeding output is similarly weighted. The repro-
ductive value of the parent is formed as the sum of the
value of her current breeding output and of her repro-
ductive value in the next stage. The latter, that is, her
residual reproductive value, is again scaled by the transi-
tion probability of remaining alive. These rules result in
a “chain” that links each individual’s current reproductive
values to future ones. The ends of the chain are linked
together, as the first phase of the period follows the last
phase.
At equilibrium, the dominant eigenvalue of Bwill equal
, and the corresponding left eigenvector gives thel=1
reproductive values as defined by the chain. As Nincludes
several stages of newborns, they are allowed to have sep-
arate unequal values. One can scale them arbitrarily, and
we have chosen the first newborn floater stage ( ) to
v
ab
equal 1.
Optimization of Reproduction
In order to fulfill the requirements of finding the optimal
allocation strategy, we need to consider explicitly the form
of density dependence or, when not modeling density de-
pendence, to accept an implicit form of density depen-
dence (Mylius and Diekmann 1995). Optimization of fit-
ness means that the intrinsic rate of increase, r, and the
net reproductive rate, R
0
, are both maximal and equal to
one; in practice, one thus needs to incorporate density
dependence, that is, in this case, the population dynamical
feedback on floaters through territoriality (see also Pa´ sztor
et al. 1996).
When in the predominant populationl(B)=1
1
through density dependence, any strategy that results in
can invade (e.g., Metz et al. 1992). All other factorsl11
being equal, increasing any of the Mor Pvalues in equa-
tion (3) will always increase the mutant’s lover ofl=1
the predominant population. However, assuming the ex-
istence of life-history trade-offs, M
Ij
and P
Ij
must be con-
strained through energy limitation. A trade-off can be rep-
resented in letting total energy depend on external food
abundance, (the “environment”), from which ter-E=[E]
ij
ritorial individuals have to allocate a fraction, , toa=[a]
Ij
reproduction. Again, irefers to the phase in which an
individual is born (when noted in upper case, it refers to
a territorial), and jrefers to the current phase of the en-
458 The American Naturalist
Figure 2: Examples of the fertility and survival functions of equation (4). Fertility is always viewed as an S-shape increase ( ) with increasingg11
energy allocated to reproduction (x
m
); examples are shown for and for with and , 5, and 7. Survival is either a saturatingm=7p=1b=10 g=3
max max
increase ( ) or an S-shaped increase ( ) with increasing energy allocated to survival (x
p
), here plotted for and , 1, and 3.«≤1«11d=2 «= 0.5
Typically, in iteroparous species.d!b
vironment. For territorials, the gains M
Ij
and P
Ij
can be
assumed to be functions of energy allocated to that fitness
component, . Floaters, inM=M(a,E), P=P(a,E)
Ij Ij Ij Ij Ij Ij
contrast, allocate all available energy into survival (P=
ij
), since they cannot reproduce.P(E)
ij
The actual reproduction and survival are assumed to be
increasing but not necessarily linear functions of energy
allocation. We study the following gain functions:
g
2(x/b)
m
M(x,m,b,g)rm12e,
[]
mmax max
«
2(x/d)
p
P(x,p,d,«)rp12e.(4)
[]
pmax max
The shape of the gains have been chosen to depend on
the parameters (b,g) and (d,«) in a way that allows both
concave and S-shaped choices of gain functions. We view
the reproductive gain, M(x
m
,m
max
,b,g), as a necessarily
S-shaped increase in reproductive output to a maximum
fecundity m
max
, with increasing energy allocated to repro-
duction ( ). Survival gain, P(x
p
,p
max
,d,«), isx=aE
mIjIj
allowed to vary from a saturating to an S-shaped increase
in survival probability, with increasing energy allocated to
survival (fig. 2; for territorials, andx=[1 2a]Ex=
pIjIj p
for floaters). Survival has an upper limit of p
max
.E
ij
In the optimization procedure, individuals in each stage
should choose their allocations, , to maximize the suma
of their own survival, weighted by their own future re-
productive value and the sum of their current fecundity,
weighted by the reproductive value of offspring. The chain
of reproductive values in equation (3) links the future
success prospects of each individual and its offspring to
its current reproductive value and, hence, shows that max-
imizing the above sum equals maximizing one’s own cur-
rent reproductive value (see also Goodman 1982).
The optimal allocation strategy, , through the cycle
∗
a
can be found iteratively by maximizing the reproductive
value of a mutant in each stage, where the predominant
determines the dynamics, N, and, hence, the reproduc-
a
tive values of offspring in each cycle phase; a mutant’s a
Ij
maximizes its current reproductive values, expressed
through the future values (app. B). Maximizing with re-
spect to each a
Ij
will give a new allocation matrix, . Dy-
0
a
namics described by the new matrix will then converge
0
a
to a new stable population vector, N
0
, and there will be a
new optimal , given by maximization of the reproductive
00
a
values. By repeating the calculations of the dynamics and
of the maximization of the reproductive values, we con-
verge at the evolutionarily stable allocation strategy (ESS)
, which cannot be invaded by any other strategy (app.
∗
a
B). At the ESS, , and maximal for
l(B)=1 l/a=0
11Ij
all i,j(see Mylius and Diekmann 1995; Pa´ sztor et al. 1996).
In this case, is the strategy that maximizes the equi-
∗
a
librium population size (see also Hastings 1978). However,
more generally, optimal life-history strategies do not nec-
essarily lead to maximization of the total population size;
generally, the population stage subject to density depen-
dence is expected to be maximized (Michod 1979; Charles-
worth 1994). In our case, this means that floater numbers
should be maximized (see also Kokko and Sutherland
1998), and this also leads to total population size maxi-
mization. We assume equally good reproductive output in
all suitable territories (which, hence, become filled and
Reproductive Values under Periodicity 459
Figure 3: A, Ratio of reproductive effort in phase aover reproductive effort in phase bdepends on the amplitude, DE,ofthe(RE[a]/RE[b])
periodicity between phases aand b. Results are given for , , , and for several values of gand «, which characterize the shapeE=85DEb=6 d=2
of the intraindividual trade-off (eq. [4]; fig. 2). Dotted lines,.Solid lines,.Circles,.Squares,.Diamonds, . The expressiong=7 g=3 «= 0.5 «=1 «=2
m
max
was set at 6 and p
max
at 0.9. B, Ratio of reproductive effort in phase aover reproductive effort in phase bto the ratio of(RE[a]/RE[b])
reproductive value of a b-born over the reproductive value of an a-born ( ), where . Parameter values are the same as in A.C, Ratio of
v/vv=1
ba ab ab
reproductive effort in phase aover reproductive effort in phase bplotted against the ratio of reproductive value of a territorial in(RE[a]/RE[b])
the a-phase over the reproductive value of a territorial in the bphase ( ). Note that, for each phase, reproductive values are the same for both
v/v
Ia Ib
a-born and b-born territorials ( ).v=v
Aj Bj
which all produce floaters) and negligible output else-
where. It is thus never optimal for an individual to reject
a “suitable” habitat. Hence, all suitable habitat should be-
come occupied. The maximization of floater numbers with
a fixed breeding population size then means that total
population also reaches its maximum possible size.
Example: Environment with Two Phases
We calculated optimal allocation strategies for two-phased
periodicity in food availability as described above and in
appendix A. For simplicity, the phase at birth does not
affect the future survival or fecundity, and individualsborn
in phase aand bare subject to the same trade-off function
(eq. [4]). We stepwise increased the amplitude of the pe-
riodicity DEaround an initial value for several parameters
describing the intraindividual trade-off (parameters gand
«in eq. [4] and fig. 2). Clearly, once phase aand bare
unequal with respect to energy available, the allocation
strategies become adjusted according to phase (fig. 3).
Here, phase ais the phase with the highest abundance of
food. Phase awould thus qualify as “favorable” and en-
courage higher reproductive effort (sensu Hirschfield and
Tinkle 1975).
It is, however, easy to find examples that either agree
or disagree with the expectation of higher reproductive
effort in the favorable phase. In the example of figure 3,
we find that it is optimal to have a higher reproductive
effort in the favorable phase (here phase a) if survival
shows an S-shaped increase with energy allocated to sur-
vival ( ). This holds for both sharp ( ) and grad-
«=2 g=7
ual ( ) increase in reproductive output with energy
g=3
allocated to reproduction. The ratio of reproductive effort
in phase aover phase bincreases with the magnitude of
environmental variation and with increasing amplitude in
the environment, E; it eventually becomes optimal to com-
pletely abandon reproduction in the unfavorable phase b.
However, our results also show solutions where repro-
ductive effort is not higher in phase a, and a variety of
strategies can evolve. When the increase in survival shows
decreasing marginal returns and the increase in repro-
ductive output is sharp ( , ), the optimal strat-
«= 0.5 g=7
egy is to have a higher reproductive effort in the bphase
(fig. 3A). In contrast, with a more gradual increase in
reproductive output ( ), the optimal strategy is to
g=3
allocate an equal fraction of the energy in phases aand
b, except with the largest amplitude in the cycle, when
allocation in phase abecomes higher.
Clearly, with an S-shaped ( ) or initially linear
«=2
( ) increase in survival, it pays more to invest in re-
«=1
production in the phase with more energy available, since
survival increases become marginal. When the increase in
survival is much slower ( ), investment in survival,«= 0.5
thus allocating energy away from reproduction, pays off
much more in phase a, especially when there is a steep
increase in reproductive gains ( ).g=7
460 The American Naturalist
Figure 4: Schematic representation of the life cycle of the Ural owl.
Territorials produce M
Ij
offspring in spring (dashed line) and survive with
a probability S
Ij
over summer (spring–autumn) and with a survival prob-
ability W
Ij
over winter (autumn–spring). Floaters obtain a territory with
probability T
j
over winter and have survival probabilities S
ij
and W
ij
over
summer and winter, respectively. Juveniles (!3 yr old) are distinguished
from adults (≥3 yr old). For simplicity, the graph represents a generalized
life cycle, where offspring born in three phases are represented in only
one diagram (compare to fig. 1, which gives the full representation with
the separate phases).
Likewise, reproductive values do not have a straight-
forward relationship to food availability. Higher foodavail-
ability facilitates the effective fecundity, M
Ij
, for the parent;
that is, the same reproductive output can be realized with
a lower reproductive effort—this obviously tends to im-
prove her own reproductive value. However, the repro-
ductive value of the offspring produced, which weighs M
Ij
(as in eq. [3]), is not necessarily higher in phase a: their
success may be lower if they face difficulties in territory
acquisition or have to start breeding in an unfavorable
year themselves (fig. 3B,3C). In the examples shown, the
reproductive value of a territorial is always higher in phase
a(fig. 3C), whereas the reproductive value for a floater is
usually higher in phase b(fig. 3B). Importantly, the re-
productive values are a result of individual prospects dur-
ing the periodic population dynamics, which themselves
result from the allocation strategies in use, hence the va-
riety of possible strategies. Predicting optimal responses
and reproductive values is not possible without knowledge
of the dynamics and the options available to each indi-
vidual in the future.
Ural Owls and the Vole Cycle
The above model visualizes the shortest possible chain with
only two environmental phases. In reality, longer periods
exist, and predictions regarding optimal allocations may
become more complex. We illustrate this by a model de-
scribing the environmental cycle in food abundance of the
Ural owl.
Data on the Ural owl have been collected from 1977
to 1998 in a study area of approximately 2,000 km
2
, with
approximately 185 nest boxes, around Lahti, southern
Finland. Here, we used data from 1987 to 1998 (four
vole cycles) in which 617 breeding events were recorded.
Practically all females and nestlings breeding in the boxes
were caught, aged, and ringed. Juveniles (!3 yr old) can
be distinguished from adults by plumage characteristics
(Pietia¨inen and Kolunen 1986). Usually, snow conditions
were severe at the time of laying (late March–early April),
and voles were only trapped in early summer (June).
Spring vole densities, in this article, are an average of
the preceding autumn’s trappings (September–October)
and of the early summer trappings (June). For more de-
tails on the study area and the methods, see Pietia¨inen
(1989).
In line with empirical findings concerning food supply
and reproductive output of birds of prey (e.g., Meijer 1988;
Daan et al. 1990; Korpima¨ki and Hakkarainen 1991), we
interpreted regional vole density as the environment for
the Ural owl. The vole cycle itself is thought to be driven
either by the interaction between mustelid predators and
voles (Henttonen et al. 1987; Hanski et al. 1991; Hanski
and Korpima¨ki 1995) or by some intrinsic cause (e.g.,
Boonstra 1994; Inchanti and Ginzburg 1998). The impact
that predation by the bird of prey community has on vole
dynamics is unclear and may be either negligible (e.g.,
Henttonen et al. 1987; Heikkila¨ et al. 1994) or of regional
importance (e.g., Norrdahl and Korpima¨ki 1996; Korpi-
ma¨ki and Norrdahl 1998). Here we consider fluctuations
in the numbers of voles as external variation in the en-
vironment, E, from the Ural owl’s point of view (Lundberg
1981; Pietia¨inen 1988). Thus, we ignore cases in which the
dynamics of the predator-prey system are affected by the
life-history decisions of either the Ural owl as the predator
or of the voles as the prey; such interactions can have
either stabilizing or destabilizing effects (Kokko and Rux-
ton 2000).
Territoriality probably forms the most important reg-
ulation of population density for Ural owls. Suitable nest
sites form a scarce resource in the Fennoscandian land-
scape (Lundberg 1979). Consequently, Ural owls are highly
site tenacious, with practically all pairs staying together
their whole reproductive life in the same territory (Lund-
berg 1979; Saurola 1987). Hence, we can assume “ideal
despotic” density dependence (Fretwell 1972; Rodenhouse
et al. 1997), with a distinction between floater and terri-
torial stages, as described above. We model a life cycle in
which territorials and floaters are traced in spring and in
autumn, with separate stages for juveniles (!3 yr old) and
for adults (≥3 yr old; fig. 4). The principles for the op-
timization of allocation, a, for this life cycle are the same
Reproductive Values under Periodicity 461
Table 1: Summary data on the vole densities in the three phases of the vole cycle and
phase-dependent life-history aspects of the Ural owl
Phase
Vole density Age distribution
Survival FledglingsSpring Autumn 1 2 ≥3
Low 2.8 51.0 19.0 53.2 ))).95 5.02 23.3 59.4
Increase 18.8 57.1 41.0 54.6 .02 .11 .87 .85 5.05 193.2 517.4
Peak 21.5 54.1 3.8 51.4 .05 .03 .92 .65 5.06 151.0 522.1
Note: Data presented here have been collected from 1987 to 1998, except for the survival values, which
are estimates for 1981–1998 for four cycles (see also Brommer et al. 1998). The age distribution represents
the proportion of breeding females in spring in the different age classes. Here, the low phase is not
represented because the number of breeding females is on average very low ( ). The survivalmedian = 7
estimates concern breeding females only. Offspring production is given as the total number of fledglings
produced by all breeding pairs. Values are given as . Vole density data was used to describemean 5SE
the environment Ein the model, and the generated output from the model was compared with the other
12 variables.
as explained above but now with six submatrices for both
spring and autumn in three phases.
As shown by Daan et al. (1996) for the kestrel, the
mortality resulting from an experimental increase in re-
production in spring takes place in the following winter.
Southern (1970) showed that tawny owls (Strix aluco) were
most likely to die over winter. Recoveries from the Ural
owl also suggest that mortality is highest in late winter (H.
Pietia¨inen, unpublished data). We compared the perform-
ance of two models presenting two extremes in the timing
of costs: the “immediate cost” model, in which reproduc-
tive effort in the spring affects survival from spring to
autumn, and the “delayed cost” model, in which allocation
in the spring affects survival from autumn to the next
spring only, that is, during the winter following repro-
duction. Because the reproductive values of all stages in
the population are chained (eq. [3]), we simply substitute
later stages, which may still depend on the allocation de-
cision a
Ij
, into the formulation of the reproductive value
that is to be maximized (app. C).
v
Ij
The environment E, interpreted as variation in food
supply, was assumed to be a function of the observed
average autumn and spring vole densities (table 1). Since
the relation between observed vole density, as recorded by
trapping, and the actual energy available for the owls was
unknown, we assumed that energy increased either linearly
with observed vole density, that is,
E=measured vole density in phase j,
Ij
(table 1) or that it showed nonlinearity:
Î
E=5#(measured density in phase j).
Ij
The former interpretation corresponds to a Type I func-
tional response. In the latter interpretation, food is scaled
such that proportionally more energy is available with
lower vole densities than with the linear interpretation,
but a roughly similar energy is available with higher vole
densities, that is, with a Type II functional response, which
incorporates the limitation of handling time with increas-
ing prey density (Holling 1959).
As the exercise above showed, the notion that repro-
ductive effort can be predicted solely from survival pros-
pects of the juveniles is not necessarily true. Nevertheless,
for boreal owls, like the Tengmalm’s owl, this is usually
assumed a priori (Hakkarainen and Korpima¨ki 1994a).
We used our modeling approach to investigate whether or
not the notion of higher reproductive effort in the increase
phase, when first-winter survival and recruitment of off-
spring is highest (Brommer et al. 1998), can be predicted
from intraindividual and intergenerational trade-offs when
following a proper definition for the reproductive values
of offspring, generated from their future prospects in a
density-dependent population. We investigated how well
the trade-offs assumed in the “immediate cost” or the
“delayed cost” model with either Type I or Type II response
could produce dynamics that resembled the observed dy-
namics in terms of the following 12 criteria: age distri-
bution of breeding females in different phases of the cycle
(six variables); survival of breeding females (three varia-
bles); and proportions of total reproductive output that
fall into low, increase, and peak years (three variables; table
1).
Since the true relationships of the trade-off between
reproduction and survival were unknown, we sought the
best descriptors of the dynamics by systematically varying,
with seven steps within the given range, the parameters b
(range [3 )20], step length 2.4), g([1 )10], step length
1.3), d([0.1 )5], step length 0.7), and «([0.1 )3], step
length 0.48). These parameters describe possible shapes of
fitness gains from energy allocations (eq. [4]; fig. 2). The
462 The American Naturalist
maximum clutch size (m
max
) was set to equal the maximum
clutch observed: 7 (Pietia¨inen 1989). The survival, now
defined as S
Ij
and S
ij
(for the survival over summer) and
W
Ij
and W
ij
(for the survival over winter), for territorials
and floaters, respectively, had a maximum (p
max
) set at
0.999, except for the “delayed cost” model, where survival
over summer, S
Ij
, was invariably set at 1. Furthermore, the
actual ratio of availability of food between territorial and
floater (J) was also unknown and was varied between 0.2
and 1.4 (step length 0.17).
The ESS was calculated for E, and all combinations of
these five parameters varied with seven in-(b,g,d,«,J)
termediate steps within the given parameter range, giving
16,807 (7
5
) possible combinations. Results were compared
with summary data on Ural owl phase-dependent life-
history data, where the fit of the proportions in the 12
output variables considered were calculated to the pro-
portions of the observed data as .
2
(expected 2observed)
As it was anticipated that a large number of parameter
combinations could produce similar results, we selected
the 100 parameter sets with the lowest sum of squares to
form a set of plausible trade-offs, both from the “imme-
diate cost” model with Type I and Type II energy intake
and from the “delayed cost” model with Type I and Type
II energy intake. The predictions from these 400 models
were then investigated for consistent patterns.
Do Ural Owls Invest in Current Reproduction
according to Current Reproductive
Value of Offspring?
The case of the Ural owl shows the need to incorporate
several aspects of biological realism to models of repro-
ductive effort. In our study area, vole numbers fluctuate
in a three-phased cycle, which can be distinguished as low,
increase, and peak phases (Norrdahl 1995). However, food
levels do not remain constant throughout one breeding
cycle (table 1). In the low and increase breeding season,
vole density increased from spring to autumn. However,
in the peak breeding season, vole density decreased from
spring to autumn because of the inevitable crash of the
vole populations after a peak. Owlets become independent
of their parents in the autumn and, consequently, first-
winter survival of juveniles varied according to the autumn
density of voles, with low survival after the peak breeding
season (Brommer et al. 1998). The mortality pattern of
adults showed a similar pattern, with the lowest survival
rates after the peak breeding season (table 1; Brommer et
al. 1998).
The reproduction of the Ural owl varied accordingly
(table 1; Pietia¨inen 1989). Although the average density
of voles in spring did not differ between increase and peak
phase ( , , ), the average number oft=0.671 df = 6 P=.53
fledglings produced per pair was significantly lower in the
peak phase (increase [ ]: 2.8 [0.07 SE]; peak [n=280 n=
]: 2.2 [1.4 SE]; , , ). A relatively275 t=4.93 df = 553 P!.001
small number of offspring was produced in the low phase
and, consequently, the age distribution of first breeders
(Brommer et al. 1998) and also of adults (table 1) showed
a gap in which there were few 1 yr olds present in the
increase phase and few 2 yr olds present in the peak phase.
In comparing the best models of each category, we find
that it was optimal in all of these four models to allocate
proportionally more energy in the increase phase (fig. 5).
The residual reproductive value of the parent decreased
from the low, through the increase, to the peak phases for
all these models, reflecting, to a large extent, the differences
in survival probabilities between phases, since territorials
do not lose their territory in life. However, the outcome
varied markedly between models with respect to the re-
productive value of a newborn. First, in the best “im-
mediate cost” models (fig. 5A,5B) and in the best “delayed
cost” model with Type I energy intake (fig. 5C), repro-
ductive value of a newborn decreased from the low (which
was defined as 1), through the increase, to the peak phase.
However, for the “delayed cost” model with Type II energy
intake, the reproductive value of a newborn was highest
in the increase phase (fig. 5D). Second, reproductive values
of offspring differed dramatically between phases in the
“immediate cost” model, in which the reproductive value
of offspring born in the peak phase was only 42% of the
value of a low-born juvenile when energy intake followed
a Type II response (fig. 5B). In contrast, the variation in
reproductive value of newborns in the “delayed cost” mod-
els was much more restricted; for example, the value of a
peak-born juvenile was a full 90% of that of a low-born
juvenile for the best “delayed cost” model with Type I
energy intake (fig. 5C).
When considering the larger group of 100 best models
per model category in more detail, we find a similar re-
lationship between residual reproductive value, as the best
models in each category (fig. 5) showed, that is, a decrease
in residual reproductive value from the low, through the
increase, to the peak phase. All 400 models predicted very
little or no effort for low years. However, the pattern of
figure 5 becomes less clear when comparing reproductive
value of newborns and reproductive effort between peak
and increase phases (fig. 6). For example, in the “imme-
diate cost” models, solutions could be found in which the
reproductive effort in the peak year is similar to or higher
than that in the increase year. Likewise, for the “delayed
cost” model with Type I energy intake, reproductive values
could also be higher for peak born than for increase-born
offspring, although reproductive effort was still higher in
the increase phase.
Turning to the fits of the respective models, we find that
Reproductive Values under Periodicity 463
Figure 5: Four best-fitting models describing the life cycle of the Ural owl. A, “Immediate cost” model with Type I energy intake. B, “Immediate
cost” model with Type II energy intake. C, “Delayed cost” model with Type I energy intake. D, “Delayed cost” model with Type II energy intake.
Graphs show the optimal reproductive effort (closed circles), the reproductive value of a newborn in the autumn of the phase under consideration
(open circles), and the residual reproductive value of the parent (squares) in each phase of the cycle. The reproductive values of increase-born offspring
and, in some models, those of low-born offspring, too, are higher than the reproductive values of peak-born offspring. Parental residualreproductive
value declines from the low, through the increase, to the peak in all four models, and reproductive effort is higher in the increase phase than in
the peak phase.
the “immediate cost” model failed to produce a satisfying
fit to the dynamics of the Ural owl throughout the pa-
rameter space (table 2; fig. 6). For example, the best-fitting
“immediate cost” model predicted survival values of adult
territorials to be highest in the increase phase and roughly
equal for low and peak phases, thus failing to predict the
improved survival after low reproduction in the low phase,
as found in the data (table 1). By contrast, the “delayed
cost” model with the best overall fit (with Type II energy
intake) could capture the dynamics well. Moreover, the
“immediate cost” model predicted that most mortality
would occur during summer, at least in the low and in-
crease phases of the cycle (table 2), whereas the natural
pattern of mortality, as established from ring recoveries of
banded breeding females, suggests a higher mortality in
winter in increase and peak phases (H. Pietia¨inen, un-
published data).
While the “delayed cost” model was clearly better at
explaining the Ural owl dynamics than the “immediate
cost” model, and, within these two categories, a Type II
response better than a Type I response, distinguishing be-
tween similar alternative parameter values for b,g,d, and,
to a lesser extent, «and Jis not possible based on our
data set. The 100 best-fitting models showed a continuous
gradual increase in model deviations, while having con-
siderable spread in parameter values. We obtained the most
464 The American Naturalist
Figure 6: Relationship between reproductive effort in the peak versus in the increase phase and the reproductive value of young born in the peak
versus in the increase years in the 100 best-fitting models of the life cycle of the Ural owl. A, Type I energy intake in the “immediate cost” model
with sum of squares and other parameter ranges , , , , andSS = [0.080 )0.095] b= [8.67 )20] g=[1)10] d= [0.92 )1.73] «= [0.58] J=
.B, Type II energy intake in the “immediate cost” model with , , , ,[0.2 )1.4] SS = [0.047)0.060] b= [8.67 )20] g= [2.5 )10] d= [1.73 )5]
, and . C, Type I energy intake in the “delayed cost” model with , ,«= [0.58 )1.07] J= [0.4 )1.4] SS = [0.014)0.029] b=[3)20] g=
, , , and . D, Type II energy intake in the “delayed cost” model with ,[1 )8.5] d= [1.73 )3.37] «= [0.58] J= [0.2 )1.4] SS = [0.012)0.019] b=
, , , , and .[8.66 )20] g=[1)10] d= [2.55 )5] «= [0.58] J= [0.4 )1.4]
consistent estimate for the parameter «, which sets the
shape of the survival curve (fig. 2) and which was typically
lower than 1; for most models, except for 22 of«= 0.58
the 100 best “immediate cost” models with Type II energy
intake that had . Thus, realistic output was most«= 1.067
likely obtained by assuming a survival function with de-
creasing marginal returns. Furthermore, all models were
relatively insensitive to the parameter J, which sets the
ratio of energy available to floaters in relation to territo-
rials; well-fitting results could be obtained with both more
Reproductive Values under Periodicity 465
Table 2: Criteria used and performance of best-fitting model for four assumptions concerning timing of trade-off and energy intake
Cost/energy intake
Age distribution of breeders Survival of breeders
Increase Peak Low Increase Peak Fledglings
12≥31 2≥3SWSWSWLIP
Observed:
a
.02 .11 .87 .05 .03 .92 .95 ).85 ).65 ).06 .53 .42
Expected:
Immediate/Type I
b
.00 .13 .88 .05 .00 .95 .73 (.72) .98 .85 (.84) .99 .89 (.71) .79 .00 .49 .51
Immediate/Type II
c
.00 .31 .69 .24 .00 .76 .65 (.63) .96 .65 (.65) .99 .76 (.55) .71 .00 .48 .52
Delayed/Type I
d
.02 .14 .86 .04 .00 .96 1.00 (.95) .95 1.00 (.94) .94 1.00 (.61) .61 .02 .54 .44
Delayed/Type II
e
.00 .11 .89 .06 .00 .94 1.00 (.95) .95 1.00 (.84) .84 1.00 (.65) .65 .00 .52 .48
Note: S, survival over summer. W, survival over winter; number of fledged young in low (L), increase (I), and peak (P) phases. Criteria are as in table 1,
but offspring production has been transformed to proportions within the cycle, to gain a comparable scale. For the model output, the survival estimates of
summer and winter are given separately. However, the total survival over the whole season (summer times winter survival), as given between parentheses,
was used for comparison with the observed data. The fit of the different models were compared using sum of squares (SS).
a
Survival values indicate .S#W
Ij Ij
b
,,,,,.SS = 0.08 b= 11.5 g= 5.5 d= 1.73 «= 0.58 J= 1.0
c
,,,,,.SS = 0.047 b= 14.3 g= 8.5 d= 2.55 «= 0.58 J= 0.6
d
,,,,,.SS = 0.014 b= 14.3 g= 2.5 d= 3.37 «= 0.58 J= 0.8
e
,,,,,.SS = 0.012 b= 11.5 g=1 d= 2.55 «= 0.58 J= 0.8
energy available and with less energy available for floaters
compared to territorials, as illustrated in table 2 and figure
6 (legend).
In conclusion, the relationship between reproductive ef-
fort and reproductive value of offspring does not seem to
be straightforward (fig. 6A,6B,6D) or may even be neg-
ative (fig. 6C), depending on the assumptions on timing
of costs and energy intake. Furthermore, allocation to re-
production is usually negligible in low phases, although
the reproductive value of offspring is typically quite high
(fig. 5). Clearly, the shortage of food itself prevents re-
production in the low phase, which underlines the limited
validity of any argumentation which solely considers the
current reproductive value of offspring.
Discussion
Our method provides an important extension in deriving
predictions of timing of reproduction and optimal repro-
ductive effort under predictable environmental conditions.
Traditionally, timing choices have be studied without tak-
ing into account varying offspring values: examples in-
clude age at maturation (Cole 1954; Caswell 1982) or the
more general question of how reproductive effort varies
with age (Schaffer 1974b; Hirschfield and Tinkle 1975;
Pianka and Parker 1975; Charlesworth and Le´on 1976).
However, when considering life-history optimization in a
more realistic perspective, allowing for variation in the
environment, one needs to recognize that offspring pro-
duced at different times may be of different value (e.g.,
Taylor 1990; Marrow et al. 1996; McNamara and Houston
1996). The simplest example is provided by “rselection”
in exponentially growing populations (e.g., Stearns 1992),
but earlier born offspring may also be more valuable
within a single season (Daan et al. 1990). Likewise, the
value of offspring may differ between seasons (Korpima¨ki
1988a; Rohner 1996; Brommer et al. 1998).
We have shown that in an environment varying with a
certain periodicity, occurring either within a single breed-
ing season or between breeding seasons, the reproductive
values of offspring are essentially different between the
phases of the period. Apart from the intraindividual trade-
off, which affects the survival within one breeding season,
such variations also affect parental residual reproductive
values indirectly, through a recursive “chaining” of future
states into the components of current parental reproduc-
tive value. Hence, reproductive effort cannot be expected
to vary linearly with offspring reproductive value; optimal
allocation strategies can be considerably more complex.
Straightforward reasoning, where costs and benefits can
be easily identified, with the current reproductive value of
the offspring and the future reproductive value of the par-
ent fixed and only the fertility and the survival components
left to vary (sensu Williams 1966) becomes impossible
when allocation decisions directly affect the population
size and thus, via density dependence, the value of the
offspring produced.
The consequences of this “chaining” are often not fully
recognized when discussing intergenerational trade-offs in
terms of reproductive value of parent and offspring. Daan
et al. (1990) showed that the first-winter survival and re-
cruitment probability of kestrel young declined through
the breeding season but did not incorporate these differ-
ences in the calculation of the future component of re-
466 The American Naturalist
productive value. In the calculation of the offspring’s re-
productive value and in the calculation of the parental
residual reproductive value, all offspring produced in the
future had reproductive value 1 (eqq. [2] and [3] in Daan
et al. 1990; Daan and Tinbergen 1997). Likewise, Hak-
karainen and Korpima¨ki (1994b) found that males of the
Tengmalm’s owl living in a environment comparable to
that of the Ural owl defended the nest more vigorously in
the increase phase, when the offspring had a higher sur-
vival probability, and concluded that parental residual re-
productive value could be ignored when considering the
owl’s reproductive effort, since the probability to breed in
the next year was higher in the increase phase. Again, a
higher probability to breed in the next phase is not nec-
essarily a good predictor of residual reproductive value,
especially when the value of the young produced is lower
in the next phase as well.
Consequences of varying offspring survival for optimal
reproductive decisions are not easily interpreted for pop-
ulations living in a density-dependent setting. Fluctuations
in population size are necessarily linked to both allocation
decisions and environmental fluctuations (Pa´ sztor et al.
1996). The modeling framework we presented here is dy-
namic, taking the coupled environment and the density
effects into account in calculating the optimal reproductive
effort. Using simple numerical examples, we can show a
wide variety of stable strategies. Deriving simplistic pre-
dictions such as increasing the reproductive effort in years
favorable for the survival of offspring (Korpima¨ki 1988a;
Lindstro¨m 1988; Hakkarainen and Korpima¨ki 1994a,
1994b; Amat 1995; Tolonen and Korpima¨ki 1996) can feel
intuitive but be misleading, unless the likely future of all
descendants and the parent can be tracked.
Despite this, results from modeling the Ural owl con-
firmed that it is possible to construct models with higher
reproductive effort in the increase phase, when offspring
survival and recruitment is higher. This idea has been fol-
lowed in empirical work on the Tengmalm’s owl (Hak-
karainen and Korpima¨ki 1994a) and can now be justified
theoretically as well. However, since many different trade-
offs were found to produce this outcome, it is difficult to
state which factors will favor higher allocation to repro-
duction in the increase year with slightly lower vole density
than in the peak year. Some alternatives, however, could
be rejected: models that deny the existence of delayed re-
productive costs never produced good fits. Our approach
of incorporating these delayed reproductive costs excluded
mortality during summer, although one could incorporate
both immediate and delayed costs (e.g., Jo¨ nsson etal. 1995;
Doebeli and Blarer 1997). For most well-fitting models,
reproductive value of young produced was somewhat
lower in the peak phase than in the increase phase, thus
facilitating the interpretation that parents do adjust effort
according to reproductive values of their offspring.
Further encouragement, in terms of consistency, can be
found in the constancy of the shape of the survival curve
with increasing energy allocated to survival; in line with
verbal reasoning (Kacelnik 1989) and experimental evi-
dence (Daan et al. 1990, 1996), most well-fitting models
showed decreasing marginal returns for the survival curve,
as opposed to an S-shaped increase. Furthermore, the
assumption of a Type II functional response of energy
availability with respect to prey density facilitated model
fits both when costs were immediate and when they were
delayed. Indeed, predation rates of tawny owls in eastern
Poland followed a Type II functional response (Jedrze-
jewski et al. 1996).
In addition, results on modeling the Ural owl’s life cycle
seemed insensitive to assumptions about the relation of
energy available to floaters compared to that available to
territorials. It is likely that floaters have less energy avail-
able than territorials, since they possibly spend more en-
ergy in trying to acquire a territory (e.g., Ens et al. 1992;
Rohner 1997) and may depend on less reliable food re-
sources. Our results indicate that measuring this precisely
is less important than knowledge of the reproductive trade-
offs faced by adults.
Our approach can demonstrate the variety of possible
strategies in externally driven environmental cycles, as well
as their potential to provide explanations for observed life-
history patterns. Several further developments are possible,
as our examples necessarily do not cover all potentially
important factors. We have, for example, not modeled any
longer-lasting effects of phase at birth on reproduction
and survival. Furthermore, we assumed that all floaters
were equal in obtaining a territory. This is mathematically
convenient, as only floaters are subjected to density de-
pendence, which aids in reaching stable population dy-
namics. However, from an empirical point of view, this
may be unrealistic in some species where, for example,
older and experienced individuals have an advantage.
Moreover, from a theoretical point of view, this limits the
consideration of the ESSs to singular strategies. Especially
in variable environments, multiple stable strategies may
be expected, but these require that the feedback environ-
ment, that is, the part of the population subjected to den-
sity dependence, is also multidimensional (Heino et al.
1998).
Regarding the Ural owl specifically, our assumption that
the owl dynamics do not affect the dynamics of voles may
be partly unjustified, and relaxing this assumption can lead
to a variety of dynamical outcomes (Kokko and Ruxton
2000). The extent to which Ural owls themselves can affect
the dynamics of their prey is currently unknown because
Reproductive Values under Periodicity 467
the owls have a territory size of 5–10 km
2
and because
their predation impact is probably only locally important.
Adding further stages to reproductive value models en-
ables us to consider further aspects of life-history evolu-
tion. Males also have reproductive values, and relaxing the
assumption of a 1 : 1 sex ratio would allow us to build
chains of reproductive values that would include both
sexes (Pen et al. 1999; H. Kokko and J. Brommer, un-
published manuscript). Also, the abundance of food,
which is available to females, is identical for all territories
in our model, although both males and territories often
vary in their quality (e.g., Newton and Marquiss 1986;
Korpima¨ki 1988b). Likewise, spatial variation in food
availability may form the major determinant in the sea-
sonal decline of clutch size with advancing laying date
(Meijer 1988; Daan et al. 1990) and should therefore be
incorporated in a full analysis of seasonal reproductive
effort. Finally, solving life-history strategies in stochasti-
cally varying environments follows, in principle, a similar
“chaining” logic (Tuljapurkar 1989; McNamara 1995,
1997).
Acknowledgments
We wish to thank J. M. McNamara and I. Pen for their
helpful comments. J.B. wishes to thank P. Brommer for
fruitful discussions on matrices and on his critics. J.B. was
financially supported by the Helsinki School of Conser-
vation and Management of Wildlife Populations (LUOVA)
Graduate School (Academy of Finland) and benefited from
a European Science Foundation travel grant to Cambridge.
H.K. was financed by the Training and Mobility of Re-
searchers program of the European Community.
APPENDIX A
In a periodically varying environment with zphases, there are a succession of population vectors ,N(t)N(t))N(t)
12 z
within one period. These population vectors are linked through transition matrices such thatL(t)L(t))L(t)
12 z
N(t)=L(t)N(t) for i!z,
i11ii
N(t11) = L(t)N(t). (A1)
1zz
Note that the matrix L
j
(t) is not necessarily fixed. Its elements are allowed to vary according to changes in external
or internal factors such as, for instance, population density or changes in allocation strategy. In this case, we consider
an externally induced period in the environment with zphases and thus also zpopulation vectors corresponding to
these. Calculation of population dynamics is always done through (A1) using the L
j
(t) matrices.
At equilibrium, and and all transitions in one period can be described by the blockN(t11) = N(t)L(t11) = L(t)
jjjj
matrix
)
00 0 L
z
)
L000
1
)
B=0L00. (A2)
2
__5 _ _
00)L0
z21
In the case of a cycle with two phases, aand b(as in fig. 1 and as discussed in the text), we have
N
Aa
N
aa
N=,
A
N
Ba
N
ba
N
Ab
N
ab
N=, (A3)
B
N
Bb
N
bb
468 The American Naturalist
NL(t)L(t)0 N
ABA A
N(t11) = (t11) = #(t), (A4)
N0L(t11) L(t)N
BABB
where we have written equation (A1) in the form of equation (1). At equilibrium, and the transitionsN(t11) = N(t)
described by the matrix in (A4) can be reduced to the block matrix B. We emphasize that matrix Bcan only be used
to describe the dynamics from tto when Nis stable (unlike the usage of Ain eqq. [1] and [A4] and in contrastt11
to Gourley and Lawrence 1977); this is because the elements within N(t), which refer to the different phases, only
correspond to elements within N(t11) when Nis stable.
At equilibrium, , wherel(B)=1
1
0L
B
B=,
L0
A
or, when written out fully:
0000PPT00
Ab ab b
()
00000P12T00
ab b
0000MT 0P1MT PT
Ab b Bb Bb b bb b
() ()()
0000M12T0M12TP12T
Ab b Bb b bb b
B= . (A5)
P1MT PT MT 00000
Aa Aa a aa a Ba a
()()()
M12TP12TM12T00000
Aa a aa a Ba a
00PPT0000
Ba ba a
()
000P12T0000
ba a
APPENDIX B
We consider cases where density dependence renders the population vector Nstable. At the ESS, population growth
rate, l, must be maximal ( ). We reach this point by maximizing l
1
(B) through the left eigenvector. Thel/a=0
1Ij
allocation strategy, a
Ij
, of a territorial born in phase iand breeding in phase jis subjected to a trade-off between the
fertility, M
Ij
, and the survival, P
Ij
. The advantage of using the left eigenvector, that is, the reproductive values, is that
changes in allocation, a
Ij
, only occur within one stage, that is, in
vl=M(a)[Tv1(1–T)v]1P(a)v,(B1)
Ij Ij j j Ij Ij
Ij J k jk Ik
with phase kfollowing phase j. Here, an upper-case first index in the left eigenvector elements (e.g., M
Ij
,,)vv
Ik Jk
indicates a territorial, and a lower-case first index indicates a floater ( ). For maximizing l
1
(B), floaters are notv
jk
considered, since they cannot choose their allocation and thus cannot alter any of the elements of the projection matrix
B, which are all given by the population dynamics. In finding the ESS, we consider the invasibility of a mutant in the
resident (predominant) population (sensu Metz et al. 1992). We may assume that only the elements dependent on a
Ij
are left to vary in (B1), since the other elements are determined by the resident population. In the maximization
procedure, we find the a
Ij
that maximizes the right-hand side of (B1), which will lead to a larger value for l(e.g.,
McNamara 1993). The reproductive values , , and and the probability of obtaining a territory, T
j
, are assumedvv v
Jk jk ik
to be determined by the resident population. Thus, the strategy , which is given by the maximization of
0
a(E,N)
res
the left eigenvector (reproductive values), can invade a population playing strategy a
res
()ifE,N
res
, except when is the maximum l
1
, which is the ESS. In
0
l{B[a(E,N), E,N]} 1l{B[a(E,N), E,N]} l=1
1res res 1res res res 1
order to find this point, the dynamics are iterated again until the population vector Nreaches equilibrium. Then
Reproductive Values under Periodicity 469
the maximization procedure is repeated. We thus iterate
00
(E,N)rB[a(E,N), E,N]ra(E,N)r(E,N)r
, and, by repeating this until and , we
00 0 000 00 ∗ ∗
B[a(E,N), E,N]ra(E,N)r(E,N)l{B[a(E,N), E,N]}=1 l/a=0
11
find the evolutionary stable allocation strategy . In practice, the iteration always converged quickly, in about
∗
a(E,N)
four to five rounds, for the models treated in this article. In general, however, such simple iterative procedures may
not always be able to find the ESS (J. M. McNamara, personal communication).
Matrix Bis irreducible, since every node in the life cycle graph has a path to every other node. Matrix Bhas an
imprimitivity index of 2, since the greatest common divisor of all path lengths back to the same node equals 2 and
thus also has two eigenvalues of the same absolute magnitude, of which one (l
1
) is real and positive (see Caswell
1989). Although only the transition matrix in (A4) and not the matrix Bcan be used to calculate the population
dynamics, maximization of Band the matrix in (A4) leads to the same result but avoids using the matrix products.
To see this, consider that is equivalent to (e.g., Taylor 1990). This, for the example discussedl/=0 v(B/)u=0
1aa
Ij Ij
above, gives
00u
A
[vv]=0,
0∗
AB
L(a)0 u
AIa B
0∗
vL(a)u=0. (B2)
AIa A
B
Because in our case , , we can rewrite (B2) asl=1 vB=v
1
0∗
vLL(a)u=0. (B3)
BA Ia A
A
A similar analysis for the transition matrix in (A4) gives
0∗ 0∗
vLL(a)u1vL(a)Lu=0. (B4)
B A Ia A A Ia B B
AB
Here, the first term equals 0, as it is the same as (B3). The second term also equals 0, as it is equivalent to (B2)
because and thus .LLu=lu(LL)Lu=(LL)u=lLu
ABB B BA BB BA A BB
APPENDIX C
We can write down directly from the generalized life cycle in figure 4 that
vl=M(a)v1S(a)v,(C1)
Ij Ij Ij Ij
(spri ng)Ij (autumn)jj (autumn)Ij
when allocation in spring affects the survival over summer, S
Ij
(“immediate cost” model). For the “delayed cost” model,
S
Ij
is set to 1, and allocation in spring affects the winter survival, W
Ij
. Because of the “chaining” process derived in
equation (3), we can likewise write
vl=M(a)v1v1W(a)v(C2)
Ij Ij Ij Ij
(spri ng)Ij (autumn)jj (aut umn)Ij (spr ing)Ik
for phase kfollowing phase j.
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Associate Editor: Steven N. Austad
