Optimal onset of seasonal reproduction in stochastic
environments: When should overwintering small rodents start
breeding?
Torbjørn Ergon
Address:
Centre for Ecological and Evolutionary Synthesis, Department of Biology, University of Oslo, P.O.
Box 1050 Blindern, 0316 Oslo, Norway
e-mail: torbjorn.ergon@bio.uio.no
phone: +47 22854608
fax: +47 22854001
Running head:
Ergon: Optimal onset of reproduction.
Typography:
Equations and mathematical symbols are inserted as “Microsoft Equation 3.0” objects or
typed in Italic Times New Roman. All other text is in Arial. Words to be set in italics are
underlined.
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SECTION
Subsection
Subsubsection
ABSTRACT
Theories for optimal life-history strategies in variable environments have until now focused
on cases where the individuals have either no information about the environment (models
maximizing geometric mean fitness) or full information about the environment (models
predicting optimal reaction norms). In this paper I investigate the optimal time for multivoltine
organisms to commence seasonal reproduction in a more general and realistic case where
animals perceive the state of their environment through cues that are measured with varying
degrees of precision. If there were only a trade-off between early reproduction and high
reproductive success, and if animals had perfect information about their environment, it
would be optimal to commence reproduction when the rate of change in reproductive
success relative to its current value equals the difference between population growth during
the reproductive and non-reproductive seasons. This implies that reproductive success at the
optimum is independent of when (but not how) the environment improves over the season.
However, because it is optimal to respond conservatively to uncertain cues, we should
expect higher reproductive success during years when breeding conditions improve early
than when they improve late. Nevertheless, a phenotypic correlation between reproductive
success and timing of reproduction will probably not be detectable in a stochastic
environment. Data from a cyclic population of field voles (Microtus agrestis) in northern
England show a negative correlation between reproductive success and timing of
reproduction among out-of-phase locations. Such a pattern may occur when there is a
convex trade-off between pre-breeding survival and timing of reproduction, or if animals
precipitate reproduction to avoid senescence when the environment improves late.
KEYWORDS: environmental stochasticity, imperfect information, Kalman filter, life-history
evolution, optimality model, population cycles.
2
INTRODUCTION
Multivoltine organisms have several generations per year and may breed repeatedly during
the reproductive season (Roff 1992). In seasonal environments, however, reproduction
typically ceases during seasons when the environmental conditions are less favorable,
usually the winter or the dry season. Individuals that endure these seasons face the problem
of when to resume breeding when the environment improves (e.g. Fairbairn 1977; Roff
1992).
Early reproduction, at a time when there are little resources and the environmental
conditions are hostile, may involve reduced fecundity and possibly complete reproductive
failure and death of the parent. In univoltine organisms with long development time of the
offspring, such as large mammals and birds, a major cost of late reproduction is poor viability
of the offspring because they have shorter time to grow or accumulate resources before the
unfavorable season (Lack 1966; Clutton-Brock et al. 1987). This cost is, however, probably
of little importance in multivoltine organisms with fast development time. Instead, multivoltine
organisms face costs of late onset of reproduction because fewer generations may complete
reproduction before the winter, and because there is a higher chance that the parent will die
before reproducing. These costs will depend on the population growth rate during the
reproductive season and the pre-breeding survival rate of the overwintering animals, both of
which are highly variable in small rodent populations (e.g. Krebs & Myers 1974; Stenseth
1999).
Indeed, small rodents in seasonal environments, and especially in populations with
multi-annual density fluctuations, show tremendous variation in the time that reproduction is
initiated. The start of the breeding season in small rodent populations typically varies over a
range of 3 to 10 weeks among years (e.g. Krebs & Myers 1974; Sharpe & Millar 1991; Ergon
2003), and individual variation within the same year may be of similar magnitude (Fairbairn
1977; Millar & Innes 1983; Lambin & Yoccoz 2001). The variation in the seasonal patterns of
reproduction between different environments or habitats may also be extensive (Millar 1984;
Bronson 1985; Bronson & Perrigo 1987; Sharpe & Millar 1991). Since this large variation in
onset of spring reproduction may be responsible for a substantial variation in the population
growth and fitness of overwintering individuals (Fairbairn 1977; Oli & Dobson 1999; Lambin &
Yoccoz 2001), it should be of great interest to understand the mechanisms responsible.
Using an optimality model I here investigate general mechanisms that may be
responsible for the variation in onset of seasonal reproduction in multivoltine organisms. I
focus on the trade-off between high success of the first breeding attempt and early
3
reproduction, but I also investigate how dependencies between pre-breeding survival and
onset of reproduction (due to trade-offs, senescence or seasonal variation in survival) will
influence the optimal strategies. In particular, I consider cases where animals use cues that
do not carry precise information about the environmental states.
Theories for optimal life-history strategies in stochastic environments have almost
invariably assumed one of two extremes: At one extreme, it is assumed that the animals
have perfect information about variations in their environment, and optimal reaction norms
are derived (e.g., McNamara & Houston 1996; Roff 2002). At the other extreme, it is
assumed that animals have no information about the environment and one studies how
environmental stochasticity affect the optimal fixed strategies (e.g., risk aversion and bet
hedging (Yoshimura & Clark 1991; Menu & Desouhant 2002; Roff 2002; Hopper et al.
2003)). Nevertheless, in many situations animals have probably evolved responses to
information (cues) that do not precisely reflect the state of the environment (e.g., ‘rules of
thumb’ (Stephens & Krebs 1986)). The approach I take here is to use a simple linear
weighting of the information in the cues, and the optimal weighting constants are found by
numerical simulations. I use this to predict norms of reaction as well as phenotypic variations
and correlations when animals respond optimally to imperfect information about their
environment. Finally, I analyze observational data on survival costs of reproduction in
fluctuating populations of field voles (Microtus agrestis, L.) in northern England, and interpret
the observed patterns in the light of the model predictions.
THE MODEL
Let T be the fraction of a year between the end of the breeding season and the onset of
reproduction under a given strategy ( 0 ≤ T ≤ 1 ). The end of the breeding season is assumed
to be independent of the strategy, so that higher T means later onset of reproduction. If
there are N t individuals following a given strategy (value of T ) just after the breeding
season in year t , then the number of descendants one year later will be
N t +1 = N t S Tp S r mmax e r (1−T )
[1]
where S p is the pre-breeding survival rate on a yearly time scale, S r is the reproductive
success defined as the fraction of maximum number of offspring plus the parent, mmax , that
survive the first breeding attempt, and r is the population growth rate during the rest of the
reproductive season. That is, a fraction S Tp of the N t individuals survive until the start of the
4
breeding season, when they each contribute S r mmax new individuals to the population that
will grow at rate r over the breeding season of length 1 − T .
Dividing by N t and taking the logarithm on both sides gives the yearly growth rate, or
fitness, W , of the strategy
⎛N ⎞
W = ln⎜⎜ t +1 ⎟⎟ = T ln( S p ) + ln( S r ) + r (1 − T ) + C
⎝ Nt ⎠
[2]
where the constant C equals ln(mmax ) , and where ln( S p ) and ln( S r ) are negative numbers
since 0 ≤ S p ≤ 1 and 0 ≤ S r ≤ 1 .
Without constraints, fitness would increase with earlier onset of spring reproduction
(lower T ) because both a higher probability of surviving until reproduction (first term of Eq.
(2) and, whenever r is positive, because of a longer reproductive season (third term of Eq.
(2). However, there is most certainly a trade-off between early reproduction (low T ) and high
reproductive success (high S r ), and possibly also between a low T and high pre-breeding
(winter) survival ( S p ). In the following I will assume that the strategy determining T has zero
genetic covariance with life-history traits other than S r and S p (i.e., r and the end of the
breeding season are independent of the strategy).
Trade-off between early reproduction and high reproductive success
The trade-off between early reproduction (low T ) and high reproductive success (high S r )
may be modeled by a function, S r = g (T ) , where 0 ≤ g (t ) ≤ 1 when 0 ≤ T ≤ 1 .
The values of T that maximize fitness, given this trade-off, are found by substituting
S r = g (T ) in Eq. 2 and setting the first derivative to zero,
∂W
= ln( S p ) + h(T ) − r = 0
∂T
[3]
where h(t ) = g ' (T ) / g (T ) . Values of T that satisfy this expression correspond to a peak in
fitness whenever the second derivative, ∂ 2W / ∂T 2 = h ' (T ) , is negative. In addition, fitness
may be maximized on the T = 0 or T = 1 boundaries.
Thus, whenever optimal onset of reproduction ( T = T * ) is not on the boundaries (i.e.,
whenever seasonal reproduction is optimal) we have that
5
h(T * ) = r − ln( S p )
[4]
and h' (T * ) < 0 , where h(T ) is the rate of change in S r relative to its current value, and
r − ln( S p ) is the difference between summer population growth and the logarithm of winter
survival. We thus obtain the following predictions, that apply whenever seasonal reproduction
occurs (i.e., T * ≠ 0 and T * ≠ 1 ):
Prediction 1: Since h' (T * ) < 0 , h(T ) will cross h(T * ) = r − ln( S p ) from above when
T increases. This means that, if the relationship between S r and T ( S r = g (T ) ) remains
unchanged, optimal onset of reproduction always will occur earlier (lower T * ) when
r − ln( S p ) is higher. An increase in r by one unit has the same effect on T * as a decrease
in ln( S p ) by the same unit. This is illustrated in Figure 1A.
Prediction 2: If the relationship between reproductive success and time of
reproduction, S r = g1 (T ) , changes to g 2 (T ) so that g 2 (T ) = g1 (T + ∆ ) (i.e., breeding
conditions improve earlier or later but change in the same manner over time), then optimal
reproductive
success
will
remain
unchanged
(i.e.,
S r* = g 2 (T2* ) = g1 (T1* ) ,
where
T2* = T1* − ∆ ). That is, optimal reproductive success ( S r* ) is independent of when S r
improves, although it is dependent on how it improves (i.e., functional form). See Figure 1B.
Prediction 3: It follows from Prediction 2 that if the spring phenology, in terms of
S r = g (T ) , is precipitated/delayed by ∆ (i.e., g 2 (T ) = g1 (T + ∆ ) ) optimal T = T * will be
precipitated/delayed by ∆ too (Fig. 1B).
Thus, there are two main mechanisms for modulation of optimal T * , as illustrated in
Figure 1: a response to variable r − ln( S p ) (Prediction 1; Fig. 1A), and a response to
variable g (T ) (Prediction 2 and 3; Fig. 1B).
The above predictions apply for any differentiable function g (T ) as long as
0 < T * < 1 . I derive more specific predictions in the example below.
Example 1:
Assume that success of the first breeding attempt, S r = g (T ) , increase according to a
logistic function over time
6
Sr = g (T ) =
a
1+ e
− b(T − c )
[5]
where a is the season-independent component of reproductive success, b is a positive
value determining how fast S r increases and c is the value of T where S r = 0.5a (which is
also the inflection point). This gives
h(T ) =
g ' (T )
b
=
g (T ) 1 + eb ( T − c )
[6]
Under this form of g (T ) the second derivative of fitness, ∂ 2W / ∂T 2 = h ' (T ) , is
always negative. Hence, any value of T = T * that satisfies h(T * ) = r − ln( S p ) (Eq. 4)
represents a peak in fitness. These forms of g (T ) and h(T ) are plotted in Figure 2.
When T * ≠ 0 and T * ≠ 1 the optimal T = T * is
⎞
1 ⎛
b
− 1⎟
T * = c + ln⎜
b ⎜⎝ r − ln( S p ) ⎟⎠
[7]
and reproductive success at optimum, S r = S r* , is
⎛ r − ln( S p ) ⎞
S r* = a ⎜⎜1 −
⎟⎟
b
⎝
⎠
[8]
Note that T * is independent of a , and S r* is independent of c when T * is not on any of the
boundaries. We may confirm that Prediction 1 to 3 hold for this special form of g (T ) by
inspecting Eqs. 7 and 8.
When r − ln( S p ) > h (T = 0) then T * = 0 (as is the case for point C1 in Fig. 2). This is
always the case when r − ln( S p ) > b since 0 ≤ (1 + e − bc ) −1 ≤ 1 . In other words, nonreproducing animals should not delay reproduction when population growth rate ( r ) is high,
pre-breeding survival ( S p ) is low and breeding conditions improve slowly (low b ). Thus,
year-round reproduction should be expected.
When r − ln( S p ) < h(T = 1) then T * = 1 (i.e., never mature in that year; C2 in Fig. 2).
This is always the case when r ≤ ln( S p ) (i.e., in years with lower population growth in the
summer than in the winter). However, the form of S r = g (T ) here used may not be realistic
7
for values of T * close to 1, as it is unlikely that g (1) is much higher than g (0) . I present a
more realistic example below.
Example 2:
In the rest of the paper I will assume a function of S r = g (T ) that declines at high T (late in
the year),
g (T ) =
(1 + e
− b1 ( T − c1 )
a
)(1 + e − b2 ( T − c2 ) )
[9]
with the corresponding
h(T ) =
g ' (T )
b1
b2
=
+
b1 ( T − c1 )
g (T ) 1 + e
1 + eb2 ( T − c2 )
[10]
(Fig. 3). This function of g (T ) may be interpreted as having three components to expected
reproductive success ( S r ): a is the season independent component, (1 + e − b1 ( T −c1 ) ) −1
determines when and how fast breeding conditions improve in the spring, and
(1 + e − b2 ( T −c2 ) ) −1 determines when and how fast breeding conditions decline in the fall.
Optimal strategies in stochastic environments
If growth rate of a strategy, W = ln( N t +1 / N t ) , varies from year to year in a stationary
process, then the long term growth of the strategy is described by the mean growth rate, W
(Caswell 2001, chap. 14.3). Hence, if S p , S r and r vary between years ( i ‘s), then the
optimal fixed strategy is given by the value of T = T * that maximizes mean fitness over many
( n ) years
W=
1 n
∑ (T ln( S p,i ) + ln( Sr ,i ) + ri (1 − T ) + C )
n i =1
[11]
Following the derivation of Eq. 4, when the trade-off between T and S r is given by
S r ,i = g i (T ) the optimal fixed strategy is given by
h(T * ) = r − ln( S p )
[12]
8
where h(T * ) is the expectation of h(T ) = g ' (T ) / g (T ) (i.e., h(T ) =
1 n
∑ hi (T ) when
n i =1
n → ∞ ), and where r and ln( S p ) are the expectations of r and ln( S p ) . In addition
h' (T * ) must be negative.
If hi (T ) varies between years only in its placement on the T -axis so that all hi (T )
can be written on the form h0 (T + ∆ i ) (see Fig. 1B), then h(T * ) will have a lower slope than
h0 (T ) , as illustrated in Figure 3, top panel. Thus, stochastic variation in h(T ) may contribute
to larger variation in T * even if the animals only respond to variation in r − ln( S p ) and not to
variation in h(T ) : The animals should initiate reproduction earlier than the mean of T *
among years when r − ln( S p ) is high (where h 0 (T ) > h(T ) ), and later than mean T * when
r − ln( S p ) is low (see Figs. 3 and 4).
Optimal response to imperfect environmental cues
The optimal responses to environmental cues depend on the reliability (or precision) of the
cues. First consider the optimal response to a cue reflecting r − ln( S p ) . In a given year i ,
ri = r + δ i( r ) and ln( S p ) i = ln( S p ) + δ i
(Sp )
the mean value is δ 1,i = δ i( r ) + δ i
(Sp )
. Hence, the deviation in r − ln( S p ) in year i from
. The voles cannot, however, measure this deviation
without error, but instead perceive the cue δ 1,i + ε 1,i . In determining the optimal strategy for
onset of reproduction, this cue may be weighted with a constant k1 where 0 ≤ k1 ≤ 1 ( k1 = 0
when no trust in the cues and k1 = 1 when full trust in the cues). Hence, in the presence of
*
such a cue, optimal Ti is the value Ti *( k1 =k1 ) that satisfy
*
h(Ti*( k1 =k1 ) ) = r − ln( S p ) + k1 (δ 1,i + ε 1,i )
{
}
{
[13]
*
*
when k1 takes the value k1* yielding values of Ti , S r ,i = Ti *( k1 =k1 ) , S r*(,ik1 =k1 )
} that maximize
fitness over many ( n ) years (see Eq. 11),
(S )
1 n ⎛⎜ Ti *( k1 = k1 ) (ln( S p ) + δ i p ) + ln( S r*(,ik1 = k1 ) ) ⎞⎟
W= ∑
*
⎟
n i =1 ⎜⎝
+ ( r + δ i( r ) )(1 − Ti *( k1 = k1 ) ) + C
⎠
*
*
[14]
which is equivalent of maximizing
9
∆W =
(
(
*
*
1 n
ln( S r*,(ik1 = k1 ) ) − Ti*( k1 = k1 ) r − ln( S p ) + δ 1,i
∑
n i =1
))
[15]
(see Fig. 4 for notation).
When k1 is high (close to 1) the expectation of Ti *( k1 ) given δ 1,i will be closer to the
theoretical
optimum
under
perfect
information
(i.e.,
the
value
of
Ti *
satisfying
h(Ti * ) = r − ln( S p ) + δ 1,i ). However, when k1 is high, the random “measurement error" ( ε 1,i )
will also have a larger influence on Ti*( k1 ) . Both a too conservative response (too low k1 ) to
reliable cues and a naive response (too high k1 ) to unreliable cues will reduce fitness.
Hence, the optimal k1 , k1* , should be high if the perceived cues are reliable (i.e., if the
variance of ε 1 is low relative to the variance of δ i ), and k1* should be low if the cues are
unreliable (high Var( ε 1 ) relative to Var( δ i )). In some sense, k1* may be seen as representing
an optimal trade-off between ”bias” and variance of Ti *( k1 ) given δ 1,i (”bias” relative to the
optimal value under perfect information). However, it is fitness ( W ) that should be
maximized and not prediction error variance that should be minimized, and due to the nonlinear relationships, this is not equivalent.
The animals may also respond to cues reflecting h(T ) (e.g. whether breeding
conditions improve early or late in the spring, see Fig. 1B). Assuming that the animals
perceive the cue of when to initiate reproduction with an error, δ 2 ,i + ε 2 ,i , and that this cue is
independent of the cue of r − ln( S p ) , optimal onset of reproduction is
Ti * = Ti *( k 2 = 0 ) + k2 (δ 2,i + ε 2,i )
[16]
where the value of k 2 is chosen so that fitness (Eq. 14) is maximized. Here, Ti*( k2 =0 ) is the
optimal value of Ti when k 2 = 0 , which is given by Eq. 13, and δ 2 ,i = Ti *( k2 =1) − Ti *( k2 =0 ) where
Ti *( k2 =1) is the optimal value of Ti when the cue is measured without error (Var( ε 2 ) = 0),
which is the value of Ti satisfying
hi (Ti ) = r − ln( S p ) + k1 (δ 1,i + ε 1,i )
[17]
(see Fig. 4).
10
Simulation results
To find the optimal responses to the environmental cues, and the fitness benefits of these
cues, the optimal weights, k1* and k 2* , may be found by searching for values that maximize
fitness in numerical simulations. For simplicity I first studied the response to cues reflecting
r − ln( S p ) assuming no response to variation in h(T ) (i.e., k 2 = 0 ), and then I studied the
optimal response to variable h(T ) assuming no response to variation in r − ln( S p ) (i.e.,
k1 = 0 ). I further assumed that h(T ) only varies in the parameter c1 (i.e., its placement on
the T -axis). In Figure 5, simulation results are shown for realistic parameter values for small
rodent populations where there is low and high variation in r − ln( S p ) , and where there is
low and high variation in c1 .
Note that there is a stronger benefit of a flexible response to variation in c1 when
r − ln( S p ) is high (bottom-right vs. bottom-left panel of Fig. 5B). This is because S r , and
hence W , is more sensitive to T in the steeper parts of g (T ) . For the same reason there
is a weaker benefit of a flexible response to variation in c1 when the slope of g (T ) , b1 , is
low (illustrated in Fig. 6). On the other hand, as also illustrated in Figure 6, there is a stronger
benefit of a flexible response to variation in r − ln( S p ) when b1 is low. That is, in order to
maximize fitness, it is more important to have information on r − ln( S p ) when breeding
conditions improve slowly than when they improve fast.
Optimal weights ( k ‘s)
The optimal weights to the cues of r − ln( S p ) , k1 , found to maximize fitness in the
simulations are very close to the theoretical weights that minimize the prediction error
variance of r − ln( S p ) . However, as shown in Figure 7, the optimal weights to the cues of c1 ,
k 2 , are substantially lower (i.e., more conservative) than the weights minimizing prediction
error variance of Ti * (see Eq. 16). In particular, when searching for a bivariate k 2 with one
value for negative cues ( k 2− ) and one value for positive cues ( k 2+ ), it appears that it is optimal
to be more conservative in responding to cues about early improvement of breeding
conditions than to cues about late improvement of breeding conditions (i.e., k 2− < k 2+ ). This is
because the fitness function (Eq. 2 is not symmetrical around T * : a one week too early onset
11
of reproduction has a higher fitness cost than a one week too late onset. The bivariate k 2
were used in the above simulations.
Phenotypic correlations and norms of reaction in stochastic environments
When individuals respond only to variable r − ln( S p ) , the expected relationship between T *
and S r* in a stochastic environment will remain positive as long as S r = g (T ) is an
increasing function, although the extent of the variation and correlation of these variables
depend on the reliability of the cues as well as the extent of variation in r − ln( S p ) and
g (T ) . In contrast, as illustrated in Figure 8, if animals respond to cues about the time that
breeding conditions improve ( c1 ), then a negative association between the expectations of
S r* and T * will occur whenever the animals do not have perfect information (i.e., k 2* < 1 ).
Because there should be less variation in T * when the cues are unreliable (due to lower
optimal k 2 ), S r* will be higher in years with early improvement of breeding conditions (low
c1 ) and lower in years with late improvement of breeding conditions (high c1 ) (Fig. 8A).
However, a negative phenotypic correlation between observed S r* and T * will not be
detectable because there will be high random variation in both S r* and T * , especially at
intermediate reliabilities of the cue (Fig. 8B).1 On the other hand, if one can measure the time
that breeding conditions improve (e.g., by the phenology of the food plants), one should
observe a negative relationship between S r and this measurement (i.e., the ”norm of
reaction” (Roff 2002, chap. 6)) when the cues are unreliable. When the cues are reliable,
there should be a positive association between T and the measurement (Fig. 8C).
Dependencies between onset of reproduction and pre-breeding survival
There may be a trade-off between pre-breeding winter survival (high S p ) and early
reproduction (low T ) if early reproduction is enabled by maintaining a physiological,
morphological or behavioral state that is disadvantageous for winter survival (e.g. large body
1
It may be shown that if k 2 takes the value that minimize the prediction error variance of Ti *
(see Fig. 7), then the expected phenotypic covariance between T * and S r* should be zero.
However, because the optimal value of k 2 that maximizes fitness is lower than this value
(Fig. 7), the expected covariance (and slope) is negative (Fig. 8A), although the correlation
will be very weak (Fig. 8B).
12
size, Ergon et al. 2004). There may also be a dependency between S p and T due to
senescence (Boonstra 1994): if survival declines with age, then the geometric mean of prebreeding survival ( S p ) of an overwintering individual will decline with time ( T ). Such a
dependency between S p and T may also simply result from seasonal variation in S p (e.g.,
survival rates in small rodent populations are often high during winter but drops to lower
levels in the spring (Krebs & Boonstra 1978; Rodd & Boonstra 1984; Boonstra & Boag 1992;
Ergon, Lambin & Stenseth 2001)), which will cause the geometric mean of S p to decrease
with higher T ).
The effects of any general form of such dependencies are difficult to investigate
analytically. In the lack of any known functional relations between S p and T , I therefore
apply a general graphical method (see e.g. Sibly 1991): Fitness isoclines in the {S p , T } plane may be calculated by viewing Eq. 2 as a function of S p and T . Plotting this function
for different values of W produces a “fitness landscape”, onto which hypothetical constraintcurves may be super-imposed (Fig. 9).
Both a convex and a concave trade-off curve, as well as a constraint curve
representing senescence (or seasonal decline in survival), are super-imposed on fitnesslandscapes under different values of r in Figure 9. Clearly, a dependency between S p and T
may greatly modify the optimal onset of seasonal reproduction ( T * ) and reproductive
success ( S r* ), and it may be adaptive to substantially “trade off” S p to obtain a low T in
response to high population growth. It is also apparent from Figure 9 that, when S p and T
are interdependent, T * becomes more sensitive to changes in population growth ( r ) at
intermediate values of r and S p . With a concave trade-off curve, the optimal strategy
switches abruptly from ‘late’ to ‘early’ as r increases. Senescence will prevent delayed onset
of reproduction at low r − ln( S p ) .
A dependency between S p and T will also modify the expected relation between T *
and S r* (Fig. 10). Senescence will force the animals to reproduce earlier than they otherwise
would when breeding conditions improve late, leading to a lower S r* when T * is high (recall
that S r* should remain constant when S p and T are independent, when there is no
response to variation in r − ln( S p ) , and when g (T ) only varies in its placement on the T -
13
axis, Figure 1B and Prediction 2). Such a negative correlation between S r* and T * will also
occur when there is a convex trade-off between S p and T because the animals should
reproduce later than they otherwise would when the environment improves early. When there
is a concave trade-off curve, S r* may increase when T * is high (Fig. 10).
Summary of model results
When considering a trade-off only between early reproduction (low T ) and high reproductive
success of the first breeding attempt (high S r ), it is optimal to commence reproduction when
the rate of change in expected reproductive success ( S r = g (T ) ) relative to its current value
( h(T ) = g ' (T ) / g (T ) ) is declining and equals the difference between population growth rate
in the reproductive season and the logarithm of survival in the non-reproductive season
( r − ln( S p ) ). Thus, there are two main mechanisms for variation in T : (1) responses to
variation in population growth rate and pre-breeding survival (variable r − ln( S p ) ; Fig. 1A),
and (2) responses to variation in the time that breeding conditions improve (variable g (T ) ;
Fig. 1B). In the first case, reproduction should start earlier in populations (or species) where
population growth is high during the reproductive season (high r ) compared to the nonreproductive season (low ln( S p ) ), whereas in populations with more stable seasonal
dynamics, reproduction should start later. Likewise, in multi-annually fluctuating populations,
breeding should start earlier in years with high r − ln( S p ) if phenotypic responses to cues
about the future population development ( r ) and/or survival chances ( S p ) have been
evolved.
If animals had perfect information about the time that breeding conditions improve ( c1
in Eq. 9), then a one week delay in improvement of breeding conditions should cause a one
week delay in the optimal time to start breeding ( T * ), and reproductive success ( S r* ) should
remain constant (given no response to variation in r − ln( S p ) ). However, because it is
optimal to be conservative in responding to cues that are unreliable, there should be less
variation in T * in a stochastic environment and S r* should tend to be higher in years when
breeding conditions improve early than when the environment improves late. A negative
relationship between observed S r* and T * will, however, not be detectable unless one has
independent information about the time that breeding conditions improve (Fig. 8). In
14
stochastic environments, there will be a larger benefit of responding to variation in r − ln( S p )
when S r improves slowly over the spring, and there will be a larger benefit of responding to
variation in the time that breeding conditions improve when S r improves fast and when
r − ln( S p ) is generally high (Fig. 5 and 6).
Optimal T and S r may be greatly altered if pre-breeding survival ( S p ) and the time
of onset of reproduction ( T ) are not independent, either due to a trade-off between high S p
and low T or due to senescence (lower S p when T is high) (Fig. 9). In the case of a convex
trade-off, and in particular senescence or seasonal decline in survival, one should expect a
negative correlation between S r* and T * when there is variation in the time that breeding
conditions improve (Fig. 10).
A CASE STUDY ON Microtus agrestis
In a fluctuating population of field voles (Microtus agrestis) in Kielder forest, Northern
England,
onset
of
spring
reproduction
is
strongly
delayed
density
dependent:
commencement of the breeding season is delayed by several weeks (around 3 weeks per
additional 100 voles·ha-1) after high population densities in the previous spring (Ergon 2003).
A field transplant experiment and energetic studies in the study system suggest that this and
related life-history traits of the overwintering individuals are due to variations in the energetic
constraints of the current winter/spring environment, and not due to variations in the structure
of the population with respect to age, physiological states or genetics (Ergon, Lambin &
Stenseth 2001; Ergon 2003; Ergon et al. 2004). Further, onset of spring reproduction is not
correlated with population growth during summer or winter in a way that would suggest that
individuals can make use of cues about their survival probability or the population growth
(Ergon 2003). Thus, we have strong evidence indicating that the variation in onset of spring
reproduction in this system is mainly due to variation in the time that the environmental
conditions for breeding improve, which is partly delayed density dependent. To investigate
the mechanisms for the variation in onset of spring reproduction in the light of the theoretical
results presented above, I have analyzed the survival costs of reproduction among female
voles in four out-of-phase sites where spring reproduction commenced at different times over
the spring. Survival probabilities of reproducing and pre-reproducing overwintered females
were estimated from capture-recapture data collected at two week intervals over the spring
(see Appendix A for a detailed description of the analysis and Lambin, Petty & MacKinnon
(2000) for details about the study system).
15
Survival was lowest at the sites where breeding commenced the latest, and
pregnant/postpartum females had lower apparent survival than pre-breeding females (Fig.
11A). Although there was no strong support for a general trend in the survival cost of
reproduction, reproducing females at the site with the latest onset of reproduction (site A)
had particularly low survival (Fig. 11B). It is possible that the poor survival of reproducing
females at site A, which was a typical ‘decline site’ (see Ergon 2003), was a result of females
being forced to reproduce while the environment was still unfavorable due to dependencies
between S p and T (e.g. senescence or a seasonal decline in pre-breeding survival, Fig.
10).
DISCUSSION
The optimal time to start seasonal reproduction depends on the condition, or state, of the
individuals and their surrounding environment at present and in the near future (McNamara &
Houston 1996). To make ”decisions’’ (in the sense of evolved physiological responses to
some stimuli) over whether to initiate or postpone reproduction, animals must rely on cues
carrying information about such state variables as body condition, food resources and social
factors in the present environment as well as in the anticipated environment at later lifehistory stages of their offspring and themselves. From a physiological point of view, many
responses to such cues are well known. For example, time of the year (date) at a given
latitude may be accurately determined by the rate of change in day length (photoperiod).
Animals perceive this cue (change in photoperiod) through the pineal gland in the brain
which produces melatonin, a hormone that affects a wide range of physiological processes
including reproductive function (Tamarkin, Baird & Almeida 1985; Mustonen, Nieminen &
Hyvärinen 2002). Another hormone also influencing reproductive function is leptin, which is
produced by fat cells and thus monitor the level of stored energy reserves in the body
(Massimiliano et al. 2001). Other hormones act as intermediaries in the link from social
stimuli (e.g. pheromones) and predator scents to the regulation of behavior, energy
acquisitioning, metabolism and reproduction (Bronson & Heideman 1994). Reproduction may
also be stimulated by nutrients and other food constituents. One such food constituent that
stimulates reproduction in many grass-eating microtines is the secondary plant compound 6MBOA, which is present in sprouting grass (see discussion on small rodents below). All
these physiological responses may interact in intricate ways to determine the onset of
seasonal reproduction in animals (reviewed in Bronson & Perrigo 1987; Bronson &
Heideman 1994). For example, ingestion of 6-MBOA accelerates puberty of juvenile
mountain voles (Microtus montanus) only under long photoperiod, whereas adult males use
16
photoperiod alone as their primary cue of when to become reproductively active (Gower &
Berger 1990). Any reproductive development in females is often hindered if the animals are
in poor nutritional condition (Bronson 1998).
Optimality models investigate the selective forces guiding the evolution of life-history
traits under given constraints, and predict the optimal trait values at different environmental
states and conditions of the individuals. Such simplifying models may be used to understand
geographical variation and differences in life-history traits between species, or to understand
optimal responses to environmental variation by the same genotype (i.e., the ‘norms of
reaction’ describing phenotypically plastic traits as a function of the environmental state
variables (Roff 2002)). Adaptive differences in fixed trait values between populations and
species in different environments may evolve without any physiological ”perception’’ of the
differences in the environments. In contrast, if individuals are to adjust their life-history
strategies according to temporal variation in the surrounding environment they must react to
some cues reflecting the state of the environment. In such cases, as illustrated in this paper,
the optimal norms of reaction (and the expected variation and co-variation of phenotypic
traits) depends on the degree these cues reflect the true state of the environment (i.e., the
reliability, or the precision, of the cues).
Precision of cues and optimal life-history traits
Some environmental states like time of season can probably be measured quite precisely
through environmental cues (e.g. photoperiod). However, cues reflecting for example
reproductive prospects for offspring are probably rather unreliable and perhaps not even
attainable. Other environmental states, such as habitat quality may be assessed with varying
and intermediate degrees of precision. Theories for optimal life-history strategies in
stochastic environments have almost invariably assumed one of two extremes: At one
extreme, it is assumed that the animals have perfect information about variations in their
environment, and optimal reaction norms are derived (e.g., McNamara & Houston 1996; Roff
2002). At the other extreme, it is assumed that animals have no information about the
environment and one studies how environmental stochasticity affect the optimal fixed
strategies (e.g., risk aversion and bet hedging (Yoshimura & Clark 1991; Menu & Desouhant
2002; Roff 2002; Hopper et al. 2003)). Given the presumably ubiquitous prevalence of
imperfect information about internal and external state variables in nature, it is perhaps
surprising that considerations regarding optimal reaction norms to imperfect information is
virtually absent from the life-history literature.
17
There are some notable theoretical works on the influence of imperfect information on
optimal foraging behavior. Some authors have considered the fitness gains and costs of
energy intake and predation risk, and specifically investigated the optimal foraging strategies
when obtaining information has a cost (Stephens & Krebs 1986; Bouskila & Blumstein 1992;
Abrams 1994; Abrams 1995; Bouskila, Blumstein & Mangel 1995). Yoccoz, Engen &
Stenseth (1993) studied optimal foraging strategies when the energy content of food items
and the search time to obtain these items cannot be determined without error. However, the
approach I have taken to study optimal life-history responses to imperfect information is to
my knowledge novel.
As I have illustrated in this paper, when cues are not precise, it is optimal to alter the
trait-values to some extent, but not fully, in the direction suggested by the cues (see Fig. 7
and 8). Hence, the optimal responses of phenotypic traits to environmental change (norms of
reaction) are not just functions of the environmental states, but also of the precision that
these states can be measured. In fact, when cues are imprecise, both the optimal reaction
norms and the expected phenotypic correlations may be very different from the predictions
when considering the extreme cases where cues are either absent or perfect.
The precisions of the cues will also greatly influence the strength of selection on the
reaction norms (see Fig. 5 and 6). Hence, the precisions of the cues used by the animals are
important for both long term (genetic selection) and short term (phenotypic) responses to
changes in the environment. Understanding what cues animals use in their reproductive
decisions and how they respond to these cues are particularly important when seeking to
predict effects of environmental change outside the range of the available data, such as
effects of climatic change (Krebs 2002; Le Maho 2002; Stenseth & Mysterud 2002). When
the environment changes rapidly (e.g. due to anthropogenic influence) phenotypic responses
to environment that have evolved under one set of environmental conditions may become
severely maladaptive even when there is a rather small change in how the environmental
state variables vary and co-vary (Stenseth & Mysterud 2002). This is due to the fact that
animals have evolved reproductive responses to latent variables (e.g. photoperiod) that covaries with some important environmental state variable (e.g. food availability) because such
cues are more precise than more direct cues and because they allow the animals to prepare
in advance of predictable changes in for example food availability. For example, if the
seasonal peak in food availability changes, but the animals time their reproduction according
to day-length, then there will be a ‘mismatch’ between the reproductive strategies and food
availability, possibly causing severe population declines (see specific examples in Stenseth
& Mysterud 2002). To predict the evolutionary change in the reaction norms, one must also
18
know the response to selection (determined by genetic variability and constraints as well as
heritabilities of the traits) in addition to the strengths of selection on the traits (Roff 2002).
Determinants of onset of reproduction in small rodents
Field voles (Microtus agrestis) in Kielder Forest, Northern England, show a delayed density
dependent pattern in onset of spring reproduction (see Case Study above). This is in
agreement with other studies of fluctuating small rodent populations: although the densitydependent pattern in the commencement of the breeding season in small rodent populations
is generally not well described, the general pattern reported in the literature is that breeding
starts earlier in the ‘increase’ and ‘peak’ phases than in the ‘decline phase’ of the population
fluctuations (Chitty 1952; Krebs & Myers 1974; Tast 1984). In the Kielder study system, we
have strong evidence indicating that variation in onset of spring reproduction is somehow
related to the time that the environmental conditions improve, rather than responses to
variation in pre-breeding winter survival and population growth (see the Case Study above).
This is further supported by the survival analysis presented in this paper: among the four
sites where detailed trapping data were available, survival costs of reproducing was highest
at the site where reproduction started the latest (site A; Fig. 11). This is predicted by my
model (see Figure 10) if females are forced to reproduce while the environment is still
unfavorable due to senescence (Boonstra 1994) or a seasonal decline in pre-breeding
survival (e.g., Krebs & Boonstra 1978; Rodd & Boonstra 1984; Boonstra & Boag 1992). The
opposite is predicted if variation in onset of spring reproduction is primarily caused by
responses to variation in pre-breeding survival probability and population growth (see Figure
1).
What component of the environment that causes the delayed density dependent
onset of spring reproduction of field voles in Kielder Forest is unknown. However, quality and
phenology of the food plants are perhaps the most likely candidates. Early reproduction is
likely to be constrained by the limited supply of energy and nutrients in the food plants during
winter/early spring (McNab 1986; Bronson 1989; Bronson & Heideman 1994). Indeed,
several food supplement field experiments have succeeded in advancing the onset of the
breeding season (reviewed in Boutin 1990). For example, Schweiger & Boutin (1995) found
that Clethrionomys rutilus initiated reproduction about 3 weeks earlier, compared to controls,
when provided unlimited sunflower seeds throughout the winter. In observational studies,
large variation in onset of spring reproduction of Microtus montanus has been linked directly
to the phenology of the food plants, which varies between years due to variations in the time
of snow melt-off (Negus, Berger & Forslund 1977). Negus, Berger & Forslund (1977) noted
19
that small amounts of sprouting green plant tissue can trigger reproduction in some microtine
species, and it was later demonstrated that the active agent is a secondary plant compound
called 6-Methoxybenzoxazolinone (6-MBOA) (Berger et al. 1981; Sanders et al. 1981). This
compound, which has no nutritional value, is thought to be abundant in all growing grasses
(Moffatt, Bennett & Nelson 1991; Nelson 1991) and thus serve as a general cue that enables
grass eating herbivores to initiate reproduction at the early stages of the plant growth season
before food becomes abundant (Negus & Berger 1998). In an early field experiment on
Microtus montanus, Negus and Berger (1977) placed sprouted wheat in the voles’ runways
and were able to precipitate onset of spring reproduction by six weeks compared to animals
in the control grids that did not initiate reproduction until the appearance of new rhizome
shoots of the common food plants. Korn and Taitt (1987) later replicated this experiment on
Microtus townsendii and found that supplements of oats coated with 6-MBOA precipitated
reproduction by four weeks compared to control sites where oats coated with the solvent only
were provided.
Although no links between population density and phenology of the food plants have
been demonstrated, such mechanisms for delayed density dependence are certainly
possible. Perennial grasses, sedges and rushes store energy in underground root stems
(rhizomes) that is used to produce new shoots (tillers) after grazing or at the start of a new
growth season in the spring (Archer & Tieszen 1983; Jónsdóttir 1991). When the plants are
repeatedly grazed during the growth season of the grasses (mainly spring and early summer)
these energy reserves may become depleted, possibly reducing tiller survival and delaying
germination in the following year (Archer & Tieszen 1983; Richards 1984; Jónsdóttir 1991;
Engel et al. 1998). Grazing during a critical period when the plants are cold hardening in the
fall may also severely reduce survival of overwintering tillers (Lawrence & Ashford 1969;
Sheaffer et al. 1992; Harrison & Romo 1994). Bergeron & Jodin (1993) investigated the
influence of intense grazing during one summer on the green biomass in the following fall
and spring by manipulating high densities of Microtus pennsylvanicus in some enclosures
and excluding voles from control enclosures. They found that the grazed plots had 15% less
green biomass in the fall and 52% less green biomass early in the growing season the
following spring, even though voles were absent from the enclosures during winter.
Of course, other environmental state variables may also contribute to the delayed
density dependence in onset of spring reproduction. For example, increased predator
densities (perceived by odors) generally reduce foraging activity of the prey (Lima 1998).
Many studies have shown such responses in small rodents (Desy & Batzli 1989; Ylönen
1994; Koskela & Ylönen 1995; Carlsen 1999; Perrot-Sinal, Ossenkopp & Kavaliers 2000).
Although it is questionable whether this can impair reproduction in the summer (Lambin et al.
20
1995; Kokko & Ranta 1996; Mappes, Koskela & Ylönen 1998), predation risk may have a
larger influence on the optimal reproductive decisions in early spring when energy
constraints are severe, there is little cover in the vegetation (in snow free areas) and when
animals that delay reproduction have a high residual reproductive value. If diseases were
important sources of the between site/year variation in onset of reproduction (e.g. Feore et
al. 1997), one should probably see larger individual heterogeneity within sites than what is
observed (see Ergon, Lambin & Stenseth 2001).
Responses to survival prospects and future population growth
Previous attempts to find optimal life-history responses to population development in other
systems have also been unsuccessful: Studies on a cyclically fluctuating population of Soay
sheep at the St. Kilda archipelago west of Scotland, have revealed that the ewes invest more
in reproduction in years of population crashes than they optimally should if they had perfect
information, and that this contributes to the severity of the declines occurring at 3-4 year
intervals (Clutton-Brock et al. 1996; Marrow et al. 1996). Assuming that the ewes have no
information about the population development, their reproductive decisions are close to the
predicted optimal (Marrow et al. 1996). One reason for the lack of flexibility in reproductive
decisions may be that Soay sheep are a primitive breed of domestic sheep, and that they
have not yet had time to adapt to the environment they inhabit (Marrow et al. 1996).
Considering the complexity in the sources to variation in population growth (Clutton-Brock &
Coulson 2002; Sibly & Hone 2002), it may also be that it is not even possible for the animals
to obtain reliable information about the population development. The population regulation
mechanisms may also change over short time-scales due to changes in the environment or
due to random switching between different dynamic attractors of the ecosystem (Hanski &
Henttonen 1996; McCauley et al. 1999), hindering evolution of responses to cues about the
population development.
One particular case where cues about survival prospects or population development
may be more reliable is when the population dynamic processes are consistently related to
habitat quality (e.g., vegetation type or micro-climate). Interestingly, Sharpe & Millar (1991)
found that Peromyscus maniculatus initiated spring reproduction earlier in habitats where
pre-breeding winter survival was lowest, which could potentially indicate that these mice
have evolved adaptive responses to habitat related variation in pre-breeding survival (see
Prediction 1 of the model). However, the observed pattern could also be due to differences in
the time that breeding conditions improve. The latter is supported by the observation that
females inhabiting the habitats where breeding commenced late had lower, and not higher,
21
reproductive success (number of weaned offspring) of the first litter compared to females in
the habitats where breeding commenced late (Sharpe & Millar 1991). The lower reproductive
success in the late breeding habitats, which is consistent with the pattern presented in the
case study of this paper, may be because there is large variation in the time that breeding
conditions improve and that the voles are conservative in their response to cues reflecting
this variation because the cues are unreliable. However, a negative correlation between
observed onset of reproduction and reproductive success will likely not be detectable in a
stochastic environment (see Fig. 8). Perhaps a more likely explanation for the pattern is that
there is a dependency between pre-breeding survival and onset of reproduction that force
the animals to reproduce earlier than they otherwise would when breeding conditions
improve late (e.g., due to trade-offs or senescence; Fig. 10).
In conclusion, the precision of environmental cues in a stochastic environment largely
influence the optimal reaction norms, the expected phenotypic correlations and how natural
populations and ecosystems respond to environmental change. I have presented an example
of how reliability (precision) of environmental cues can be incorporated into life-history
models by a simple simulation approach, studying the response to cues about only one state
variable at a time. This approach may be expanded to study the optimal responses to a set
of dependent cues reflecting several state variables. Detailed studies at the individual level
and studies of physiological mechanisms are obviously necessary to fully understand how
organisms respond to changes in the environment. However, to link these mechanisms to
evolved life-history strategies and population dynamics it is also necessary to consider
models of the type presented in this paper.
ACKNOWLEDGEMENTS
I thank Rolf Ergon, Xavier Lambin and Nils Chr. Stenseth for fruitful discussions and
comments at various stages of writing this paper. Rolf Ergon contributed with ideas relating
to the weighting of uncertain cues and thoroughly checked the consistency of the
mathematics and simulations.
LITERATURE CITED
Abrams, P. A. 1994. Should prey overestimate the risk of predation? The American Naturalist
144:317-328.
Abrams, P. A. 1995. Overestimation versus underestimation of predation risk: a reply to
Bouskila et al. The American Naturalist 145:1020-1024.
22
Archer, S., & L. Tieszen. 1983. Effects of simulated grazing on foliage and root production
and biomass allocation in an arctic sedge (Eriophorum vaginatum). Oecologia 58:92102.
Berger, P. J., N. C. Negus, E. H. Sanders, & P. D. Gardner. 1981. Chemical triggering of
reproduction in Microtus montanus. Science 214:69-70.
Bergeron, J.-M., & L. Jodin. 1993. Intense grazing by voles (Microtus pennsylvanicus) and its
effect on habitat quality. Canadian Journal of Zoology 71:1823-1830.
Boonstra, R., & P. T. Boag. 1992. Spring declines in Microtus pennsylvanicus and the role of
steroid hormones. Journal of Animal Ecology 61:339-352
Boonstra, R. 1994. Population cycles in microtines: the senescence hypothesis. Evolutionary
Ecology 8:196-219.
Bouskila, A., & D. T. Blumstein. 1992. Rules of thumb for predation hazard assessment:
predictions from a dynamic model. The American Naturalist 139:161-176.
Bouskila, A., D. T. Blumstein, & M. Mangel. 1995. Prey under stochastic conditions should
probably overestimate predation risk: a reply to Abrams. The American Naturalist
145:1015-1019.
Boutin, S. 1990. Food supplementation experiments with terrestrial vertebrates: patterns,
problems and the future. Canadian Journal of Zoology 68:203-220.
Bronson, F. H. 1985. Mammalian reproduction: An ecological perspective. Biology of
Reproduction 32:1-26.
Bronson, F. H. 1989, Mammalian reproductive biology. Chicago, The University of Chicago
Press.
Bronson, F. H. 1998. Energy balance and ovulation: small cages versus natural habitats.
Reproduction Fertility and Development 10:127-137.
Bronson, F. H., & P. D. Heideman. 1994. Seasonal regulation of reproduction in mammals,
Pages 541-583 in E. Knobil, & J. D. Neill, (eds.) The physiology of reproduction. New
York, Raven Press.
Bronson, F. H., & G. Perrigo. 1987. Seasonal Regulation of Reproduction in Muroid Rodents.
American Zoologist 27:929-940.
Burnham, K. P., & D. R. Anderson. 2002, Model selection and inference: a practical
information-theoretic approach. New York, NY, Springer-Verlag.
Burnham, K. P., D. R. Anderson, G. C. White, C. Brownie, & K. H. Pollock. 1987. Design and
analysis of fish survival experiments based on release-recapture data. American
Fisheries Society, Monograph 5, Bethesda, Maryland.
Carlsen, M. 1999. The effects of predation risk on body weight in the field vole, Microtus
agrestis. Oikos 87:277-285.
23
Caswell, H. 2001, Matrix Population Models; Construction, analysis and interpretation.
Sunderland, Massachusetts, Sinauer Associates.
Chitty, D. 1952. Mortality among voles (Microtus agrestis) at Lake Vyrnwy, Montgomeryshire,
in 1936-9. Philosophical Transactions of the Royal Society of London (Ser. B)
236:505-552.
Clutton-Brock, T. H., & T. Coulson. 2002. Comparative ungulate dynamics: the devil is the
detail. Philosophical Transactions of the Royal Society of London 357:1285-1298.
Clutton-Brock, T. H., M. Major, S. D. Albon, & F. E. Guinness. 1987. Early development and
population dynamics in red deer. I. Density-dependent effects of juvenile survival.
Journal of Animal Ecology 56:53-67.
Clutton-Brock, T. H., I. R. Stevenson, P. Marrow, A. D. MacColl, A. I. Houston, & J. M.
McNamara. 1996. Population fluctuations, reproductive costs and life-history tactics in
female Soay sheep. Journal of Animal Ecology 65:675-689.
Desy, E. A., & G. O. Batzli. 1989. Effects of food availability and predation on prairie vole
demography: a field experiment. Ecology 70:411-421.
Engel, R. K., J. T. Nichols, J. L. Dodd, & J. E. Brummer. 1998. Root and shoot responses of
sand bluestem to defoliation. Journal of range management 51:42-46.
Ergon, T. 2003. Fluctuating life-history traits in overwintering field voles (Microtus agrestis).
PhD Dissertation. University of Oslo, Norway. (ISSN 1501-7710).
Ergon, T., X. Lambin, & N. C. Stenseth. 2001. Life-history traits of voles in a fluctuating
population respond to the immediate environment. Nature 411:1043-1045.
Ergon, T., J. R. Speakman, M. Scantlebury, R. Cavanagh, & X. Lambin. 2004. Optimal body
size and energy expenditure during winter: Why are voles smaller in declining
populations? The American Naturalist 163 (3): 442-457.
Fairbairn, D. J. 1977. Why breed early? A study of reproductive tactics in Peromyscus.
Canadian Journal of Zoology 55:862-871.
Feore, S. M., M. Bennett, J. Chantrey, T. Jones, D. Baxby, & M. Begon. 1997. The effect of
cowpox virus infection on fecundity in bank voles and wood mice. Proc. R. Soc. Lond
B 264:1457-1461.
Gower, B. A., & P. J. Berger. 1990. Reproductive responses of male Microtus montanus to
photoperiod, melatonin, and 6-MBOA. Journal of Pineal Research 8:297-312.
Hanski, I., & H. Henttonen. 1996. Predation on competing rodent species: A simple
explanation of complex patterns. Journal of Animal Ecology 65:220-232.
Harrison, T., & J. T. Romo. 1994. Regrowth of smooth bromegrass (Bromus inermis; Leyss)
following defoliation. Canadian Journal of Plant Science 74:532-537.
Hopper, K. R., J. A. Rosenheim, T. Prout, & S. J. Oppenheim. 2003. Within-generation bet
hedging: a seductive explanation? Oikos 101:219-222.
24
Jónsdóttir, I. S. 1991. Effects of grazing on tiller size and population dynamics in a clonal
sedge (Carex begelowii). Oikos 62:177-188.
Kokko, H., & E. Ranta. 1996. Evolutionary optimality or delayed breeding in voles. Oikos
77:173-175.
Korn, H., & M. J. Taitt. 1987. Initiation of early breeding in a population of Microtus
townsendii (Rodentia) with the secondary plan compound 6-MBOA. Oecologia
(Berlin) 71:593-596.
Koskela, E., & H. Ylönen. 1995. Suppressed breeding in the field vole (Microtus agrestis): An
adaptation to cyclically fluctuating predation risk. Behavioral Ecology 6:311-315.
Krebs, C. J. 2002. Two complementary paradigms for analysing population dynamics.
Philosophical Transactions of the Royal Society of London 357:1211-1219.
Krebs, C. J., & R. Boonstra. 1978. Demography of the spring decline in populations of the
vole, Microtus townsendii. Journal of Animal Ecology 47:1007-1015.
Krebs, C. J., & J. H. Myers. 1974. Population cycles in small mammals. Advances in
Ecological Research 8:267-399.
Lack, D. 1966, Population studies of birds. Oxford, Oxford University Press.
Lambin, X., R. A. Ims, H. Steen, & N. G. Yoccoz. 1995. Vole Cycles. Trends in Ecology &
Evolution 10:204-204.
Lambin, X., S. J. Petty, & J. L. MacKinnon. 2000. Cyclic dynamics in field vole populations
and generalist predation. Journal of Animal Ecology 69:106-118.
Lambin, X., & N. G. Yoccoz. 2001. Adaptive precocial reproduction in voles: reproductive
costs and multivoltine life-history strategies in seasonal environments. Journal of
Animal Ecology 70:191-200.
Lawrence, T., & R. Ashford. 1969. Effect of stage and height of cutting on the dry matter
yield, and the persistence of intermediate wheatgrass, bromgrass and reed
canarygrass. Canadian Journal of Plant Science 49:321-332.
Le Maho, Y. 2002. Nature and function. Nature 416:21.
Lima, S. L. 1998. Nonlethal effects in the ecology of predator-prey interactions. Bioscience
48:25-34.
Mappes, T., E. Koskela, & H. Ylönen. 1998. Breeding suppression in voles under predation
risk of small mustelids: laboratory or methodological artifact? Oikos 82:365-369.
Marrow, P., J. M. McNamara, A. I. Houston, I. R. Stevenson, & T. H. Clutton-Brock. 1996.
State-dependent life history evolution in Soay sheep: dynamic modeling of
reproductive scheduling. Philosophical Transactions of the Royal Society of London
351:17-32.
Massimiliano, C., E. Fabbrini, A. M. Isidori, A. Aversa, & A. Fabbri. 2001. Leptin and
reproduction. Trends in Ecology & Evolution 12:65-72.
25
McCauley, E., R. M. Nisbet, W. W. Murdoch, A. M. deRoos, & W. S. C. Gurney. 1999. Largeamplitude cycles of Daphnia and its algal prey in enriched environments. Nature
402:653-656.
McNab, B. K. 1986. The influence of food habits on the energetics of eutherian mammals.
Ecological Monographs 56:1-19.
McNamara, J. M., & A. I. Houston. 1996. State-dependent life histories. Nature 380:215-221.
Menu, F., & E. Desouhant. 2002. Bet-hedging for variability in life cycle duration: bigger and
later-emerging chestnut weevils have increased probability of a prolonged diapause.
Oecologia 132:167-174.
Millar, J. S. 1984. Reproduction and survival of Peromyscus in seasonal environments,
Pages 253-266 in J. F. Merritt, (ed.) Winter ecology of small mammals, Special
Publication Carnegie Museum of Natural History.
Millar, J. S., & D. G. L. Innes. 1983. Demographic and life cycle characteristics of montane
deermouse. Canadian Journal of Zoology 61:574-585.
Moffatt, C. A., S. A. Bennett, & R. J. Nelson. 1991. Effects of photoperiod and 6-methoxy-2benzoxazolinone on male-induced estrus in prairie voles. Physiology & Behavior
49:27-32.
Mustonen, A. M., P. Nieminen, & H. Hyvärinen. 2002. Melatonin and the wintering strategy of
the tundra vole, Microtus oeconomus. Zoological Science 19:683-687.
Negus, N. C., & P. J. Berger. 1977. Experimental triggering of reproduction in a natural
population of Microtus montanus. Science 196:1230-1231.
Negus, N. C., & P. J. Berger. 1998. Reproductive strategies of Dicrostonyx groenlandicus
and Lemmus sibiricus in high-arctic tundra. Canadian Journal of Zoology 76:391-400.
Negus, N. C., P. J. Berger, & L. G. Forslund. 1977. Reproductive strategy of Microtus
montanus. Journal of Mammalogy 58:347-353.
Nelson, R. L. 1991. Maternal diet influences reproductive development in male prairie vole
offspring. Physiology & Behavior 50:1063-1066.
Nichols, J. D., J. E. Hines, K. H. Pollock, R. L. Hinz, & W. A. Link. 1994. Estimating breeding
proportions and testing hypotheses about costs of reproduction with capturerecapture data. Ecology 75:2052-2065.
Oli, M. K., & F. S. Dobson. 1999. Population cycles in small mammals: the role of age at
sexual maturity. Oikos 86:557-565.
Perrot-Sinal, T., K. P. Ossenkopp, & M. Kavaliers. 2000. Influence of a natural stressor
(predator odor) on locomotor activity in the meadow vole (Microtus pennsylvanicus):
modulation
by
sex,
reproductive
condition
and
gonadal
hormones.
Psychoneuroendocrinology 25:259-276.
26
Richards, J. H. 1984. Root growth response to defoliation in two Agropyron bunchgrasses:
field observations with an improved root periscope. Oecologia 64:21-25.
Rodd, H. F., & R. Boonstra. 1984. The spring decline in the meadow vole, Microtus
pennsylvanicus: the effect of density. Canadian Journal of Zoology 62:1464-1473.
Roff, D. A. 1992, The Evolution of Life Histories; Theory and Analysis. New York, Chapman
& Hall.
Roff, D. A. 2002, Life history evolution. Sunderland, MA, Sinauer Associates, Inc.
Sanders,
E.
H.,
P.
D.
Gardner,
P.
J.
Berger,
&
N.
C.
Negus.
1981.
6-
Methoxybenzoxazolinone: A plant derivative that stimulates reproduction in Microtus
montanus. Science 214:67-69.
Schweiger, S., & S. Boutin. 1995. The effects of winter food addition on the population
dynamics of Clethrionomys rutilus. Canadian Journal of Zoology 73:419-426.
Sharpe, S. T., & J. S. Millar. 1991. Influence on the variation in initiation of breeding in
Peromyscus maniculatus. Canadian Journal of Zoology 69:698-705.
Sheaffer, C. C., D. K. Barnes, D. D. Warnes, W. E. Lueschen, H. J. Ford, & D. R. Swanson.
1992. Seeding-year cutting affects winter survival and its association with fall growth
score in alfalfa. Crop Science 32:225-231.
Sibly, R. M. 1991. The life-history approach to physiological ecology. Functional Ecology
5:184-191.
Sibly, R. M., & J. Hone. 2002. Population growth rate and its determinants: an overview.
Philosophical Transactions of the Royal Society of London 357:1153-1170.
Stenseth, N. C. 1999. Population cycles in voles and lemmings: density dependence and
phase dependence in a stochastic world. Oikos 87:427-461.
Stenseth, N. C., & A. Mysterud. 2002. Climate, changing phenology, and other life history
traits: Nonlinearity and match-mismatch to the environment. Proceedings of the
National Academy of Sciences of the United States of America 99:13379-13381.
Stephens, D. W., & J. R. Krebs. 1986, Foraging theory. Princeton, New Jersey, Princeton
University Press.
Tamarkin, L., C. J. Baird, & O. F. X. Almeida. 1985. Melatonin: a coordinating signal for
mammalian reproduction? Science 227:714-720.
Tast, J. 1984, Winter success of root voles, Microtus oeconomus, in relation to population
density and food conditions at Kilpisjärvi, Finnish Lapland. J. F. Merritt, (ed.) Winter
ecology of small mammals, Special Publication Carnegie Museum of Natural
History:59-66.
White, G. C. 2002. Discussion comments on: the use of auxiliary variables in capturerecapture modeling. An overview. Journal of Applied Statistics 29:103-106.
27
White, G. C., & K. P. Burnham. 1999. Program MARK: Survival estimation from populations
of marked animals. Bird Study 46 Supplement:120-138.
Ylönen, H. 1994. Vole cycles and antipredatory behaviour. Trends in Ecology & Evolution
9:426-430.
Yoccoz, N. G., S. Engen, & N. C. Stenseth. 1993. Optimal foraging: The importance of
environmental stochasticity and accuracy in parameter estimation. American
Naturalist 141:139-157.
Yoshimura, J., & C. W. Clark. 1991. Individual adaptations in stochastic environments.
Evolutionary Ecology 5:173-192.
28
APPENDIX I - SURVIVAL ESTIMATION
Capture-recapture data were collected at four 1 ha trapping grids at two-week intervals from
February to May (5 to 6 primary sessions at each site). Using only overwintered animals, the
data contained capture histories of 535 individuals and a total of 1427 captures at the
primary trapping sessions (see Ergon, Lambin & Stenseth (2001) for details). At each
capture, the reproductive condition of individually marked animals was recorded. It can be
determined that females have given birth if they are lactating and have small gap between
their pubic bones. Females were classified as pregnant/postpartum (‘P’) in the interval they
gave birth and in subsequent intervals, and as immature (‘I’) in the preceding intervals. All
females that were observed to have given birth during any given interval had gained ≥ 5 g
before this interval, and no females that had gained ≥ 5 g since last capture were observed
to not give birth in the following interval. Hence, all captured females that had gained ≥ 5 g
since last capture were classified as ‘P’ in the following interval and onwards.
Survival of ‘I’ and ‘P’ females were estimated with multi-state capture-mark-recapture
models (Nichols et al. 1994) using Program MARK (White & Burnham 1999), where survival
parameters depend on the state (‘I’ and ‘P’) at the start of the intervals, and where individuals
may change state in the end of the intervals. Males (only one state) were included in the
analysis, as this will increase precision of the estimates if survival and/or recapture
probabilities of the two sexes have any common structure. The parameters describing the
probabilities of transition from ‘I’ to ‘P’ (ψ ‘s) were constrained to be a logistic function of
sampling date ( T ) at the four sites. Both models where the transition probabilities were
constrained to have a common slope (ψ (site + T )) as well as models where the slope
varied between sites (ψ (site × T )) were considered. All other transition parameters were
fixed to zero.
Contingency tables in Program RELEASE (Burnham et al. 1987) showed no lack of fit
due to transients or trap-response, and the data did not appear overdispersed (grouping by
‘sex × site’, combined GOF test: Chi-sq. = 53.7, df = 45, p = 0.18, (White 2002)).
Models fitted to the data without using multi-states (Appendix Table I) provides strong
evidence for different survival ( φ ‘s) between sites and between sexes. There is also strong
evidence for different recapture probabilities (p’s) between the sexes. I therefore used a
‘ φ (sex + site);p(sex)’ model as a ”base model" to investigate the additive effects of
reproductive state on both survival and recapture probabilities (Appendix Table II).
29
Among the multi-state models, the best model according to the AICc-criterion is the
‘ φ (sex + site);p(sex + state); ψ (site + T )’ model, but the model where a ‘state’ effect on
survival is added, ‘ φ (sex + state + site);...’, is only marginally worse in terms of AICc
(∆AICc=0.58, one more parameter). The model with different ‘state’ effects at the four sites,
‘ φ ((sex + state) × site);...’, had substantially less support, but suggest that postpartum
females in the typical decline site, site A (see text), had particularly low survival (odds-ratio
for survival of ‘P’ females vs. ‘I’ females in site A is 0.17, 95% c.i.: [0.05, 0.58]). The
estimated survival probabilities are plotted in Figure 11, and parameter estimates are given
in Appendix Table III.
30
Appendix Table I: AICc-weightsa of models without multi-states. Rows show candidate
models for recapture probability (p), and columns show candidate models for survival ( φ ). ‘+’
denote additive effects and ‘×’ denote interaction effects. Bottom row and right column show
the sums of the weights for each set of models.
p\φ
.
sex
site
site×tb
sex + site
sex + site×t
∑
.
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
sex
<0.01
<0.01
<0.01
<0.01
0.01
0.01
0.02
site
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
site×tb
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
sex + site
0.08
0.46
0.04
0.04
0.13
0.21
0.97
sex + site×t
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
<0.01
∑
0.09
0.47
0.04
0.04
0.14
0.22
1.00
a
Approximate probabilities that each model is the Kullback-Liebler best model in the set (i.e., model with the
lowest expected prediction error (Burnham & Anderson 2002 p. 75)).
b
Time-dependence (t) was only included as an interaction effect with ‘site’ because sites were not trapped on
exactly the same dates.
31
Appendix Table II: AICc-weights of models incorporating effects of reproductive states. Two
left-most columns represent ‘parallel slope’ models for transition probabilities (‘I’ to ‘P’), and
two right-most columns represent ‘different slope’ models. ‘state’ has two levels: immature
females (I) and postpartum females (P). See Appendix Table I and text for explanation.
φ\p
sex + site
sex + state + site
(sex + state)×site
∑
ψ(site + T)
sex
sex + state
0.19
0.30
0.08
0.22
0.01
0.04
0.28
0.56
ψ(site×T)
sex
sex + state
0.04
0.06
0.02
0.04
<0.01
0.01
0.05
0.11
∑
0.58
0.36
0.06
1.00
32
Appendix Table III: Parameter estimates from the ‘ φ (sex + state + site);p(sex + state);
ψ (site + T )’ model.
Parameter
type
Contrast
intercept (‘I’ females, site A)
state ‘P’
males
Survivala
(φ)
site B
site C
site D
intercept (‘I’ females)
Recapture
state ‘P’
probability (p)
males
intercept (site A, Jan. 1)
site B
Transition
site C
probabilityb
(ψ)
site D
Tc
a
Two-weekly survival
Logit scale
estimate [95% c.i.]
1.40 [0.99, 1.81]
-0.36 [-0.92, 0.20]
-0.65 [-1.01, -0.29]
0.86 [0.37, 1.34]
0.18 [-0.23, 0.59]
0.57 [0.13, 1.00]
1.22 [0.83, 1.61]
0.72 [-0.002, 1.44]
0.82 [0.30, 1.35]
-18.7 [-23.5, -13.9]
3.46 [2.11, 4.81]
0.47 [-0.59, 1.52]
2.64 [1.43, 3.86]
0.17 [0.13, 0.22]
Probability scale
estimate [95% c.i.]
0.8 [0.73, 0.86]
Odds-ratio
estimate [95% c.i.]
0.70 [0.40, 1.22]
0.52 [0.36, 0.75]
2.35 [1.45, 3.82]
1.20 [0.80, 1.80]
1.76 [1.14, 2.73]
0.77 [0.70, 0.83]
2.05 [1.00, 4.22]
2.28 [1.35, 3.84]
0.00 [0.00, 0.00]
31.9 [8.3, 123.0]
1.60 [0.56, 4.58]
14.1 [4.2, 47.5]
1.19 [1.14, 1.24]
Probability that a vole in state ‘I’ at time T -7 days will have moved to state ‘P’ before time T +7 days given
that it survived.
b
c
Slope on a daily scale, equivalent to an increase in the transition probability from 0.05 to 0.95 over 34 days
(95% c.i.: [27, 46] days). When fitting the model, the time covariate ( T ) was given on a yearly scale as this
gives more stable convergence (see MARK help file).
33
FIGURE LEGENDS
Figure 1: Two main mechanisms for modulation in optimal onset of spring reproduction: A,
Higher r ≤ ln( S p ) (horizontal lines in upper panel) will lead to an earlier optimal onset of
reproduction ( T * ) and, if S r = g (T ) is increasing, a lower reproductive success at the
optimum ( S r* ) (see Prediction 1). B, An earlier improvement of breeding conditions (i.e.,
g 2 (T ) = g1 (T + ∆ ) ) will lead to a lower T * ( T * reduced by ∆) and S r* will remain unchanged
(Prediction 2 and 3). At optimum, h(T ) = g ' (T ) / g (T ) (top panels) equals r − ln( S p ) and is
decreasing ( h' (T * ) < 0 ) (see Text).
Figure 2: Example 1. g (T ) (lower panel; Eq. 5) and h(T ) = g ' (T ) / g (T ) (upper panel; Eq.
6) with three values of b (A,B,C). Thick horizontal lines in upper panel (labeled 1 and 2)
show two values of r − ln( S p ) . Labels in lower panel show the optimality points given by
h(T ) = r − ln( S p ) (asterisks in upper panel; see Fig. 1). C1 and C2 are on the T = 0 and
T = 1 boundaries (see text). Parameter values are: a = 0.9 , b = {20,9,3} , c = 0.5 and
r − ln( S p ) = {6.5,0.3} .
Figure 3: Example 2. Lower panel: g (T ) given by Eq. 9. Upper panel: h(T ) = g ' (T ) / g (T ) ,
Eq. 10. The solid lines show h(T ) and g (T ) for a typical year ( h0 (T ) = h (T ) |c1 =c1 and
g 0 = g (T ) |c1 =c1 ), while the dotted lines show g (T ) and h(T ) for c1 = c1 ± 2SD( c1 ) . The
stippled line show the expectation of h(T ) , h(T * ) , which determine the optimal fixed value
of T (see text). The boxed area in the upper panel is plotted in Figure 4. Parameter values
are realistic for small rodents: a = 0.95 , b1 = 51 (increase in S r = g (T ) from 0.05a to 0.95a
over 6 weeks), c1 = 0.41 , SD( c1 ) = 0.038 = 14 days , b2 = −18 and c2 = 0.88 .
Figure 4: Notation. Ti *
( k1 ,k2 )
show the expectations of Ti *
( k1 ,k2 )
under different values of k1
and k 2 (here k1* = k 2* = 0.5 ). The plotted area is marked out in Figure 3.
34
Figure 5: Simulation results. A. Simulations were repeated for four scenarios: low/high
variance of in the time that breeding conditions improve, c1 (top/bottom), and low/high
variance in r − ln( S p ) (left/right). Horizontal error-bars show ± 2SD( c1 ) , and stippled lines
show ± 2SD( r − ln( S p )) . h(T ) is given in Eq. 10 and Figure 3. Parameter values are given
below. B. Fitness benefits of flexible strategies (”value of information’’; y-axis) depending on
reliability of cues (x-axis). Simulations over 10,000 years were repeated with different values
of Var (ε 1 ) and Var (ε 2 ) , and the values of k1 = k1* and k 2 = k 2* that maximize fitness were
found by a numerical search. In simulations with k1 = k1* (filled symbols) k 2 was fixed to
zero, and in simulations with k 2 = k 2* (open symbols) k1 was fixed to zero (see Fig. 4). X-axis
(Reliability of cue, R2) is the proportion of the variance of the cue that is due to respectively
Var( c1 ) (open symbols) and Var( r − ln( S p ) ) (filled symbols). Left y-axis is the gain in mean
fitness from a flexible strategy compared to a fixed strategy. Right y-axis shows the number
of years it takes for the proportion of individuals following a flexible strategy to increase from
0.1% to 99.9% of the population (asexual clones and 100% heritability). C. SD(T ) between
years in the different simulations expressed in units of days. D. SD( S r ) between years.
Parameter values: Parameter values for g (T ) are the same as in Figure 3 except b1 = 30.6
(an increase in S r
from 0.05a
to 0.95a
over 10 weeks).
SD(c1 ) low = 3.5 days ,
SD(c1 ) high = 14 days , rlow = 0.72 , ln( S p ) low = −3.42 , SD( r ) low = 1.59 , SD(ln( S p )) low = 0.89 ,
Cor( r, ln( S p )) low = 0.21 ,
rhigh = 1.10 ,
ln( S p ) high = −5.04 ,
SD( r ) high = 2.91 ,
SD(ln( S p )) high = 1.71 , Cor( r, ln( S p )) high = 0.21 . Values of c1 and {r, ln( S p )} were drawn
from normal/multi-normal distributions.
Figure 6: Influence of how fast breeding conditions improve ( b1 ) in a stochastic environment.
A. Simulations were run with a low value of b1 (left column; b1 = 15.3 , an increase in S r from
0.05a to 0.95a over 20 weeks) and with high b1 (right column; b1 = 76.6 , an equivalent
increase in S r over 4 weeks). B. Fitness benefits of a flexible strategy (top), SD(T ) (middle)
and SD( S r ) (bottom) depending on the reliability of the cues. See Figure 5 for explanation
(note different scales on y-axes). Parameters except b1 have the same values as in Figure 5
with high Var( c1 ) and high Var( r − ln( S p ) ) (bottom-right panels).
35
Figure 7: Optimal values of k 2 found in the simulations presented in the right column of
Figure 6. ‘+’ denote the optimal k 2 ‘s for positive δ 2 ,i (see Fig. 4) and ‘-’ are the optimal k 2 ‘s
for nagative δ 2,i . Solid line is the theoretical weights that minimize the prediction error
variance of Ti * (Eq. 16): k 2 = (σ c21 + (δ 2, 0 ) 2 ) /(σ c21 + σ ε22 + (δ 2, 0 ) 2 ) , where σ c21 is the variance
of c1 , σ ε22 is the variance of ε 2 , and δ 2 ,0 is given in Figure 4.
Figure 8: Relationships between optimal onset of reproduction ( T * ) and reproductive
success ( S r* ) when there is stochastic variation in the time that breeding conditions improve
( c1 ). A,
h(T )
(upper panels) and the linearizing transformation of
S r = g (T ) ,
y = ln( S r /( a − S r )) = b1 (T − c1 ) (lower panels). Parallel lines show the functions for three
values of c1 : mean ± 2SD. Horizontal error bars in upper panels show the expectation of
*
T *( k2 =k2 ) ± k 2 2 SD(ε 2 ) (see Fig. 4), with the corresponding error bars along the y (T ) -lines in
the lower panels. Reliability of cues are given above the plots (see Fig. 5). Solid line in lower
panel connects the expectations of y * and T * . The expected regression line (stippled line)
will have a lower slope because there is random variation is not only in the y-direction. B,
Phenotypic correlations: values of y * and T * for 100 simulated years (see Fig. 5). C, Norms
of reaction: T * and y * (y-axes) plotted against the simulated values of c1 . Parameter values
as in the lower right panels of Figure 5 (assuming no response to variation in r − ln( S p ) ).
Figure 9: Fitness isoclines (stippled contours) and optima under different dependencies
(solid lines) between pre-breeding survival rate ( S p ) and onset of reproduction ( T ).
Asterisks show the optima on the given curves representing different constraints: A,
senescence (or seasonal decline in survival); B, a convex trade-off; and C, a concave tradeoff. Lines with plotted circles for given values of S p (intervals of 0.015) show the optima
when pre-breeding survival ( S p ) and onset of reproduction ( T ) are independent. Population
rate of increase in the reproductive season ( r ) varies between panels (increasing by row;
values on a monthly scale above plots). Fitness isoclines (plotted at intervals of 0.4) are
given by Eq. 2 where S r = g (T ) is given by Eq. 9 with the same parameter values as in
Figure 5. Pre-breeding survival (y-axis) and values of r are given on a monthly scale S 1p/ 12
and r / 12 ).
36
Figure 10: Effects of variation in the time that breeding conditions improve ( c1 ) on {T * , S r* }
under different dependencies (constraints) between pre-breeding survival ( S p ) and onset of
reproduction ( T ). A, Constraints and fitness isoclines (see Fig. 9) plotted for four values of
c1 (1 to 4 by rows): 0.41 ± 28/365 (panel 1 and 4) and 0.41 ± 14/365 (panel 2 and 3). Other
parameter values are the same as in the lower-left panel of Figure 9. B, Phenotypic
correlations between S r* and T * at the optimality points (asterisks in the upper panels).
Letters A to C denote type of constraint (see Fig. 9), and numbers represent values of c1
(panel 1 to 4). Interpretation: At lines with plotted circles in the upper panels (showing
optima when S p and T are independent), S r* for a given value of S p (y-axis) is independent
of c1 (i.e, the same across panels; see Prediction 2 and Fig. 1B). Hence, S r will decline
towards the left (lower T ) and increase towards the right, but remain the same on the dotted
line for a given S p in all panels. Because the optima on constraint-curve A (senescence) and
B (convex trade-off) will move towards the left relative to a fixed point on the dotted line when
c1 increases, S r* declines when c1 increases, and a negative correlation between S r* and
T * will result. This effect is strongest for curve A because the fitness-isoclines are steeper to
the left of the dotted line. In contrast, under a concave trade-off curve (C) S r* may increase
when T * increases.
Figure 11: Survival of immature (subscript ‘I’) and pregnant/postpartum (subscript ‘P’)
females at four study sites (A to D) plotted against estimated dates when 50% of the females
at the sites are postpartum (estimated by logistic regression; see Ergon, Lambin & Stenseth
2001). Error bars show 95% confidence intervals. A, Estimates from a model with additive
effects of reproductive state and sampling site on survival. B, Estimates from a less
constrained model where the state effect is allowed to vary freely between sites. See
Appendix A for details.
37
Ergon; MS: ECO-29292
h(T) and r - ln(Sp )
A
B
12
r3 - ln(Sp)3
10
h(T) = g’(T)/g(T)
8
6
r2 - ln(Sp)2
h3 (T)
h2 (T)
h1 (T)
4
r1 - ln(Sp)1
2
0
1.0
Sr = g(T)
S*r,1
0.8
0.6
S*r,2
0.4
0.2
S*r,3
g3 (T)
g2 (T)
g1 (T)
0.0
T3*
T2*
T1*
T3*
T2*
T1*
T
Fig. 1
38
Ergon; MS: ECO-29292
20
A
h(T)
15
10
B
1
5
C
2
0
1.0
A2
B2
Sr = g(T)
0.8
C2
A1
0.6
0.4
B1
0.2
C1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
T
Fig. 2
39
Ergon; MS: ECO-29292
50
40
h(T)
30
20
10
0
-10
1.0
g(T)
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
T
Fig. 3
40
Ergon; MS: ECO-29292
15
_
h0 (T) h (T)
__ (k = 1, k = 0)
2
Ti* 1
ri -ln(Sp) i
h(T)
10
__ (k = k* = 0)
Ti* 1 1, k2
δ 1,i
_ _____
r -ln(Sp)
5
h i(T)
T0
__ (k = 0, k = 1)
2
T *(k1= 0, k2= 0)
Ti* 1
__ (k = 0, k = k*
2 2)
Ti* 1
| δ 2,0 |
δ 2,i
|
0
0.40
0.45
0.50
0.55
T
Fig. 4
41
Ergon; MS: ECO-29292
h(T)
0
Low Var(r - ln(S p))
High Var(r - ln(S p))
-10
High Var(c 1)
20
High Var(c 1)
10
__
__
W (k=k*) - W(k=0)
10
Response to
cue of c 1
Response to
cue of r - ln(S p)
0.1
Low Var(c1)
Low Var(c1)
20
0.05
150
300
450
∞
0
0.1
150
0.05
300
450
0
Low Var(r - ln(S p))
High Var(r - ln(S p))
-10
∞
0
0.2
0.4
0.6
0.2
0.4
0.6
0.0
0.4
0.0
0.4
0.8
Reliability of cue (R2 )
T
C
D
15
0.15
10
0.10
5
SD(S r )
SD(T) 365
0.8
0
15
0.05
0.0
0.15
10
0.10
5
0.05
0
0.0
0.0
0.4
0.8
0.0
0.4
0.8
2
Reliability of cue (R )
0.0
0.4
0.8
0.0
0.4
0.8
2
Reliability of cue (R )
Fig. 5
42
Selection time
(years from 0.1% to 99.9%)
B
A
Ergon; MS: ECO-29292
A
70
h(T)
50
30
10
-10
g(T)
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.2
0.4
0.6
__
__
W (k=k*) - W(k=0)
50
0.25
Response to
cue of c1
Response to
cue of r - ln(Sp)
0.20
0.15
100
0.10
150
0.05
∞
SD(T) 365
0.0
Selection time
(years from 0.1% to 99.9%)
T
B
25
20
15
10
5
0
SD(Sr )
0.20
0.15
0.10
0.05
0.0
0.0
0.4
0.8
0.0
0.4
0.8
Reliability of cue (R 2 )
Fig. 6
43
1.0
Ergon; MS: ECO-29292
0.8
+
+
+
+
k2
0.6
+ + + - -
-
-
+ +
+ + -
-
0.4
+ +
+ - -
0.0
0.2
+ +-
+0.0
+-
+-
+ +- -
0.2
0.4
0.6
0.8
1.0
Reliability of cue (R2 )
Fig. 7
44
A
____
Sr
y = ln a - Sr
( )
h(T)
R2 = 0.1
R2 = 0.5
R2 = 0.9
14
12
10
8
6
4
2
4
2
0
-2
0.35
0.45
0.55 0.35
0.45
0.55 0.35
0.45
0.55
Optimal onset of reproduction (T*)
B
R2 = 0.1
R2 = 0.5
R2 = 0.9
y*
4
2
0
-2
0.35
0.45
0.55 0.35
0.45
0.55 0.35
0.45
0.55
Optimal onset of reproduction (T*)
C
R2 = 0.1
0.6
R2 = 0.5
R2 = 0.9
T*
0.5
0.4
0.3
y*
4
2
0
-2
0.3
0.4
0.5
0.6 0.3
0.4
0.5
0.6 0.3
0.4
0.5
0.6
Time that breeding conditions improve (c1)
Fig. 8
45
Ergon; MS: ECO-29292
0.80
0.70
C
0.70
C
0.3
0.4
0.5
0.6
0.7
0.8
0.3
0.4
0.5
0.90
A
0.7
0.8
0.7
0.8
A
0.80
B
0.80
B
0.6
r = 0.3
1.00
r = 0.1
1.00
0.90
Monthly Sp
A
B
0.80
B
0.70
C
0.70
C
0.3
r=0
1.00
A
0.90
0.90
1.00
r = -0.2
0.4
0.5
0.6
0.7
0.8
0.3
0.4
0.5
0.6
T
Fig. 9
46
Ergon; MS: ECO-29292
A
0.85
A
B
0.70
C
0.4
0.5
3: c1 = 0.45
1.00
0.4
0.5
0.6
4: c1 = 0.49
0.85
A
B
C
0.70
0.85
C
0.6
A
0.70
B
1.00
0.70
0.85
A
Monthly Sp
2: c1 = 0.37
1.00
1.00
1: c1 = 0.33
0.4
0.5
B
C
0.6
0.4
0.5
0.6
T
0.95
B
0.90
B1 B2
B3
C3
C4
B4
A2
0.80
Sr*
0.85
C2
C1
A1
0.75
A3
0.70
A4
0.40
0.45
0.50
0.55
0.60
T*
Fig. 10
47
Ergon; MS: ECO-29292
A
14 days survival
0.9
Model: Φ (state + site),
BI
BP
DI
DP
B
∆ AICc = 0.58
Model: Φ (state × site), ∆ AICc = 4.11
BI BP
CI
CP
AI
DP
DI
AP
CI
CP
AI
0.7
0.5
AP
0.3
13 Apr
20 Apr
28 Apr
1 May
13 Apr
20 Apr
28 Apr
1 May
Date when 50% are postpartum
Fig. 11
48