Appendix A Population response to a localized disturbance
Basic model
Assume a single, time-invariant, disturbance at location xd , with all model parameters having
constant values elsewhere. At equilibrium, Eq. Error! Reference source not found. in the text
implies that for all x  xd ,
0
e
N *( x)
 R  e0 N *( x)  m N *( x)  0
t
LD
x
 x y
dy .
LD 
 N ( y) exp 
*
0
(A.1)
Differentiating Eq. (A.1) with respect to x yields
0    e0  m 
x
dN * x  e0 *
e
 x 
 y 

N  x   20 exp    N * y  exp   dy .
dx
LD
L
LD
 D0
 LD 
From Eq. (A.1), the final term (identified by the bracket) is equal to 
(A.2)
R   e0  m  N * x 
.
LD
Making this substitution, defining and substituting N * x   N H* 1  n  x   , and re-arranging
(A.2) yields the differential equation
dn  x 
1
  n  x
dx
LR
(A.3)
where N H*  Rm 1 is the spatially homogeneous steady state (discussed in the text) and
e
 e 
LR  LD 1  0   LD 0 if e0  m .
m
 m
( .4)
Thus, downstream of any disturbance, the equilibrium population density approaches its spatially
homogeneous value exponentially with a decay constant 1/LR.
Model with general one-sided dispersal kernel
An extension of the above argument shows that the approximate result in Eq. ( .4) holds for any
one-sided kernel h(u) in Eq. (1), provided h(u )  0 at least exponentially as u   (i.e., we
exclude some “fat-tailed” kernels). The model assumes zero mortality in transit, implying

 h(u)du  1 , and the assumption on the tail guarantees that the mean distance traveled per jump
0

is LD   uh(u )du . Linearizing Eq. Error! Reference source not found. about the spatially
0
homogeneous steady state (see Chapter 5, Nisbet and Gurney 2003) yields the linear equation
x
0  e0n( x)  mn( x)  e0  h( x  y)n( y )dy .
( .5)
0
With the general kernel, the decay in population density downstream of a disturbance is
no longer a pure exponential, but can be represented as a sum or integral of decaying exponential
terms, each of which is a solution of Eq. ( .5). If a typical term is n( x)  exp  sx  , then we
define the response length to be the reciprocal of the smallest real value of s for which the
exponential form is a solution. Substituting s  LR 1 , inserting the exponential solution in Eq.
( .5), and rearranging terms, shows that LR is obtained by solving the equation
x
u 
e0  m
  h(u ) exp   du .
e0
 LR 
0
We approximate Eq. ( .6) by replacing the upper limit in the integral by infinity and setting
exp  su   1  su . Then, Eq. ( .6) implies
( .6)

e0  m
L
  k (u ) 1  su  du  1  sLD  1  D .
e0
LR
0
From this it follows that LR  LD
( .7)
e0
, which is the desired result.
m
Density dependence
Suppose R, m, and e0 are functions of local density. Then, Eq. (A.1) is replaced by
x
 x y
1
0  R N ( x)  m N ( x)  e0 N ( x) N ( x) 
e0 N * ( y ) N * ( y ) exp 
 dy

LD 0
 LD 

*
  
*



*

*

( .8)
Recall that the spatially homogeneous equilibrium is denoted by N H* . We use R , e0 , and m to
denote the spatial mean value of the recruitment rate, the per capita emigration rate, and the
mortality rate, respectively. Finally, we define
 de 
 dR 
 dm 
RN*  
, e*N   0 
, m*N  
.


 dN  N  N H *
 dN  N  N H *
 dN  N  N H *
( .9)
Linearizing Eq. ( .8) about the spatially homogenous equilibrium yields the following equation
for n(x) (defined above in Eq. (A.3)):


0  RN*  m  N H* m*N  e0  N H* e*N

n  x 
1
LD
x
 e
0
0
 x y
 N H* e*N n  y  exp 
 dy .
 LD 

( .10)
This equation is similar in form to Eq. (A.1) and the response length can be calculated by the
same reasoning as was used for that equation, yielding the result

e0  N H* e*N
LR  LD 1 
*
*
*
 m  N H mH  RH

.

( .11)
Appendix B Equilibrium population responses to spatial environmental variability in
parameters describing demography
As in the text, consider first variation in the recruitment rate alone. At equilibrium, Eq. (1) takes
the form
0
e
N *( x)
 R( x)  e0 N *( x)  m N *( x)  0
t
LD
x
 x y
dy .
LD 
 N ( y) exp 
*
0
(B.1)
By differentiating this equation with respect to x, and then proceeding as in appendix A, it can be
shown that the equilibrium population density obeys the differential equation

dN  x 
1
1  dR  x 
  N  x 
 R  x .
 LD
dx
LR
mLR 
dx

(B.2)
The structure of solution to an equation of this form is the sum of two components (details in
Nisbet & Gurney 2003, pp 27-29) – a “complementary function” involving the initial condition
(x = 0), and a “particular integral” solution that is independent of the initial conditions and
represents the persisting response, i.e. the spatial variation far from x = 0. Here we are only
concerned with the particular integral.
With the sinusoidal form of variation in recruitment assumed in Eq.
Error! Reference source not found., there are two “brute force” ways of deriving the solution
(see Nisbet & Gurney 2003, pp 29-39). The first option exploits the fact that the particular
integral is unique; we assume the solution has the form given in Eq.
Error! Reference source not found. and with some tedious trigonometry show that it is indeed
a solution if Eq. Error! Reference source not found. is valid. The second (mathematically
equivalent) option is to perform the analogous calculation using complex exponentials. The most
general approach, applicable to arbitrary spatial variation and not just sinusoids, is to use Fourier
analysis (for an ecologically oriented introduction, see Nisbet & Gurney 2003). The spatial
distribution of recruitment and the resulting equilibrium population distribution are expressed as
a sum or integral of sinusoids with different spatial frequencies, k:

1
R( x) 
2
 R(k ) exp ikx  dk

and
1
N ( x) 
2

 N (k ) exp ikx  dk .
(B.3)

The spatial frequency k is related to the more easily interpreted spatial wavelength LE used in the
text by the relationship k  2 LE . The functions R(k ) and N (k ) are the Fourier transforms of
R(x) and N(x) over the spatial domain 0  x   , and are defined by


0
0
R  k    R  x  exp  ikx  dx and N  k    N  x  exp  ikx  dx .
(B.4)
The Fourier transform can be interpreted as a complex number whose modulus represents the
amplitude and whose argument represents the phase of a sinusoidal pattern of environmental
variability. The motivation for its use in the our work is that, although the spatial distributions of
recruitment and of the population are related by a differential equation, their Fourier transforms
are related by an algebraic equation that can be readily interpreted. If R and N H* denote the spatial
means of R(x) and N(x) respectively, we can write the relation between the Fourier transforms of
R(x) and N(x) in the form
N (k )
R(k )
.
 TR (k )
*
NH
R
(B.5)
In Eq. (B.5), the transfer function TR(k) is a complex function whose modulus represents the ratio
of the proportional amplitudes of sinusoids of spatial frequency k; this is the ratio “b/a” used in
the text. By Fourier transforming Eq. (B.2), it can be shown with a little algebra – and noting that
the transform of the derivative of a function is ik times the transform of the function (Nisbet &
Gurney 2003, Appendix F) – that
TR  k  
with modulus
1  LD ik
,
1  LR ik
(B.6)
 k  LR  LD  
1  L2D k 2
and argument tan 1 
. The expression for b/a in Eq.
2
2
2 
1  LR k
 1  LD LR k 
Error! Reference source not found. of the text follows immediately from the modulus (noting
that k  2 LE ). The argument represents the phase shift (i.e. displacement in space of the peaks
or troughs, expressed in radians) between the sinusoidal variations in recruitment and in the
population. The downstream displacement LL is obtained by multiplying this lag by LE 2 .
Analogous analyses can be performed to describe equilibrium population responses to
spatial variation in the per capita mortality rate m(x), but one new complication occurs. In this
case, Eq. (B.1) is replaced by
 x y
1
N *( x)
 R  e0 N *( x)  m( x) N *( x)  e0  N *( y )
exp  
 dy .
t
LD
 LD 
0
x
0
(B.7)
One term in this equation (indicated by the underbrace) involves a product that impedes progress
to a simple algebraic formula like Eq. (B.5). To avoid this problem, we linearize Eq. (B.7) about
the spatial mean values of the per capita mortality rate and the population density (for examples
of such linearization, see Chapter 5 of Nisbet & Gurney 2003). If we define small deviations of
the per capita mortality rate from its mean value as m( x)  m 1   ( x)  and the resulting
population fluctuations as N * ( x)  N H* 1  n( x)  , then by substituting these definitions and
Fourier transforming the linearized form of Eq. (B.7) and rearranging terms, we find that the
transfer function for mortality fluctuations is
Tm (k ) 
1  LD ik
n( k )

,
 (k )
1  LR ik
(B.8)
which is identical to Eq. (B.6), apart from the negative sign that arises because a uniform (k = 0)
increase in m causes a decrease in population density.
Appendix C Equilibrium population responses to spatial environmental variability in
parameters describing dispersal
Eq. Error! Reference source not found. in the text can be derived as a limiting situation for a
more general model that uses a pair of coupled partial differential equations that separately
describe dispersing ND(x) and sessile N(x) sub-populations (Lutscher et al. 2005). At steady state,
these equations take the form
0
0
N * x 
t
N D*  x 
t
 R  x    m  x   e0  x   N * x    N D*  x 
v  x 

  v  x  N D*  x    N D*  x 
   x  N D*  x   e0  x  N * x 
x
x
(C.1)
where v is the advection speed, μ is the rate of settlement from the dispersal to the sessile mode,
and the average dispersal length LD equals v  . Equation Error! Reference source not found.
is obtained as the limiting case where    and v   but the ratio v  retains a fixed finite
value LD. This corresponds to a situation where organisms spend an infinitesimally small portion
of their lifetime in transit between locations on the benthos. Mathematically, the limit is
problematic if v and/or μ vary spatially, so in this appendix we work with the system in Eq. (C.1)
, taking any required limits at the end of calculations.
Consider small deviations from spatial population averages defined by
N * x   N 1  n  x   and
N D*  x   N D 1  nD  x  
(C.2)
caused by one or more of the parameters varying spatially,
e0  x   e0 1    x   ;   x    1    x   ; v  x   v 1   x   .
Linearizing Eq. (C.1) yields
(C.3)
0  e0 N  n  x     x    mNn  x    N D  nD  x     x  
 
 n
0  v N D  D 
   N D  nD  x     x    e0 N  n  x     x   .
x 
 x
(C.4)
By taking Fourier transforms and rearranging terms, we can derive a transfer function that relates
variation in the per capita emigration rate to variation in the population distribution of sessile
individuals (assuming  ( x)  0 and  ( x)  0 ),
T (k ) 
e LD ik
( L  LD )ik
n( k )
 0
 R
.
 (k )
m 1  LR ik
1  LR ik
(C.5)
The proportional amplitude ratio, plotted in Fig 3d, is
T (k ) 
LD k
2 ( LR  LD )
b e0


.
2
2
a m 1  k LR
LE 2  4 2 LR 2
(C.6)
 L 
The downstream displacement is  tan 1  E  . If we follow the conventional rule that the
 2 LR 
(multi-valued) inverse tangent function is assigned a value in the range   / 2,  / 2  , this would
imply an ecologically impossible upstream displacement of an effect from its cause. The
downstream displacement LL is obtained by adding one wavelength with the result that
LL  LE 
 L 
LE
tan 1  E  . This formula was used to calculate Fig. 3e.
2
 2 LR 
There are two components of LD that may vary spatially: the settlement rate μ, and the
advection speed v. Equation (C.4) can be used to derive the transfer function for both sources of
variation. Both transfer functions are to proportional to N D , which implies T  k   0 as N D  0 .
Appendix D Additional background on parameter estimation for Leuctra nigra in Broadstone
Stream
Broadstone Stream lacks fish, thus we assume that L. nigra mortality is due entirely to predation
by the two common predators in this stream, the caddisfly Plectrocnemia conspersa and the
alderfly Sialis fuliginosa. Speirs et al. (2000) reported that P. conspersa had an average density
of 119 m-2 and an attack rate on L. nigra of 2.77 x 10-5 m2 day-1. S. fuliginosa has an average
reported density of 44 m-2 and an attack rate while foraging on L. nigra of 1.92 x 10-5 m2 day-1.
Assuming an additive effect of both predators on L. nigra mortality and constant predator
densities, we obtain an L. nigra mortality rate m = 0.00414 day-1.
Winterbottom et al. (1997b) present the ratio of the density of colonists that settled in a
cleared experimental area over one week to background benthic densities (their “Index of
Mobility”) across a range of daily discharges D. Assuming the surrounding system was in
equilibrium (i.e., immigration and emigration in the surrounding habitat were in balance), but
that no organisms left the experimental arena (in which case immigration rate can be estimated
from the number of organisms accumulated in the experimental area), the Index of Mobility
provides a per capita emigration rate in units of week-1. Winterbottom et al.’s data suggest an
exponential relationship,
e0  eB 0 exp  D  ,
(D.1)
yielding a baseline per capita emigration value of eB0 = 0.98 week-1, which equals 0.14 d-1, and η
= 71.55 seconds m-3. Speirs & Gurney (2001) report that the average cross-sectional area of
Broadstone Stream is 0.1932 m2. Recasting η as responsiveness to velocity Vs (in m s-1), rather
than discharge, yields η = 13.82 seconds m-1.
Elliott (1971) observed that the dispersal kernel for L. nigra was well-described by an
exponential distribution, which was approximately directly proportional to stream velocity at
LD  35 VS ,
where VS is the stream velocity in m s-1 and LD is the average dispersal length in meters.
(D.2)