Online Supplementary Materials
Our model of adaptation-by-time (ABT) consists of a recursion equation that tracks
the joint breeding value distribution of reproductive date (x) and a trait (z) that is under
selection that varies with date. Here we refer to this selected trait as “body size” simply
for ease of presentation. We make the simplifying assumption that this joint distribution,
2
2
p(x,z) , is bivariate Gaussian with means x and z , variances v x and v z , and a
correlation between breeding values for the two traits denoted by  . In population
genetic terms,  is a measure of linkage disequilibrium between the traits. We start by

supposing that the actual reproductive date, y, of an individual
with a breeding value for
2

reproductive
date, x, is normally distributed with a mean of x and a variance of v ;
 (y  x) 2 
1
 (y, x) 
exp
.
2
2 v

 2v 



Given a particular
breeding value distribution, we can then calculate the mean breeding
value for body size on each reproductive date, y, as:
z (y) 
 zp(x,z) (y, x)dxdz ,
 p(x,z) (y, x)dxdz
(1)

z (y) denotes the mean value of z on reproductive date y. Carrying out
where the notation
the integration in (1) yields

z ( y)  z 
vxvz y  x 
v x  v
2
2

vz 2
h  y  x   h 2  y  x b ,
vx
(2)
where b is the regression coefficient of y on x (i.e., b  v z /v x ) and h2 is the heritability
2
2
2
of reproductive date (i.e., h 2  v x /(v x  v ) ). Equation (2) reveals that the extent of

local adaptation to a given date (i.e., the difference between the “local” average body size
 population; z (y)  z ) depends on the strength of the linkage
and that of the entire
disequilibrium,  , the time difference between that date the population mean breeding

value for reproductive date ( y  x ), and the heritability of reproductive date.
We census the population in the non-reproductive stage and denote the joint

(reproductive date/body
size) breeding value distribution in generation t by pt (x,z) . The
first event in the life cycle is the initiation of reproduction, where all individuals in the
reproductive date, y,
population get distributed to specific reproductive dates. The actual
of an individual with a breeding value for reproductive date, x, is given by equation (1).
Therefore, the joint distribution of breeding values on any given date, y, (which we
denote by q(x,z;y) ) is

q(x,z;y) 

pt (x,z) (y, x)
.
pt (x,z) (y, x)dxdz
(3)
 event in the life cycle is selection on body size, which we model as an
The next
optimum size that varies with date (i.e., (y) ). To make the calculations tractable, we

assume that the fitness of an individual of body size z on reproduction date y is given by
the Gaussian function
 z   (y)2 
.
w z z, y   exp
2 z 



(4)
 joint distribution of breeding values on date y after selection on body size
Therefore, the
(denoted by q s (x,z;y) ) is

qs (x,z;y) 
q(x,z;y)w(z, y)
.
 q(x,z;y)w(z, y)dxdz
(5)
 in the life cycle is actual reproduction. For simplicity, we assume that
The next event
reproduction is asexual, and therefore the joint distribution of breeding values remains
unchanged. If we were to allow for sexual reproduction, the joint means of the breeding
value distribution would remain unchanged, but the variation around these means might
be altered.
The final event in the life cycle is the mixing of all individuals back into the
overall population during the non-reproductive stage. Specifically, we suppose that the
contribution of different reproductive dates to the overall population (denoted by g(y)) is
described by a Gaussian function with its maximum at date  and a width of  x :
 (y  
) 2 
w x (y)  exp
.
 2 x 


(6)
This function is weighted by the reproductive activity on each date: i.e., the proportion of
the population reproducing on a given date: g(y) 
 p (x,z)(y, x)dxdz. As a result, the
t
joint distribution of breeding values in the next generation (measured in the nonreproductive stage) is

pt 1(x,z) 
 g(y)q (x,z;y)w (y)dy .
s
x
(7)
 (7) can be combined into a single integro-difference equation for
Equations (3) through
the dynamics of the joint breeding value distribution. This equation predicts how both
body size and reproductive date evolve from one generation to the next. Assuming an
equilibrium breeding value distribution is attained, pˆ (x,z) , we can plug this equilibrium
equation into equation (1) to calculate the mean body size on each date, z ( y ) , and see
how this compares to the optimum for thatdate, (y) .
In order to maintain tractability, we now make some further simplifying
 body size is a linear function of reproductive
assumptions. First, we suppose that optimal
date. Without loss of generality, we then standardize body size so that the optimal value
is zero on the date having the largest reproductive contribution to the population as a
whole ( y   ), with this date also set at zero (   0 ). We therefore have (y)  y where
 is the slope of the temporal cline in optimal body size and equation (6) becomes

 w (y)  exp[ 2 /2 ]
x
x . The resulting breeding value distribution will remain Gaussian,


and we make the additional simplifying assumption that the variances of this distribution
are constant through time. Thus we track only the dynamics of the means x and y and
the correlation  .


The above recursion can be used to obtain recursions for the means and the

correlation.
These calculations are straightforward but tedious (unpubl. results). They
reveal that, at equilibrium, we have xˆ  0 and zˆ  0 ; the mean breeding value for
reproductive date in the population as a whole is zero, as is the mean breeding value for

 mean breeding value for reproductive date should
body size. This is expected because the
evolve to match the date in the season that yields the largest reproductive returns (which
we have standardized to zero). Similarly, the population average for body size should
settle at zero as well, because (1) most of the population reproduces on the date y = 0, (2)
optimal body size at this date is zero, (3) reproduction by other individuals is symmetrical
distributed around the mean date, and (4) optimal body size increases or decreases
linearly as reproductive date deviates from y = 0.
ˆ , is very complex (unpubl.
The expression for the equilibrium correlation, 
results) and sheds little intuitive light on how adaptation occurs across time. We can gain
 for the case where selection on body size
more insight by approximating this equilibrium
is weak (i.e.,  z is large). In this case we have

ˆ

v x  x
.
v z v 2   x
(8)
Substituting 
this into equation (2) and simplifying yields the following equation for the
equilibrium average body size as a function of reproductive date
z ( y)  h 2
y
v
2
x
,
(9)
1
where h2 is the heritability of reproductive date.
This equation can be directly compared to results for adaptation across spatial
clines. The equilibrium prediction for García-Ramos & Kirkpatrick’s (1997) spatial
model under a set of assumptions that most closely matches those of the present temporal
model is
z ( y) 
y
d
 1
 n2
2
,
(10)
where  is the strength of stabilizing selection around the optimum at each spatial
location (large values correspond to weak selection),  d is the dispersal distance (large
2
values correspond to high dispersal), and  n is the width of the population density
2

distribution across space (larger values correspond to a population density that is more
uniform across space).

Comparison of the temporal (equation 9) and spatial (equation 10) models reveals
that adaptation in both is similar function of the optimal trait cline (  ), dispersal ( v or
2
 d 2 ), and the width of the density distribution (  x or  n 2 ). The spatial model also


includes the effects of stabilizing selection on the trait (  ), which was present in our



model before we assumed it was weak to obtain an intuitive solution. The main
difference between the two models is that adaptation along temporal clines is also a
function of the heritability of reproductive time, whereas the spatial model does not
include a heritability of reproductive location.
Literature cited
García-Ramos G, Kirkpatrick M (1997) Genetic models of adaptation and gene flow in
peripheral populations. Evolution, 51, 21–28.