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Online Supplement S1. – Stability analysis of the RDA producer-consumer model
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The RDA model is given by eq. (1) in the text,
P  x, t 
 I P  f  P  x, t   C  x , t 
t
production
3
losses to consumption
C  x, t 

2
 I C  mC  x, t   v  P  x, t   C  x, t    2  D  P  x, t   C  x, t  
t
x
x
recruitment
mortality
advection
. (S1.1)
diffusion
4
We approximate eq. (S1.1) in terms of small, linear deviations from the spatially uniform steady
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state (P*, C*). Substituting deviations P  x, t   P*  p  x, t  and C  x, t   C*  c  x, t  and
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ignoring non-linear terms yields
p
  f   P*  C * p  f  P*  c
t
.
2
2
c
*
* p
* c
*
*  p
*  c
 mc  v  P  C
 v  P   D  P  C
 DP  2
t
x
x
x 2
x
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We define the Fourier transforms for the spatial dimension x
p k, t  
9

 p  x, t  e
 ikx
dx

c k,t  
and

where k is the spatial frequency, and apply to eq. (S1.2), yielding
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
 f   P*  C *
 p
n
 Jn, where n    and J  
t
 v  P*  C *ik  D  P*  C *k 2
c 

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The matrix J has a characteristic polynomial

 c  x, t  e
 ikx
dx
(S1.3)

10
13
(S1.2)

 f  P* 

 . (S1.4)
m  v  P*  ik  D  P*  k 2 
 2  m  v  P*  ik  D  P*  k 2  f   P*  C * 



 f   P*  C * m  v  P*  ik  D  P*   f  P*  v  P*  C *ik  D  P*  C *k 2

(S1.5)
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where the roots 1,2  1,2  i1,2 are the eigenvalues of J. Using routine but tedious algebra, we
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solve for the real and imaginary parts of the eigenvalues λ1,2,
1

1
1,2   m  D  P*  k 2  f   P*  C * 
2

1

1
1,2  vk  sgn  
2

where
 2   2   


2
 2   2   
,


2
(S1.6)

 


  D  P*  k 4  2 D  P*  k 2 m  f   P *  C *  m  f   P *  C *  k 2 v 2  4 D   P *  f  P * 
2


2

  2vk D  P*  k 2  m  f   P*  C *  4v  P*  C *kf  P* 
2
The terms outside of the radicals in real parts ρ of the eigenvalues are negative. Therefore, the
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only eigenvalue that can contribute to instability has a real part whose radical terms are led by a
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positive sign, which is λ1.
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The system eq. (S1.4) is always stable when deviations from the homogenous steady-
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state are spatially uniform with frequency k = 0. Setting k = 0 and performing routine algebra,
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we find that ω1,2 = 0 and the eigenvalue λ1 = max{-m, -f´(P*)C*}. Thus, the system is always
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stable and overdamped.
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When deviations become infinitely frequent in space (i.e., k→∞), the system also
becomes stable providing D(P*) > 0 and D´(P*) > 0. For eq. (S1.4),
lim Tr  J    D  P*  and lim Det  J   f   P*  C * D  P*   f  P*  D  P*  .
k 
k 
(S1.7)
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Routh-Hurwitz conditions for a 2 x 2 matrix require Tr(J) < 0 and Det(J) > 0 for stability which
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are both clearly met in the limit of k→∞. In contrast, the system is not guaranteed to be stable
14
when D(P*) = D´(P*) = 0 as k→∞. In this case,
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 v  P*  C * f  P* 
v  P*  f   P*  C *  v  P*  C * f  P*  
lim 1  k   max 
 m, 

k 
v  P* 
v  P* 


2
(S1.8)
1
The system will therefore become unstable when
 v  P *  C * f  P * 
v  P* 
 m.
In total, these results indicate that the system in eq. (1) with advection only (D(P*) =
2
3
D´(P*) = 0) will have one critical point kv where ρ1 crosses the zero axis when unstable. When
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eq. (1) has both advection and diffusion and is unstable, there will be a second critical point kd
5
where ρ1 crosses the zero axis and therefore a maximum value of ρ1 at some point kmax.
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Further considerations of producer-dependence in consumer dispersal.
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We have shown above that the RDA model is stable in the absence of dispersal (i.e., when v(P*)
9
= D(P*) = 0). Additionally, the RDA model lacks activator-inhibitor feedbacks in the absence of
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dispersal, as the signs of the Jacobian in eq. (S1.4) in this case are
 
J=
.
 0 
11
(S1.9)
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A simple heuristic alteration of the RDA model demonstrates how producer-dependence
13
in consumer dispersal mediates the ‘activating’ feedback on consumer densities. Consider now
14
the RDA model with advection and diffusion terms replaced by a producer-dependent emigration
15
term,
P  x, t 
 I P  f  P  x, t   C  x, t 
t
production
16
losses to consumption
C  x, t 
 IC
 mC  x, t   E  P  x, t   C  x, t 
t
recruitment
+ immigration
mortality
.
(S1.10)
emigration
17
The term IC now includes immigration from nearby sites as well as recruitment from outside the
18
system. This system is also similar to the general IDE eq. (12) with the convolution integral
3
1
describing dispersal being replaced with a constant. We now follow the same steps that took us
2
from (S1.1) to (S1.4), which yields a Jacobian matrix J with signs
3
 
J=
.
 
4
The change in the lower left entry from zero to ‘+’ shows that producer-driven dispersal in
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consumers has an ‘activating’ effect. However, Tr(J) < 0 and Det(J) > 0 for eq. (S1.11) above,
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demonstrating stability. Thus, producer-driven dispersal appears necessary, but not sufficient,
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for instability. When the emigration and immigration are made spatially explicit through
8
advection, the lower right entry becomes complex, which is what allows it to exert a de-
9
stabilizing influence on the system via the dominant eigenvalue eq. (S1.6).
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(S1.11)
1
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Online Supplement S2. – Spatiotemporal dynamics in the RDA model
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Development of spatial patterns over time in the RDA producer-consumer model eq. (1).
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Dynamics are calculated as in Fig. 1, including functional forms and parameter values.
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Producers and consumers exhibit small, quasi-random deviations from their uniform space
6
equilibria at t = 0. These deviations quickly grow into regular waves that occur with spatial
7
frequency kmax ≈ 25 rad/unit x. Since the mortality rate in the particular simulation is m = 0.1 t-1,
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the average consumer lifetime is ten model time steps, and patterns at the dominant spatial
9
frequency arise within this timeframe. The persistent pattern has largely formed within about
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two consumer lifetimes, while remaining small differences in the amplitudes of the consumer
11
waves reflect transient effects of other spatial frequencies.
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See supporting video clip (filename: ele_1727_sm_videoS1)
ele_1727_sm_videoS
1.gif
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1
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Online Supplement S3. – Details on estimation of plausible parameter values
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Hillebrand (2009) conducted a meta-analysis of phytoplankton control by grazing consumers in
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aquatic systems. Results are presented as log response ratios of grazed to ungrazed biomass. We
5
assume that the removal of the phytoplankton subject to grazing can be described by
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dP
  CP; P  t   P  0  exp   C *t  .
dt
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We further assume that phytoplankton are at equilibrium during grazer addition, P  0   P*
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meaning that the log response ratio LRR is
 P t  
*
LRR  log 
   C t
 P  0 
9
(S3.1)
C*  0
(S3.2)
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where t is equal to the experimental duration. Hillebrand (2009) reports LRR values that range
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from no effect up to -5.5 with an average of ~ -0.9. Similarly, experimental durations appear to
12
range from 1 hour to two weeks with a median of about 4 days. This yields values of
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proportional consumption that could range anywhere from 0 to 132 d-1 with a typical value of
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~0.23 d-1.
15
,
We estimate the effect of producer biomass on consumer per-capita emigration rates
16
using two sets of sources. The first uses short duration and small spatial scale experimental data
17
on Baetis mayflies in Kohler (1985; Fig. 4, patchy and uniform habitats), Forrester et al. (1999;
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Fig. 3, all treatments), and Roll et al. (2005; Fig. A, C, and D, all treatments). In all cases, per-
19
capita emigration rates were observed over a range of stream periphyton biomass densities and
20
suggested negative trends. (Chironomids were also studied in Forrester et al. and Roll et al. but
21
exhibited no discernible dependence of per capita emigration on producer biomass.) Biomass
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1
densities were reported in cm2 and converted to length units using the reported width of
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experimental stream channels in each study. The exponential function describing per-capita
3
emigration eq. (20) was fit to each dataset by minimizing the residual sum of squares (Kohler: e0
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= 0.6, α = 57; Forrester et al.: e0 = 0.09, α = 0.0055; Roll et al.: e0 = 0.03, α = 0.81). To obtain a
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range of proportional per-capita emigration sensitivities αP*, we assumed that equilibrium
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producer biomass P* for each study was likely to fall within the range of reported experimental
7
values and multiplied estimates of α by the minimum and maximum of these. This yielded the
8
values 0.29-17 for Kohler, 0.029-2.6 for Forrester et al., and 0.11-2.8 for Roll et al.
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The second source uses larger scale observation data on 17 macroinvertebrate taxa
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collected from a set of experimental stream channels as described in (Diehl et al. 2008). Diehl et
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al. estimated “response lengths” using emigration rates from each stream channel eC, where
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response length 
eC
e
LC  G LD , eG  e0 exp   P*  ,
m
m
(S3.3)
13
and the channel length LC = 50 m (full details given in Diehl et al. 2008, Appendix B). Diehl et
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al. provided a guesstimate of m = 0.01 d-1 that equals a long average lifespan of 100 days to
15
obtain response lengths that range from 1-218 m. The stream channels used by Diehl et al. had
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average flow velocities of 0.21 m/s; average dispersal distances LD for stream organisms vary
17
widely (see Discussion), but typical values in this velocity range are around 5 m. Using this
18
assumed value and the already stated per capita mortality estimate yields estimates for eG of
19
0.002-0.436 d-1. Of the 17 taxa studied by Diehl et al., 7 had values of eG < 0.02 while the other
20
10 had values of eG > 0.07, and four of these had values of eG > 0.15.
21
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Potential values of the maximum per capita emigration rate e0 from Diehl et al. inhabit a
several order of magnitude range depending on the value of αP* used. The discrepancy in e0
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1
estimates between the small scale experimental studies of Kohler, Forrester et al., and Roll et al.
2
and the observational data of Diehl et al. highlight the difficulties of indirectly estimating
3
maximum emigration rates. Most short term experimental and observational studies follow
4
organisms over shorter periods—allowing individual organisms to drift at most once—and lack a
5
“no resource” treatment (Kohler is an exception in the latter regard). Thus, estimates of e0 are
6
artificially constrained to be small, whereas it is possible that consumers with high drift
7
propensities would do so quite frequently in a completely resource denuded environment.
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Literature Cited
Diehl S., Anderson K. & Nisbet R.M. (2008). Population responses of drifting stream
invertebrates to spatial environmental variability: New theoretical developments. In:
Aquatic insects: Challenges to populations (eds. Lancaster J & Briers RA). CABI
Publishing, p. 158-183.
Forrester G.E., Dudley T.L. & Grimm N.B. (1999). Trophic interactions in open systems: Effects
of predators and nutrients on stream food chains. Limnol. Oceanogr., 44, 1187-1197.
Hillebrand H. (2009). Meta-analysis of grazer control of periphyton biomass across aquatic
ecosystems. J. Phycol., 45, 798-806.
Kohler S.L. (1985). Identification of stream drift mechanisms: An experimental and
observational approach. Ecology, 66, 1749-1761.
Roll S.K., Diehl S. & Cooper S.D. (2005). Effects of grazer immigration and nutrient enrichment
on an open algae-grazer system. Oikos, 108, 386-400.
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