Optimal onset of seasonal reproduction in multivoltine organisms:
When should overwintering small rodents start breeding?
Torbjørn Ergon
Abstract
Multivoltine organisms that refrain from breeding in certain seasons must make the decision of
when to resume reproduction when the environmental conditions improve. Early reproduction
(low T ) has the advantage of a longer breeding season and a higher survival to reproduction, but
has the cost of a lower reproductive success (Sr = g(T )). I present a theoretical model showing
that, at optimal onset of reproduction (T ∗ ), rate of change in reproductive success relative to
its current value (g0 (T ∗ )/g(T ∗ )) should equal the difference between population growth in the
reproductive season and ln(survival) in the non-reproductive season, r − ln(Sp ). It follows that,
if other factors remain unchanged, 1) reproduction should start earlier when r − ln(Sp ) is high,
2) optimal reproductive success (Sr∗ = g(T ∗ )) depends on how, but not when, breeding conditions
improve, and 3) a one-week delay in the time that breeding conditions improve will lead to a
one-week delay in T ∗ . I further demonstrate how the optimal reaction norms and phenotypic
correlations of T ∗ and Sr∗ change when the environmental state variables cannot be perceived
without error, and when pre-breeding survival (Sp ) and T are dependent. Data on field voles in
northern England show that onset of spring reproduction varies with more than 6 weeks between
years and locations, and reproduction starts later when population density in the previous spring
was high. Correlations between onset of spring reproduction and population trends, as well as
estimates on survival costs of reproduction, suggest that this variation is not due to responses to
variable r − ln(Sp ), but rather due to variation in the time that breeding conditions improve.
Key-words: life-history evolution, optimality model, onset of spring reproduction, seasonality,
small rodent population cycles, imperfect information.
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 (CluttonBrock et al. 1987; Lack 1966). 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 pre-breeding survival rate
of the overwintering animals and the population growth rate during the reproductive season,
which are highly variable in small rodent populations (e.g. Krebs and Myers 1974; Stenseth
1999).
Indeed, small rodents in seasonal environments, and especially in populations with multiannual density fluctuations, show tremendous
variation in the time that reproduction is initiated. The start of the breeding season in
small rodent populations typically vary over a
range of 3—8 weeks between years (e.g. Krebs
and Myers 1974; Sharpe and Millar 1991), and
individual variation within the same year may
be of similar magnitude (Fairbairn 1977; Lambin and Yoccoz 2001; Millar and Innes 1983).
The variation in the seasonal patterns of reproduction between different environments or
habitats may also be extensive (Bronson 1985;
Bronson and Perrigo 1987; Millar 1984; Sharpe
and Millar 1991). Since this large variation in
onset of spring reproduction may be responsible for a substantial variation in the population
growth and the fitness of overwintering individuals (Fairbairn 1977; Lambin and Yoccoz 2001;
Oli and Dobson 1999), it should be of great interest to understand the mechanisms responsible. In particular, in order to predict responses
to environmental change (e.g. due to climatic
change) we must know what environmental cues
animals use in their reproductive decisions and
how they react to these cues (Le Maho 2002;
Stenseth and Mysterud 2002).
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 tradeoff between high success of the first breeding attempt and early reproduction, but I also investigate how dependencies between pre-breeding
survival and onset of reproduction (due to tradeoffs, 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, and I predict norms of reaction as well as phenotypic variations and correlations when animals respond optimally to imperfect information (cues) about their environment.
Finally, I analyze data on onset of spring reproduction and survival 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
Nt 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
Nt+1 = Nt SpT Sr mmax er(1−T )
(1)
where Sp is the pre-breeding survival rate, Sr
is the reproductive success defined as the frac-
2
tion 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 SpT of the Nt individuals survive until
the start of the breeding season, when they each
contribute Sr mmax new individuals to the population that will grow at rate r over the breeding
season of length 1 − T .
Dividing by Nt and taking the logarithm on
both sides gives the yearly growth rate, or fitness, W , of the strategy
µ
¶
Nt+1
W = ln
= T ln(Sp )+ln(Sr )+r(1−T )+C
Nt
(2)
where the constant C = ln(mmax ), and where
ln(Sp ) and ln(Sr ) are negative numbers since
0 ≤ Sp ≤ 1 and 0 ≤ Sr ≤ 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 Sr ), and possibly also between a low
T and high pre-breeding (winter-) survival (Sp ).
In the following I will assume that the strategy
determining T has zero genetic covariance with
life-history traits other than Sr and Sp (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 Sr ) may
be modeled by a function, Sr = 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 Sr =
g(T ) in eq. (2) and setting the first derivative
to zero,
∂W
= ln(Sp ) + h(T ) − r = 0
∂T
(3)
where h(T ) = g0 (T )/g(T ).
Values of T
that satisfy this expression correspond to a
peak in fitness whenever the second derivative,
∂ 2 W/∂T 2 = h0 (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
(4)
h(T ∗ ) = r − ln(Sp )
and h0 (T ∗ ) < 0, where h(T ) is the rate of
change in Sr relative to its current value, and
r −ln(Sp ) is the difference between summer population growth and the logarithm of winter survival (equaling growth rate of a homogeneous
population of non-breeding individuals). We
thus obtain the following predictions, that apply whenever seasonal reproduction occurs (i.e.,
T ∗ 6= 0 and T ∗ 6= 1):
Prediction 1: Since h0 (T ∗ ) < 0, h(T ) will
cross h(T ∗ ) = r − ln(Sp ) from above when T increases. This means that, if the relationship between Sr and T (Sr = g(T )) remains unchanged,
optimal onset of reproduction will always occur
earlier (lower T ∗ ) when r − ln(Sp ) is higher. An
increase in r by one unit has the same effect on
T ∗ as a decrease in ln(Sp ) by the same unit. See
figure 1A.
Prediction 2: If the relationship between
reproductive success and time of reproduction,
Sr = g1 (T ), changes to g2 (T ) so that g2 (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., Sr∗ = g2 (T2∗ ) = g1 (T1∗ ),
where T2∗ = T1∗ − ∆). That is, optimal reproductive success (Sr∗ ) is independent of when Sr
improves, although it is dependent on how it improves. See figure 1B.
Prediction 3: It follows from Prediction
2 that if the spring phenology, in terms of
Sr = g(T ), is precipitated/delayed by ∆ (i.e.,
g2 (T ) = g1 (T + ∆)) optimal T = T ∗ will be
precipitated/delayed by ∆ too (fig. 1B).
Thus, there are two main mechanisms for optimal modulation in T ∗ , as illustrated in figure 1:
a response to variable r − ln(Sp ) (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
3
h(T) and r - ln(Sp )
A
B
12
r3 - ln(Sp)3
10
h(T) = g'(T)/g(T)
8
r2 - ln(Sp)2
h3 (T)
h2 (T)
h1 (T)
6
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
Figure 1: Two main mechanisms for modulation in optimal onset of spring reproduction: A, Higher
r − ln Sp (horizontal lines in upper panel) will lead to an earlier optimal onset of reproduction (T ∗ ) and,
if Sr = g(T ) is increasing, a lower reproductive success at the optimum (Sr∗ ) (see Prediction 1 ). B, An
earlier improvement of breeding conditions (i.e., g2 (T ) = g1 (T + ∆)) will lead to a lower T ∗ (T ∗ reduced
by ∆) and Sr∗ will remain unchanged (Prediction 2 and 3 ). At optimum, h(T ) = g0 (T )/g(T ) (top panels)
equals r − ln(Sp ) and is decreasing (h0 (T ∗ ) < 0) (see Text).
derive more specific predictions in the example in fitness. These forms of g(T ) and h(T ) are
below.
plotted in figure 2.
Example 1: Assume that success of the first
When T ∗ 6= 0 and T ∗ 6= 1 the optimal T = T ∗
breeding attempt, Sr = g(T ), increase according is
µ
¶
b
1
to a logistic function over time
−1
(7)
T ∗ = c + ln
b
r − ln(Sp )
a
(5) and reproductive success at optimum, S = S ∗ ,
Sr = g(T ) =
r
1 + e−b(T −c)
r
is
where a is the season-independent component of
a(r − ln(Sp ))
(8)
Sr∗ = a −
reproductive success, b is a positive value deterb
mining how fast Sr increases and c is the value
These expressions are plotted as functions of b in
of T where Sr = 0.5a (which is also the infliction
figure 3. Note that T ∗ is independent of a, and
point). This gives
Sr∗ is independent of c when T ∗ is not on any of
the boundaries. We may confirm that Prediction
b
g0 (T )
=
(6) 1—3 hold for this special form of g(T ).
h(T ) =
b(T
−c)
g(T )
1+e
When r − ln(Sp ) > h(T = 0) then T ∗ = 0
Under this form of g(T ) the second derivative (as is the case for point C1 in fig. 2). This
the case when r − ln(Sp ) > b since
of fitness, ∂ 2 W/∂T 2 = h0 (T ), is always nega- is always
¢−1
¡
6 1. In other words, nontive. Hence, any value of T = T ∗ that satisfy 0 6 1 + e−bc)
h(T ∗ ) = r − ln(Sp ) (eq. (4)) represent a peak reproducing animals at T = 0 should not de-
4
1.0
A
20
0.8
h(T)
T*
15
B
0.8
C
S*r
2
0.6
0.4
1.0
A2
0.2
B2
0.8
Sr = g(T)
r - ln(Sp ) = 7
1.0
5
C2
A1
0.6
0.0
c = 0.3
c = 0.7
1
3
5 7
11
19
31 49 79 135
249
b
0.4
0.2
c = 0.3
r - ln(Sp ) = 1
0.0
1
0
c = 0.5
0.4
0.2
10
c = 0.7
0.6
306.2
102.1
43.7
20.4
9.9
5.2
2.8
1.5
0.8
Weeks from Sr = 0.05a to Sr = 0.95a
B1
C1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
T
Figure 2: Example 1. g(T ) (lower panel; eq. (5))
and h(T ) = g 0 (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(Sp ). Labels in lower panel show the optimality
points given by h(T ) = r−ln(Sp ) (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(Sp ) =
{6.5, 0.3}.
Figure 3: Example 1. Optimal onset of reproduction, T ∗ (eq. 7), and reproductive success of the
first breeding attempt, Sr∗ (eq. 8), plotted as functions of b for three values of c and two values of
r − ln(Sp ) (values given in the plot). T ∗ increases to
a peak at b = (LambertW(e−1 )+1)−1 (r − ln(Sp )) ≈
4.59(r − ln(Sp )), indicated by vertical lines, and
then declines asymptotically towards c. Sr∗ increases
asymptotically towards a (except when T ∗ = 0).
Lower x-axis show b at the scale of number of weeks
between g(T ) = 0.05a and g(T ) = 0.95a.
(late in the year),
g(T ) =
lay reproduction when population growth rate
(r) is high, pre-breeding survival (Sp ) is low
and breeding conditions improve slowly (low b).
Thus, year-round reproduction should be expected.
When r − ln(Sp ) < h(T = 1) then T ∗ = 1
(C2 in fig. 2). This is always the case when
r 6 ln(Sp ) (i.e., in years with lower population
growth in the summer than in the winter). However, the form of Sr = g(T ) here used may not
be realistic 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.
a
(1 + e−b1 (T −c1 ) )(1 + e−b2 (T −c2 ) )
(9)
with the corresponding
b1
b2
g 0 (T )
=
+
b
(T
−c
)
b
1
1
g(T )
1+e
1 + e 2 (T −c2 )
(10)
(fig. 4). This function of g(T ) may be interpreted as having three components to expected
reproductive success (Sr ): 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. If b2 has a high negative
value and c2 is high, then the last component is
Example 2: In the following I will assume a close to 1 except when T is high (close to c2 ) (see
function of Sr = g(T ) that declines at high T fig. 4). Thus, when T is small (in the spring),
h(T ) =
5
is given by (following the derivation of eq. 4)
50
40
~(T ∗ ) = r − ln(Sp )
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
Figure 4: Example 2. Lower panel: g(T ) given by
eq. (9). Upper panel: h(T ) = g0 (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 g0 (T ) = 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 ), ~(T ), which determine the optimal fixed value
of T (see text). The boxed area in the upper panel is
plotted in figure 5. Parameter values are realistic for
small rodents: a = 0.95, b1 = 51 (increase in Sr =
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.
this function may be approximated by the one
given in Example 1 (eq. (5)).
Optimal strategies in stochastic
environments
(12)
where ~(T ) is the expectation
Pn of h(T ) =
g 0 (T )/g(T ) (i.e., ~(T ) = n1 i=1 hi (T ) when
n → ∞), and where r and ln(Sp ) are the
expectations
of r and ln(Sp ). In addition
1 Pn
0
h
(T
)
must be negative.
i=1 i
n
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
~(T ) will have a lower slope than h0 (T ), as illustrated in figure 4. Thus, stochastic variation
in h(T ) may contribute to larger variation in T ∗
even if the animals only respond to variation in
r − ln(Sp ) and not to variation in h(T ). The animals should initiate reproduction earlier than
the mean of T ∗ among years when r − ln(Sp ) is
high (where h0 (T ) > ~(T )), and later than mean
T ∗ when r − ln(Sp ) is low (see fig. 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(Sp ). In a given
(S )
(r)
year i, ri = r + δ i and ln(Sp )i = ln(Sp ) + δ i p .
Hence, the deviation in r − ln(Sp ) in year i from
(S )
(r)
the mean value is δ 1,i = δ i − δ i p . The voles
cannot, however, measure this deviation without error, but instead perceive the cue δ 1,i +ε1,i .
In determining the optimal strategy for onset of
reproduction, this cue may be weighted with a
constant k1 where 0 6 k1 6 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,
∗(k =k∗ )
optimal Ti is the value Ti 1 1 that satisfy
∗(k1 =k1∗ )
) = r − ln(Sp ) + k1 (δ 1,i + ε1,i ) (13)
If Sp , Sr and r vary between years (i’s), then
the optimal fixed strategy is given by the value of when k1 takes the value k1∗ yielding values of
∗
∗
T = T ∗ that maximize mean fitness over many {Ti , Sr,i } = {T ∗(k1 =k1 ) , S ∗(k1 =k1 ) } that maxii
r,i
(n) years
mize fitness over many (n) years (see eq. (2)),
~(Ti
³
´
n

∗(k =k∗ )
(S )
1X
Ti 1 1 ln(Sp ) + δ i p
(T ln(Sp,i ) + ln(Sr,i ) + ri (1 − T ) + C)
n
n i=1
1 X
∗(k1 =k1∗ )

W =
+´ln(S
) ´
³ r,i
n i=1  ³
(11)
∗(k1 =k1∗ )
(r)
1 − Ti
+C
+ r + δi
Hence, when the trade-off between T and Sr is
(14)
given by Sr,i = gi (T ) the optimal fixed strategy
W =




6
_
h 0 (T) h (T)
15
ri -ln(Sp) i
h(T)
10
__
(k = k* = 0)
Ti* 1 1, k2
δ 1,i
_ _____
r -ln(Sp)
5
when to initiate reproduction with an error,
δ 2,i + ε2,i , and that this cue is independent of
the cue of r − ln(Sp ), optimal onset of reproduction is
h i(T)
__
(k = 1, k2 = 0)
Ti* 1
T*
(k1 = 0, k 2 = 0)
∗(k2 =0)
__
(k = 0, k2 = 1)
Ti* 1
__
(k = 0, k2 = k*
2)
Ti* 1
T0
| δ 2,0 |
δ 2,i
|
0
0.40
0.45
0.50
0.55
T
Figure 5: Notation. Ti∗
(k1 ,k2 )
show the expectations of Ti∗ (k1 ,k2 ) under different values of k1 and k2
(here k1∗ = k2∗ = 0.5). The plotted area is marked
out in figure 4.
Ti∗ = Ti
+ k2 (δ 2,i + ε2,i )
(16)
where the value of k2 is chosen so that fitness
∗(k =0)
is the
(eq. 14) is maximized. Here, Ti 2
optimal value of Ti when k2 = 0, which is given
∗(k =1)
∗(k =0)
by eq. (13), and δ 2,i = Ti 2 −Ti 2
where
∗(k =1)
is the optimal value of Ti when the cue
Ti 2
is measured without error (Var(ε2 ) = 0), which
is the value of Ti satisfying
hi (Ti ) = r − ln(Sp ) + k1 (δ 1,i + ε1,i )
(17)
(see fig. 5).
Simulation results: To find the optimal responses to the environmental cues, and the fitness benefits of these cues, values of k1∗ and k2∗
which is equivalent of maximizing
may be found by searching for values that maxÃ
!
∗
imize fitness in numerical simulations. For sim∗(k
=k
)
1
n
1
)
ln(S
1X
r,i
³
´
plicity
I first studied the response to cues reflect∗
∆W =
∗(k =k )
n i=1
−Ti 1 1 r − ln(Sp ) + δ 1,i
ing r − ln(Sp ) assuming no response to variation
(15) in h(T ) (i.e., k2 = 0), and then the optimal response to variable h(T ) assuming no response to
(see fig. 5 for notation).
When k1 is high (close to 1) the expectation variation in r − ln(Sp ) (i.e., k1 = 0). I further
∗(k )
of Ti 1 given δ 1,i will be closer to the theoreti- assumed that h(T ) only varies in the parameter
cal optimum under perfect information (i.e., the c1 (i.e., its placement on the T -axis). In figure 6,
value of Ti∗ satisfying ~(Ti∗ ) = r − ln(Sp ) + δ 1,i ). simulation results are shown for realistic paramHowever, when k1 is high, the random “measure- eter values for small rodent populations where
ment error” (ε1,i ) will also have a larger influ- there is low and high variation in r − ln(Sp ),
∗(k )
ence on Ti 1 . Both a too conservative response and where there is low and high variation in c1 .
Note that there is a stronger benefit of a flexi(too low k1 ) to reliable cues and a naive response
ble
response to variation in c1 when r − ln(Sp ) is
(too high k1 ) to unreliable cues will reduce fithigh
(bottom-right vs. bottom-left panel of fig.
∗
ness. Hence, the optimal k1 , k1 , should be high
6B).
This is because Sr , and hence W , is more
if reliable cues can be perceived (i.e., if the varisensitive
to T in the steeper parts of g(T ). For
ance of ε1 is low relative to the variance of δ 1 ),
the
same
reason there is a weaker benefit of a
∗
and k1 should be low if the cues are unreliable
flexible
response
to to variation in c1 when the
1
(high Var(ε1 ) relative to Var(δ 1 )).
,
is low (illustrated in fig. 7).
slope
of
g(T
),
b
1
The animals may also respond to cues reflectOn
the
other
hand,
as also illustrated in figure 7,
ing h(T ) (e.g. whether breeding conditions imthere
is
a
stronger
benefit
of a flexible response
prove early or late in the spring, see fig. 1B).
)
when
b1 is low. That
to
variation
in
r
−
ln(S
p
Assuming that the animals perceive the cue of
is, in order to maximize fitness, it is more im1 In some sence, k∗ may be seen as representing an opportant to have information on r − ln(Sp ) when
1
∗(k )
timal trade-off in the “bias” and variance of Ti 1 given breeding conditions improve slowly.
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 of r−ln(Sp )
that should be minimized.
Optimal k’s: The optimal weights to the cues
of r − ln(Sp ), k1 , found to maximize fitness in
7
A
B
h(T)
0
Low Var(r - ln(S p))
-10
High Var(r - ln(S p))
High Var(c 1)
20
High Var(c 1)
10
0.05
150
300
450
∞
0
0.1
150
0.05
300
450
0
Low Var(r - ln(S p))
-10
0.2
0.4
0.6
High Var(r - ln(S p))
0.2
0.4
∞
0
0.6
0.0
0.4
0.8
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
Selection time
(years from 0.1% to 99.9%)
10
Response to
cue of c1
Response to
cue of r - ln(Sp)
0.1
Low Var(c1)
__
__
W (k=k*) - W(k=0)
Low Var(c 1)
20
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
Reliability of cue (R²)
0.8
0.0
0.4
0.8
0.0
0.4
0.8
Reliability of cue (R²)
Figure 6: 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(Sp ) (left/right).
Horizontal error-bars show ±2SD(c1 ), and stippled lines show ±2SD(r − ln(Sp )). h(T ) is given in eq.
(10) and figure 4. 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(ε1 ) and Var(ε2 ), and the values of k1 = k1∗ and k2 = k2∗ that maximize fitness
were found by a numerical search. In simulations with k1 = k1∗ (filled symbols) k2 was fixed to zero,
and in simulations with k2 = k2∗ (open symbols) k1 was fixed to zero (see fig. 5). 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(Sp )) (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(Sr ) between years. Parameter values: Parameter values for g(T ) are the same as in figure 4 except
b1 = 30.6 (an increase in Sr from 0.05a to 0.95a over 10 weeks). SD(c1 )low = 3.5 days, SD(c1 )high = 14
days, rlow = 0.72, ln(Sp )low = −3.42, SD(r)low = 1.59, SD(ln(Sp ))low = 0.89, Cor(r, ln(Sp ))low = 0.21,
r high = 1.10, ln(Sp )high = −5.04, SD(r)high = 2.91, SD(ln(Sp ))high = 1.71, Cor(r, ln(Sp ))high = 0.21.
Values of c1 and {r, ln(Sp )} were drawn from normal/multi-normal distributions.
8
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
T
__
__
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%)
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 (R²)
Figure 7: 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 Sr from 0.05a to 0.95a over 20
weeks) and with high b1 (right column; b1 = 76.6, an equivalent increase in Sr over 4 weeks). B. Fitness
benefits of a flexible strategy (top), SD(T ) (middle) and SD(Sr ) (bottom) depending on the reliability of
the cues. See figure 6 for explanation (note different scales on y-axes). Parameters except b1 have the
same values as in figure 6 with high Var(c1 ) and high Var(r − ln(Sp )) (bottom-right panels).
the simulations are very close to the theoretical
weights that minimize the prediction error variance of r − ln(Sp ). However, as shown in figure
8, the optimal weights to the cues of c1 , k2 , are
substantially lower (i.e., more conservative) than
the weights minimizing prediction error variance
of Ti∗ (see eq. 16). In particular, when search-
ing for a bivariate k2 with one value for negative
cues (k2− ) and one value for positive cues (k2+ ),
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., k2− < k2+ ). This is because the fitness func-
1.0
9
0.8
+ +
+
0.6
+
+
+
+
-
+ + -
-
-
0.4
k2
+ - + + + -
0.2
0.0
+
+0.0
+-
+-
+-
+ ++- -
0.2
0.4
0.6
0.8
1.0
Reliability of cue (R²)
Figure 8: Optimal values of k2 found in the simulations presented in the right column of figure 7.
‘+’ denote the optimal k2 ’s for positive δ 2,i (see
fig. 5) and ‘—’ are the optimal k2 ’s for nagative
δ 2,i . Solid line is the theoretical weights that minimize the prediction error variance of Ti∗ (eq. 16):
(T )
k2 = (σ2c1 + (δ 0 )2 )/(σ2c1 + σ 2ε2 + (δ 2,0 )2 ), where σ 2c1
is the variance of c1 , σ 2ε2 is the variance of ε2 , and
δ 2,0 is given in figure 5.
tion (eq. (2)) is not symmetrical around T ∗ : a
one week too early onset of reproduction has a
higher fitness cost than a one week too late onset. The bivariate k2 were used in the above
simulations.
Correlations between T ∗ and Sr∗ in
stochastic environments: When individuals
respond only to variable r − ln(Sp ), the expected relationship between T ∗ and Sr∗ in a
stochastic environment will remain positive as
long as Sr = 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(Sp ) and g(T ). In contrast, as illustrated
in figure 9, if animals respond to cues about the
time that breeding conditions improve (c1 ), then
a negative association between the expectations
of Sr∗ and T ∗ will occur whenever the animals do
not have perfect information (i.e., k2∗ < 1). Because there should be less variation in T ∗ when
the cues are unreliable (due to lower optimal k2 ),
Sr∗ will be higher in years with early improve-
ment of breeding conditions (low c1 ) and lower
in years with late improvement of breeding conditions (high c2 ) (fig. 9A). However, a negative
phenotypic correlation between observed Sr∗ and
T ∗ will not be detectable because there will be
high random variation in both Sr∗ and T ∗ , especially at intermediate reliabilities of the cue
(fig. 9B)2 . 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
Sr and this measurement (i.e., the “norm of reaction”) when the cues are unreliable. When the
cues are reliable, there should be a positive association between T and the measurement (fig.
9C ).
Dependencies between onset of
reproduction and pre-breeding survival
There may be a trade-off between prebreeding winter survival (high Sp ) 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 size,
Ergon et al. 2003). There may also be a dependency between Sp and T due to senescence:
if survival declines with age, then the geometric
mean of pre-breeding survival (Sp ) of an overwintering individual will decline with time (T ).
Such a dependency between Sp and T may also
simply result from seasonal variation in Sp (e.g.,
survival rates in small rodent populations are often high during winter but drops to lower levels
in the spring (Boonstra and Boag 1992; Ergon
et al. 2001; Krebs and Boonstra 1978; Rodd and
Boonstra 1984)), which will cause the geometric
mean of Sp 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 Sp and T , I therefor apply a general
graphical method (see e.g. Sibly 1991): Fitness
2 It may be shown that if k takes the value that mini2
mize the prediction error variance of Ti∗ (see fig. 8), then
the expected phenotypic covariance between T ∗ and Sr∗
should be zero. However, because the optimal value of
k2 that maximize fitness is lower than this value (fig. 8),
the expected covariance (and slope) is negative (fig. 9A),
although the correlation will be very weak (fig. 9B).
10
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)
Figure 9: Relationships between optimal onset of reproduction (T ∗ ) and reproductive success (Sr∗ ) when
there is stochastic variation in the the time that breeding conditions improve (c1 ). A, h(T ) (upper panels)
Sr
and the linearizing transformation of Sr = g(T ), y = ln( a−S
) = b1 (T − c1 ) (lower panels). Parallel lines
r
show the functions for three values of c1 : mean ± 2SD. Horizontal error bars in upper panels show the
∗
expectation of T ∗(k2 =k2 ) ± k2 2SD(ε2 ) (see fig. 5), with the corresponding error bars along the y(T )-lines
in the lower panels. Reliability of cues are given above the plots (see fig. 6). Solid line in lower panel
connect 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. 6). 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 6 (assuming no response to
variation in r − ln(Sp )).
isoclines in the Sp —T -plane may be calculated by
viewing eq. (2) as a function of Sp and T . Plotting this function for different values of W produces a “fitness landscape”, onto which hypothetical constraint-curves may be super-imposed
(fig. 10).
Both a convex and a concave trade-off curve,
as well as a constraint curve representing senescence (or seasonal decline in survival), are superimposed on fitness-landscapes under different
values of r in figure 10. Clearly, a dependency
between Sp and T may greatly modify the optimal onset of seasonal reproduction (T ∗ ) and reproductive success (Sr∗ ), and it may be adaptive
11
B
0.70
C
0.70
0.3
0.4
0.5
0.6
0.7
0.8
0.3
0.4
0.5
0.90
A
B
0.6
0.7
0.8
0.7
0.8
r = 0.3
1.00
1.00
r = 0.1
A
B
0.80
0.80
0.90
Monthly Sp
A
0.80
B
C
C
0.70
0.70
C
0.3
r=0
1.00
0.90
A
0.80
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
Figure 10: Fitness isoclines (stippled contours) and optima under different dependencies (solid lines)
between pre-breeding survival rate (Sp ) 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 trade-off. Lines with plotted circles for given values of Sp (intervals of
0.015) show the optima when pre-breeding survival (Sp ) 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
Sr = g(T ) is given by eq. (9) with the same parameter values as in figure 6. Pre-breeding survival (y-axis)
1/12
and values of r are given on a monthly scale (Sp
and r/12).
to substantially “trade off” Sp to obtain a low T
in response to high population growth. It is also
apparent from figure 10 that, when Sp and T
are interdependent, T ∗ becomes more sensitive
to changes in population growth (r) at intermediate values of r and Sp . With a concave tradeoff curve, the optimal strategy switches abruptly
from ‘late’ to ‘early’ as r increases. Senescence
will prevent delayed onset of reproduction at low
r − ln(Sp ).
dent, when there is no response to variation in
r−ln(Sp ), and when g(T ) only varies in its placement on the T -axis, figure 1B and Prediction 2 ).
Such a negative correlation between Sr∗ and T ∗
will also occur when there is a convex trade-off
between Sp 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, Sr∗ may increase when
T ∗ is high (fig. 11).
A dependency between Sp and T will also
modify the expected relation between T ∗ and
Summary of model results
Sr∗ (fig. 11). Senescence will force the animals
When considering a trade-off only between
to reproduce earlier than they otherwise would
when breeding conditions improve late, causing early reproduction (low T ) and high reproduca lower Sr∗ when T ∗ is high (recall that Sr∗ should tive success of the first breeding attempt (high
remain constant when Sp and T are indepen- Sr ), it is optimal to commence reproduction
12
A
A
0.85
0.85
A
B
0.70
0.4
0.5
0.6
0.4
0.5
0.6
4: c1 = 0.49
1.00
3: c1 = 0.45
A
0.85
A
0.85
B
C
0.70
C
1.00
Monthly Sp
2: c1 = 0.37
1.00
1.00
1: c1 = 0.33
B
0.70
C
0.70
C
B
0.4
0.5
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*
Figure 11: Effects of variation in the time that breeding conditions improve (c1 ) on {T ∗ , Sr∗ } under
different dependencies (constraints) between pre-breeding survival (Sp ) and onset of reproduction (T ). A,
Constraints and fitness isoclines (see fig. 10) plotted for four values of c1 (1—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 10. B, Sr∗ and T ∗ at the optimality points (asterisks in the upper panels). Letters A—C
denote type of constraint (see fig. 10), and numbers represent values of c1 (panel 1—4). Interpretation:
At lines with plotted circles in the upper panels (showing optima when Sp and T are independent), Sr∗ for
a given value of Sp (y-axis) is independent of c1 (i.e, the same across panels; see Prediction 2 and fig. 1B).
Hence, Sr will decline towards the left (lower T ) and increase towards the right, but remain the same on
the dotted line for a given Sp 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,
Sr∗ declines when c1 increases, and a negative correlation between Sr∗ 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) Sr∗ may increase when T ∗ increases.
13
when the rate of change in expected reproductive success (Sr = g(T )) relative to its current value (h(T ) = g 0 (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(Sp )). Thus, there are two main mechanisms for variation in T : responses to variation in population growth rate and pre-breeding
survival (variable r − ln(Sp ); fig. 1A) and 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 non-reproductive season (low ln(Sp )), 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(Sp ) if phenotypic responses to cues about the future population development (r) and/or survival chances
(Sp ) 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 (Sr∗ ) should
remain constant (given no response to variation
in r − ln(Sp )). 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 Sr∗ should
tend to be higher in years when breeding conditions improve early than when the environment
improves late. A negative relationship between
observed Sr∗ and T ∗ will, however, not be detectable unless one has independent information
about the time that breeding conditions improve
(fig. 9). In stochastic environments, there will
be a larger benefit of responding to variation
in r − ln(Sp ) when Sr improves slowly over the
spring, and there will be a larger benefit of responding to variation in the time that breeding
conditions improve when Sr improves fast and
when r − ln(Sp ) is generally high (fig. 6 and 7).
Optimal T and Sr may be greatly altered if
pre-breeding survival (Sp ) and the time of onset
of reproduction (T ) are not independent, either
due to a trade-off between high Sp and low T
or due to senescence (lower Sp when T is high)
(fig. 10). In the case of a convex trade-off, and in
particular senescence, one should expect a negative correlation between Sr∗ and T ∗ when there
is variation in the time that breeding conditions
improve (fig. 11).
A case study on Microtus agrestis
I now assess the general mechanisms for variation in onset of spring reproduction, as illustrated by the model, within a population
of field voles (Microtus agrestis, L.) in Kielder
forests on the border between England and Scotland. In this region, which is largely covered
by spruce plantations, field voles are confined
to distinct grassland clear-cuts surrounded by
dense tree stands that lack ground vegetation
and are hence uninhabitable for voles. The subpopulations of voles inhabiting these clear-cuts
fluctuate asynchronously (Lambin et al. 1998;
MacKinnon et al. 2001), enabling replicated
short-term studies of density dependence and
between-year fluctuations in life-history traits.
Studies of wintering voles and onset of spring reproduction are also made easy by the fact that
there is no permanent snow cover during winter.
For details on the study system see Lambin et
al. (2000).
Correlations with population density and
growth
Data on proportions of overwintering female
voles that were lactating (nursing young) at different times during the spring were obtained at
18 different sites over 1 to 5 years at each site.
These data represent a wide range of population densities and growth rates (fig. 12). In
figure 13, estimates of the dates that 50% of
overwintered females were postpartum (lactating) in the spring are plotted against population density estimates at different lags as well as
population growth rates. Although there may
be substantial variation in onset of spring reproduction between study sites, there is also large
variation in this trait between years within sites
(e.g. site D, E and I). It appears that onset of
spring reproduction is more strongly related to
densities in the past, especially in the previous
14
200
Q
P
P
Q
M
L
N
K
O
C
M
M
L
N
E
O
E
0
K
C
S
F
1995
N
L
M
C
N
L
O
Q
P
E
M
N
P
Q
M
L
L
C
H
N
F
K
O
I
E
O
K
S
1996
C
E
K
K
H
O
G
F
C
I
F
S
F
1997
D
E
E
F
G
H
FJI
G
H
C
A
B
A
S
F
1998
F
Q
E
J
A
H
E
I
F
G
C
D
H
B
A
R
R
D
B
E
Q
I
Q
R
D
C
A
G
H
FJ
E
M
J
B
A
F
G
H
I
D
C
S
F
1999
S
May 01
Apr 15
300
B
I
G
E
100
Density (voles/ha)
I
B
D
J
Date when 50% are postpartum
J
C
Apr 01
400
G
F
2000
Year / Season
0
100
200
300
400
Density previous autumn (voles/ha)
Figure 12: Density trajectories at the study sites
(A—R). ‘S’ is spring (March/April) and ‘F’ is fall
(September/October). Shaded regions show the
main reproductive season. Methods: Density estimates in 1995 and at the sites A—D, Q and R
in year 2000 were obtained from calibrated ’vole
sign indices’ (Lambin et al. 2000). All other density estimates were obtained from closed capturemark-recapture models (for description see (Ergon
et al. 2001) (sites A—D and R), (Graham and Lambin 2002) (site E—J), and (MacKinnon 1998) (sites
C and K—Q).
spring (fig. 13A), than densities at present (fig.
13C ): breeding starts early in years when densities in the previous year are low. Reproduction
does not, however start early in the spring after
population declines in the previous reproductive
season (fig. 14). Hence, onset of spring reproduction is more strongly correlated with population densities in the previous spring than in the
previous fall (fig. 13A vs. B).
Onset of spring reproduction does not seem to
occur earlier when population growth during the
preceding winter (fig. 13E ) is low or when population growth during the following summer (fig.
13G) is high, as should be expected if the variation in onset of spring reproduction is caused by
an optimal response to variable winter survival
and/or population growth, r − ln(Sp ) (see Prediction 1 and fig. 1A above). Although populations that initiate breeding early have a higher
growth rate over the following spring (fig. 13F ),
there does not appear to be any association between onset of spring reproduction and popula-
Figure 14: Onset of spring reproduction (y-axis)
plotted against density in the previous fall (x-axis).
Size and filling of symbols denote population growth
rate during the previous reproductive season: filled
circles are declining populations while open circles
are increasing populations, and the size (area) of
the symbols are proportional to the absolute value
of the growth rate. See figure 13 for details.
tion growth rate over the entire breeding season
(fig. 13G).
Survival costs of reproduction
I investigated the model predictions with respect to relationships between reproductive success of the first breeding attempt (Sr∗ ) and the
time of reproductive commencement (T ∗ ; see
Summary of model results) by estimating survival probabilities from capture-recapture data
of reproducing and pre-reproducing overwintered females at four of the sampling sites (sites
A—D in 1999, fig. 13; see Appendix A for a description of the analysis). Although litter size
may also vary, survival during the breeding attempt is probably a major source to variation in
reproductive success of voles in the spring (see
Fairbairn 1977; Lambin and Yoccoz 2001).
As seen in figure 15A, survival was lowest
at the sites where breeding commenced the
latest, and pregnant/postpartum females had
lower apparent survival than pre-breeding females. There was no strong support for a gen-
15
B
May 15
A
I4
A4
G5
P2
C4
Q2
corr = 0.30 [-0.03,0.56]
Apr 15
J4
R5 H3
H5
L1
L2
150
200
100
May 15
E5
I4
E4 A4
J5P2 C4
Q2
J4
D4
Apr 15
L1
M1
C2
N2
G4
L2
D5
M2 B4
I5
F5 R5
K2 H5 H3
G5
C4
-0.2
-0.1
0.0
0.1
r prev. summer
0.2
C2
J5G5
E5
I4
A4
P2
Q2
J4
E4
300
400
E5
A4
50
L1
M1
M2N2
I5 G4
F5 H3C2
H5 L2
E3 F3
-0.2
-0.1
0.0
r prev. winter
K2
D5
I3
0.1
D5
E3
100
150
Present density
G
corr = 0.00 [-0.32,0.31]
N1 E5 I4
E4
G5O1 J5P2
C4
Q2
J4
D4L1
N1
I4
E4
J5 P2
G5
O1 C4
Q2
L1 J4 D4
M1
G3N2
M2 B4
G4
I5
F5
H3
H5
K2 L2
G3
B4
0.3 -0.3
O1
N2
I5
G4
H3 F5 H5
L2
corr = -0.38 [-0.62,-0.08]
N1
J4
L1
F3 I3
J3
200
E4
P2
Q2
G3M2
K2
F
D4
G3
D4
M1B4
I5
I4
G5
O1 J5
C4
E3
corr = -0.25 [-0.55,0.09]
I3
E3 F3
Mar 15
F5
E
N1
A4
G5
Density previous autumn
corr = -0.26 [-0.51,0.06]
O1
E5N1
C4J5
J4
D4
Density previous spring
D
corr = -0.01 [-0.33,0.29]
I4
G3
B4
M2N2
G4
C2
R5
H3 H5
K2
L2
D5
I3
F3
G4 I5
F5C2
100
E4
P2
Q2
A4
L1
M1
D4
E3
50
E5
N1
O1
D5
I3 F3
Mar 15
E5
N1
E4
O1 J5
M1N2
G3 M2B4
K2
Date when 50% are postpartum
C
corr = 0.57 [0.31,0.73]
C2
I5
A4
N2 G3 M2 M1
F5 H3
H5
L2
B4
G4
K2
C2
D5
F3 I3 E3
J3
-0.4
0.0
0.2
F3
E3
0.4
r next spring
0.6
0.8
-0.1
0.0
I3
J3
0.1
0.2
0.3
r next summer
Figure 13: Estimated dates (±SE) of when 50% of overwintering female field voles in Kielder forest
have given birth for the first time in the spring (y-axis) plotted against delayed and present population
densities (top panels) and population growth rates in different seasons (bottom panels). Panels (x-axes)
represent: A, Population density (voles/ha) in the previous spring (March/April); B, Density in the
previous fall (September/October); C, Density in the present spring; D, Monthly population growth rate
over the previous summer (March/April to September/October); E, Growth rate over the preceding winter
(September/October to March/April); F, Growth rate over the following spring (March/April to June); G,
Growth rate over the following breeding season (March/April to September/October). Values above the
plots are Pearson correlation coefficients (95% bootstrap confidence limits in brackets; 10,000 resamples).
Plotted labels represent site (A—R) and year (1=1996 to 5=2000) (see fig. 12). Error bars show standard
errors (smaller than the symbol when not visible). Methods: Dates that 50% of the overwintering females
were post-partum were estimated by logistic regression models of proportion lactating on sampling date
(estimate = −intercept/slope; see Ergon et al. (2001)). All data were obtained at one to five sampling
occasions between 15 February and 1 June. Because some of the sites×year’s were sampled at only one or a
few sampling occasions, it was only possible to fit an additive model (i.e., it is assumed that the slope in the
regression is the same at all sites). The slope on a logit-scale was estimated to 0.154 day−1 (SE = 0.0111),
which is equivalent to an increase in the proportion lactating from 5% to 95% over 38 days (95% c.i.:
[33,45] days). Standard errors (plotted error bars) were obtained by bootstrapping with 2000 resamplings
of individuals (i.e., not observations; some individuals were captured at more than one occasion but could
not make the transition to postpartum more than once). Note that the fall censuses may sometimes have
been undertaken before the end of the breeding season, and the spring censuses may have been undertaken
after the start of the breeding season. Thus, estimates may not be accurate (particularly estimates of
population growth in the winter, panel E). Density estimates were also obtained at a local scale (that may
not be representative for the larger scale), and there may be substantial error in the estimates obtained
by ‘vole sign indices’. See figure 12 for methods of density estimation.
16
A
14 days survival
0.9
Model: Φ (state + site),
BI
BP
DI
DP
B
∆ AICc = 0.58
CI
CP
0.7
Model: Φ (state × site), ∆ AIC c = 4.11
BI BP
AI
DI
DP
AI
CI
CP
AP
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
Figure 15: Survival of immature (subscript ‘I’) and pregnant/postpartum (subscript ‘P’) females at four
study sites (A—D) plotted against estimated dates when 50% of the females at the sites are postpartum.
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.
eral trend in the survival cost of reproduction
(i.e., in the difference between survival of pregnant/postpartum and immature females). However, reproducing females at the site with the
latest onset of reproduction (site A) had particularly low survival (fig. 15B). It is possible that
the poor survival of reproducing females at site
A, which was a typical ‘decline site’ (see Discussion), was a result of females being forced to
reproduce while the environment was still unfavorable due to dependencies between Sp and T
(e.g. senescence, fig. 11).
In summary, neither the correlations between
observed onset of spring reproduction and population growth (fig. 13) nor the differences in
survival of reproducing and pre-reproducing females (fig. 15) suggest that these voles adjust
onset of spring reproduction according to cues
about their survival chances or the future population growth (Prediction 1 and fig. 1A). It
is therefore more likely that the about 7 weeks
range in variation in onset of spring reproduction
is caused by variation in the time that breeding
conditions improve in the spring (Prediction 3
and fig. 1B), which appears to be delayed density dependent (fig. 13A and 14).
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
and Houston 1996). To make “decisions”3 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 life-history 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 (Mustonen et al.
2002; Tamarkin et al. 1985). Another hormone
also influencing reproductive function is leptin,
3 “Decisions” here means evolved physiological responses to some stimuli, and do not necessarily involve
any cognitive acts.
17
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 and 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 6-MBOA, which is
present in sprouting grass (see below ). All these
physiological responses may interact in intricate
ways to determine the onset of seasonal reproduction in animals (reviewed in Bronson and
Heideman 1994; Bronson and Perrigo 1987). For
example, ingestion of 6-MBOA accelerates puberty of juvenile mountain voles (Microtus montanus) only under long photoperiod, whereas
adult males use photoperiod alone as their primary cue of when to become reproductively active (Gower and 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, 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
Most theoretical models on optimal lifehistory strategies or behavior in variable environments assume one of two extremes: At one
extreme, it is assumed that the animals have
perfect information about changes in their environment, and optimal reaction norms are derived (e.g., McNamara and 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 (Roff
2002; Yoshimura and Clark 1991)). 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 and Krebs
1986)). Some environmental states like energy
availability and time of season can probably be
measured quite precisely through environmental cues (e.g. photoperiod). However, cues reflecting other environmental states such as reproductive prospects for offspring and future descendants (i.e., population growth) are probably
rather unreliable and perhaps not even attainable.
There are some notable theoretical works on
the influence of imperfect information on optimal foraging behavior, balancing the fitness
gains and costs of energy intake and predation
risk (Abrams 1994; Abrams 1995; Bouskila and
Blumstein 1992; Bouskila et al. 1995; Stephens
and Krebs 1986). These authors consider discrete patches of various qualities with respect to
predation risk and energy availability, and ask
the question of where it is optimal to be foraging. Specifically, they investigate optimal foraging strategies when obtaining information has a
cost: due to these costs animals should be tolerant towards imperfect information (cues) as long
as the cues are not too unreliable (i.e., within a
‘tolerance zone’), and it is discussed whether animals should overestimate or underestimate the
risk of predation. However, none of these studies
focus on deducing the norms of reaction and phenotypic variation and co-variation that should
18
be observed under different reliabilities of environmental cues.
In this paper I have considered continuous environmental state variables that the animals can
“measure” (through cues) with varying degrees
of precision. 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. 8 and 9). 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 of which these states can be measured.
The precisions of the cues will not only influence the optimal reaction norms (and the expected phenotypic variations and co-variations),
but this will also greatly influence the strength
of selection on the reaction norms (see fig. 6
and 7). 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 and Mysterud 2002). It is especially important to understand the phenotypic
responses to environmental change when the environment changes rapidly (e.g. due to anthropogenic influence) because the norms of reaction
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 covary (Stenseth and Mysterud 2002). This is due
to the fact that animals have evolved reproductive responses to latent variables (e.g. photoperiod) that co-varies 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 anticipated 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 popula-
tion declines (see specific examples in Stenseth
and Mysterud 2002). In order to predict the
evolutionary change in the reaction norms, one
must also 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).
My modeling, investigating optimal onset of
seasonal reproduction, illustrate the importance
of the precision of the cues used. When the variation in the time that breeding conditions improve can be measured without error, and when
the major dependency between life-history traits
is the trade-off between early reproduction and
high success of first reproduction, then the expected success of first reproduction should be
constant and the time of reproduction should
track the variation (one-to-one) in the time that
breeding conditions improve (see fig. 1). However, if the animals cannot measure precisely
the time that breeding conditions improve, then
there will be (if the animals behave optimally)
less variation in the initiation of reproduction
and the expected reproductive success should
be higher the earlier the animals choose to initiate reproduction (although a negative correlation will not be detectable in the case of my
specific model; see fig. 9).
I found that onset of spring reproduction
in field voles in the Kielder forest differed by
more than six weeks between years and locations. Such large variation in this trait seems
to be commonplace in small rodent populations
(see introduction). I did not find any correlations between onset of reproduction and population growth rates during summer and winter
in the direction predicted by the model, indicating that the variation in reproductive decisions are not mainly determined by responses to
variation in pre-breeding survival or future population growth. One may also rule out differences in the population structure (with respect
to e.g. genotypes, age, or maternal effects) to
be cause of the variation because a large transplant experiment in this study system showed
that life-history trait values (including onset of
reproduction) converged to the values prevailing at the target sites when voles were moved
between sites during mid-winter (Ergon et al.
2001). Thus, the large variation in onset of
19
spring reproduction is probably mainly pertaining to a response to variation in the time that
breeding conditions improve. The large variation observed also indicates that voles must be
able to detect this time rather precisely. This
may indicate that field voles (and their specialist predators) are robust against climate related changes in the spring phenology (time that
breeding conditions improve). However, in general, to predict the effects of persistent environmental change one must consider how the accuracies as well as the precisions of the environmental cues are affected by the environmental
change. For example, warmer winters could hypothetically cause certain plant compounds to
increase above a threshold level that would induce the animals to initiate reproduction at a
time when food was not sufficiently abundant.
Detailed knowledge at the physiological level is
clearly desirable (see Le Maho 2002).
Determinants of onset of reproduction in
small rodents
Although the density-dependent 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 (Krebs and
Myers 1974). The observations presented in this
paper are in agreement with this pattern: field
voles initiated reproduction early when densities in the previous year were low but increasing.
My data further indicate that variation in onset
of spring reproduction was mainly pertaining to
variation in the time that breeding conditions
improved (see above).
Early reproduction is likely to be constrained
by the limited supply of energy and nutrients
in the food plants during winter/early spring
(Bronson 1989; Bronson and Heideman 1994;
McNab 1986). Indeed, several food supplement field experiments have succeeded in advancing the onset of the breeding season (reviewed in Boutin 1990). For example, Schweiger
and 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 et al. 1977). It has long been known
that small amounts of sprouting green plant tissue can trigger a fast reproductive response in
some microtine species (Negus et al. 1977),
and that the active agent is a secondary plant
compound called 6-Methoxybenzoxazolinone (6MBOA) (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 et al. 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 (Negus
and Berger 1998). In an early field experiment
on Microtus montanus, Negus and Berger (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.
The above suggest that grazing induced delays in the time that plant growth is initiated
in the spring may be responsible for the delayed density dependent onset of spring reproduction found in my study system. Although
no links between population density, phenology of the food plants and onset of spring reproduction of small rodents have been demonstrated, such mechanisms 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 and 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 follow-
20
ing year (Archer and Tieszen 1983; Engel et al.
1998; Jónsdóttir 1991; Richards 1984). Grazing during a critical period when the plants are
cold hardening in the fall may also severely reduce survival of overwintering tillers (Harrison
and Romo 1994; Lawrence and Ashford 1969;
Sheaffer et al. 1992). Bergeron and 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 (Carlsen 1999;
Desy and Batzli 1989; Koskela and Ylönen 1995;
Perrot-Sinal et al. 2000; Ylönen 1994). Although it is questionable whether this can impair
reproduction in the summer (Kokko and Ranta
1996; Lambin et al. 1995; Mappes et al. 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 (Ergon et al.
2003). 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.
Responses to survival prospects and
future population growth
As illustrated by my model, it is optimal to
commence reproduction earlier in the spring,
and hence obtain a lower expected reproductive
success, when pre-breeding survival is lower and
when population growth in the following reproductive season is higher. However, neither the
correlations between observed onset of reproduc-
tion and population growth rates (see fig. 13)
nor the estimated differences in survival costs
of reproduction (see fig. 15) indicated that the
voles adjust their reproductive strategies according to information (cues) about variation in survival prospects or 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 and Coulson 2002; Sibly
and 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 and 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 and Millar (1991) found
that Peromyscus maniculatus initiated spring
reproduction earlier in habitats where prebreeding winter survival was lowest, which could
indicate that these mice have evolved adaptive
responses to habitat related variation in prebreeding survival. However, the observed pattern could also be due to differences in the time
21
that breeding conditions improve. The latter
is supported by the observation that females
inhabiting the habitats where breeding commenced early had higher, rather than lower, reproductive success (number of weaned offspring)
of the first litter compared to females in the
habitats where breeding commenced late. This
is also consistent with the pattern I have presented in this paper: among the four sites where
detailed trapping data were available, reproductive success in terms of survival costs of reproducing was highest at the site where reproduction started the latest (site A; fig. 15). A study
on energy expenditure using doubly-labeled water also revealed that voles at this late breeding site expended more energy despite having
a smaller body mass, indicating more severe
energetic constraints at this site (Ergon et al.
2003). The lower reproductive success in the
late breeding habitats observed by Sharpe and
Millar (1991), or at the late breeding site presented in 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. However, in the case of my specific
model, animals must be more conservative than
what is optimal to cause a negative correlation
between observed onset of reproduction and reproductive success (see fig. 9). Perhaps a more
likely reason for the lower reproductive success
at the habitats/sites where reproduction commenced late is that there is a dependency between pre-breeding survival and onset of reproduction forcing the animals to reproduce earlier
than they otherwise would when breeding conditions improve late (e.g., due to trade-offs or
senescence; fig. 11).
In conclusion, variation in the time that spring
reproduction commence in the populations of
field voles studied in this paper seems to be
mainly due to variation in the time that breeding
conditions improve (in terms of expected reproductive success of the first breeding attempt). I
did not find any evidence that the voles adjust
their reproductive decision according to future
survival prospects or population growth, probably because cues about variation in these environmental state variables are very unreliable.
Precision of cues and the consequences of lack of
perfect information largely influence the optimal
reaction norms, the expected phenotypic correlations and how natural populations and ecosystems respond to environmental change. Nevertheless, this seems to be a rather neglected
theme in the theory of life-history evolution and
population dynamics. 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 responce 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.
I am grateful to James L. MacKinnon and Isla
M. Graham for providing some of the data presented in figure 12—14.
22
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26
Appendix A
Capture-recapture data were collected at four 1 ha trapping grids at two-week intervals from
February to May (5—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 et al. 2001 for details). At each capture, the reproductive condition of individually marked animals was recorded. Females were classified as 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 interval i, had gained > 4 g before this
interval. Hence, all captured females that had gained > 4 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 and 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 test: Chi-sq. = 53.7, df = 45, p = 0.18, (White 2002)).
Models fitted to the data without using multi-states (table 1) 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 (table 2).
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 15,
and parameter estimates are given in table 3.
27
Table 1: 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.
pÂφ
·
sex
site
site×ta
sex + site
sex + site×t
P
·
< 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
P
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 and 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.
Table 2: 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 table
1 and text for explanation.
φÂp
sex + site
sex + state + site
(sex + state) × site
P
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
P
0.58
0.36
0.06
1.00
28
Table 3: Parameter estimates [95% c.i.] from the ‘φ(sex + state + site);p(sex + state);ψ(site + T)’ model.
Parameter
type
Survivala
(φ)
Recapture
probability
(p)
Transition
probabilityb
(ψ)
a Two-weekly
Contrast
intercept (‘I’ females, site A)
state ‘P’
males
site B
site C
site D
intercept (‘I’ females)
state ‘P’
males
intercept (site A, Jan. 1)
site B
site C
site D
Tc
logit scale
estimate
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
0.8 [0.73, 0.86]
Odds-ratio
estimate
0.70
0.52
2.35
1.20
1.76
[0.40,
[0.36,
[1.45,
[0.80,
[1.14,
1.22]
0.75]
3.82]
1.80]
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
1.60
14.1
1.19
[8.3, 123.0]
[0.56, 4.58]
[4.2, 47.5]
[1.14, 1.24]
survival.
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.
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).
b Probability