Supplementary Material
Appendix I. Details of the Cellular Automata Model.
Although prior studies provide clear evidence for gross increases or decreases in
the vital rates over spatial gradients, the precise functional forms of most parameters are
not known. Therefore, we assumed the simplest functional relationships supported by
available data, and then use the model to generate qualitative trends over an idealized
two-dimensional landscape, rather than make specific numerical predictions. In the
Discussion we consider the effects of choosing different functional forms for key
parameters.
In the cellular automata (CA) model, a mussel bed is represented as a rectangular
lattice of cells, where each cell is a potential site for the location of a mussel. Each cell is
identified by its ( x, y ) coordinate in a w  h lattice, where the horizontal dimension
represents an alongshore gradient in wave energy and the vertical dimension represents a
gradient of tidal height. To facilitate comparisons with lattices of different sizes, the x
and y coordinates are scaled to vary from zero to one. Each cell can take on integer state
values representing mussel size. The state of cell in position ( x, y ) at time t is
S xy (t )  0, 1,
, s , xy  , where 0 is an empty cell, 1 is a cell with a newly recruited
mussel, and s , xy is the asymptotic maximum size of a mussel at a particular x, y
coordinate, which varies with shore level and wave energy. Model dynamics are
described as stochastic transitions among states, where the transition probabilities for any
given cell are determined by the current state of the cell, the cell’s position in the lattice
(gradient effects), and the states of the surrounding cells (“neighborhood effects”). The
transition probabilities represent position-specific mussel recruitment, growth to larger
sizes, size dependent predation, and background mortality, which includes mortality for
physical stress.
Spatial patterns of mussel recruitment are perhaps the least understood aspect of
mussel ecology. It is known, however, that M. californianus recruitment probabilities
often increase with wave energy along a given shore level, reaching a maximum on wave
beaten shores (Menge 1992; Robles 1997; Moya 2005). To estimate a 2-D surface of
mussel recruitment, vertical transects sampling naturally occurring M. californianus
recruits (1-5 mm long) were placed at regular alongshore intervals on a shore with a
horizontal gradient of wave energy. Samples were made late winter of three successive
years, the season with the highest recruitment of M. californianus on the Catalina site
(Robles 1997). The recruitment surface showed a peak occurring in the mid-intertidal
zone at the wave-exposed extreme and decreasing towards lower wave energies and more
extreme shore levels (Figure S1). The CA model assumes that recruitment rates to empty
cells peak at mid-shore levels and high wave energies. Along the vertical axis,
recruitment rate declines from the peak mid-shore value to higher and lower shore levels
according to a Gaussian. Along the horizontal axis, recruitment rate declines
exponentially with decreasing wave energy. The gradient effects in recruitment were
summarized by the following function:
 xy  1e (1 x )e y  y
m
2
(2 )
,
(A1)
where   ln 1  0  ,  0 and  1 are minimum and maximum recruitment rates along
the wave energy gradient, ym is the shore level where recruitment rates reach a peak, and
 is a parameter which specifies the rate of recruitment decline with shore level as one
moves away from the mid-tidal peak.
Mussels have indeterminate, environmentally influenced growth (Sebens 1987).
Growth rates and maximum (“terminal”) sizes increase with longer immersion times (i.e.
lower shores levels; Dehnel 1956; Garza 2005), and higher nutrient flux (i.e. higher wave
energies; Leigh et al. 1987; Dahlhoff and Menge 1996). In the model, mussel growth
rates were expressed with the von Bertalanffy function:
S xy    s , xy  S xy  .
(A2)
The expected growth increment of a mussel, S xy , is proportional to the difference
between the size of a mussel, S xy , and its maximum asymptotic size, s ,xy , where  is
the growth rate. Maximum asymptotic size depends on flow rate and immersion time and,
therefore, increases with increasing wave energy and decreasing shore level. The CA
model uses a simple product of linear trends along the horizontal and vertical dimensions:
s, xy  s0   s1  s0  x 1  y  ,
(A3)
where s0 and s1 are the minimum and maximum asymptotic sizes on the lattice.
The CA model includes two possible sources of mussel mortality: background
mortality and predation. For simplicity, the same low rate of background mortality 0 is
assumed for all mussels independent of their size and location in the lattice. Mortality
rates due to predation are the product of the per capita attack rate of predators,  xy , and
the predator density, P (t ) . Predation rates decrease with lengthening periods of tidal
emersion (i.e. vertically towards higher shore levels; Paine 1974; Menge 1992; Robles et
al. 1995, Garza 2005) and greater hydrodynamic stress (i.e. horizontally from sheltered to
wave-exposed areas; Menge 1983). The CA model uses a simple linear trend from a
maximum predation rate 1 at the lowest wave energy and lowest shore level to a
minimum rate 0 at the highest wave energy and highest shore level:
 xy  0  1  0  2  x  y  2 .
(A4)
All other rates are assumed to be constant within the lattice.
Rates of recruitment and predatory mortality are modified by neighborhood
effects. Local densities of mean mussel biomass were computed for neighborhoods of
size r as a simple average of the size of the mussels in the grid of (2r  1)  (2r  1) cells
centered at ( x, y ) :
S xy ,r (t )   2r  1
2
xr
y r
 S
i  x r j  y r
ij
(t ) .
(A5)
At the sides and corners of the lattice, the local densities were calculated as the mean of
the subset of cells within the neighborhood.
Our observations and field experiments (Moya 2005) indicate that while M.
californianus can settle out on any rough surface, and recruitment rates are higher under
adults than algae or bare rock (Robles unpublished data). The CA model assumes that
rates of recruitment to empty cells increase exponentially with the density of surrounding
mussel biomass:
 xy (t )   xy e
aS xy ,b ( t )
,
(A6)
where a is a parameter representing the strength of the recruitment neighborhood effect,
b is the size of the recruitment neighborhood, and  xy is given by equation (A1).
Experimental studies (Bertness and Grosholtz 1985; Robles et al. 2009; Fong and
Robles, unpublished data) indicate that younger, smaller mussels are shielded from
predators by the larger less vulnerable members of an aggregation. Thus, the likelihood
of predation decreases as a mussel becomes surrounded by larger, less vulnerable
individuals. The CA model assumes that large mussels are more resistant to predation and
that mussels surrounded by large conspecifics are less susceptible to attack. This
neighborhood effect is modeled as a negative exponential decrease in predator attack rate
with an increase in the density of neighborhood mussel biomass. The total mortality rate
of a mussel in cell ( x, y ) is given by
 xy (t )  0   xy P(t )e
 cS xy ,d ( t )
,
(A7)
where c is the strength of the predation neighborhood effect, d is the size of the
predation neighborhood, 0 is the background mortality rate, and  xy is given by
equation (A4).
Experimental evidence shows that sea stars congregate in episodes massive
recruitment of small mussels and disperse as these preferred prey are reduced in
abundance relative to the larger mussels (Robles et al. 1995). This aspect of predation
constitutes a numerical response. In the CA model, predator density is an additional
global variable. The density of predators is modeled as an immigration-emigration
process. It is assumed that predators enter the system at a constant rate I and exit the
system at a per capita rate EP (t ) that is inversely proportional to the overall per capita
consumption rate of prey:
EP (t ) 
( wh)
1
S
x, y
e0
xy
(t ) xy e
 cS xy ,d ( t )
,
(A8)
where the sum in the denominator is over all the cells in the lattice. The parameter e0 is
the constant of inverse proportionality between mussel biomass consumed and rate of
emigration.
In the CA model, mussel recruitment, growth, and mortality are treated as
stochastic events. For mussel recruitment and mortality, this was done by choosing an
iteration time step t and assuming rates are constant within that time step. This
assumption yields an exponential function for the probabilities. The probability of a
recruitment event into an empty cell ( x, y ) is given by


Pr S xy (t  t )  1 S xy (t )  0  1  e
 xy ( t ) t
.
(A9)
.
(A10)
Similarly, the probability that a mussel in cell ( x, y ) dies is given by


Pr S xy (t  t )  0 S xy (t )  0  1  e
  xy ( t ) t
Each mussel that dies is assigned a cause of death: background mortality with probability
0  xy (t ) or predatory mortality with probability 1  0  xy (t ) . The total biomass of
mussels consumed by predators in each time step, Cˆ (t ) , is used in the computation of
predator emigration rates (see below). If a mussel escapes a random mortality event, then
it may experience a growth event in that time step. While this assumed ordering of
survival followed growth seems arbitrary, from a practical consideration, growth occurs
slowly on the time scale of our model iterations, so that this assumption is of little
importance. A truncated Poisson distribution is used to grow mussels:
 e  xy xyj
,
for 0  j  s , xy  i,

 j!
 s ,xy i 1  xy k
e xy

Pr S xy (t  t )  i  j S xy (t )  i  1  
, for j  s , xy  i,
k
!
k

0


0,
for j  s , xy  i,



(A11)
where xy  S xy t is the expected growth increment over the time interval t .
Predator immigration and emigration are handled as a stochastic birth-death
process. However, predator movements occur on a much faster time scale than changes in
mussel biomass. We, therefore, found it necessary to implement a second, shorter time
step of   t n for predator immigration and emigration events. During the time
interval t , predators randomly enter and leave the system every  based on mussel
biomass densities from the previous t time step; i.e. mussel densities are not updated
during the n time steps of size  . Without this assumption, the expected number of
predator immigrants is whI t , which may, in fact, exceed the theoretical equilibrium
number of predators unless t is very small. On the fast time scale, the predator
immigration probabilities are given by a Poisson distribution:
e whI  (whI  ) j
,
Pr (t   )  j 
j!
(A12)
where  (t   ) is the number of predators entering the system. For emigration, the
probability of an individual predator leaving the system was calculated using equation
(A8) where the summation in the denominator was replaced with Cˆ (t ) , the actual
biomass of mussels eaten during the previous time interval t . The probability that a
predator emigrates during the time interval  equals
ˆ
eP  1  e t e0 whP (t ) C (t ) ,
(A13)
and the probabilities for the number of emigrating predators is given by the binomial
distribution:
 whP(t )  j
whP ( t )  j
,
Pr (t   )  j  
 eP 1  eP 
 j 
(A14)
where (t   ) is the number of predators leaving the system and whP (t ) is the
number of predators at the beginning of the time interval (product of the lattice size and
the predator density). At the end of each time interval  , the new predator density
equals
P(t   )  P(t )   (t   )  (t   )  (wh) .
Equation (A15) is iterated n times for every t time step.
Parameter values used in the simulations appear in Table S1.
(A15)
Table S1. Variables and Parameters of the Cellular Automata Model.
Symbol
Units
Description
Time
t
days (d)
Time
t
1–8d
mussel bed time step; varies†

2 4  2 12 d
predator time step; varies†
2 7  214 (unitless)
number of predator time steps per mussel bed time step; varies†
1 cell
each cell holds a single mussel
x
(unitless)
horizontal location of a mussel in the lattice, scaled 0 to 1
y
(unitless)
vertical location of a mussel in the lattice, scaled 0 to 1
n
Space
area
w
10000 or 16000
h
250
cell
width (number of cell widths) of the mussel bed lattice; 16000
cell cells on single within-site lattice; 10000 cells on each between-site
lattice
height (number of cell heights) of the mussel bed lattice
Mussel Size
S xy (t )
r
mm
size of a mussel in position x, y
1 (unitless)
neighborhood radius; number of cells in each direction
S xy ,r (t ) mm
mean size of mussels within radius r of position x, y (Eq A5)
Mussel Recruitment
 xy
d1
mussel recruitment rate into empty cell at position x, y without
neighborhood effects (Eq A1)
0
0.0 d1
minimum recruitment rate along the wave energy gradient
1
1.0 d1
maximum recruitment rate along the wave energy gradient
ym
0.5 (unitless)
tidal height location of mussel recruitment peak

0.01 (unitless)
shore level width of mussel recruitment curve around peak
a
0.05 mm1
enhancement of recruitment with neighboring mussels
b
1 (unitless)
recruitment neighborhood radius
 xy (t )
d1
mussel recruitment rate into empty cell at position x, y with
recruitment neighborhood effects (Eq A6)
Mussel Growth
s ,xy
mm
asymptotic size of a mussel in position x, y (Eq A3)
s0
30 mm
minimum asymptotic size of a mussel within lattice gradients
s1
200 mm
maximum asymptotic size of a mussel within lattice gradients

0.0007 d1
decrease in mussel growth rate with size
S xy
mm d1
mean size change per unit time of mussel in position x, y (Eq A2)
 xy
mm
expected mussel growth increment
Mussel Mortality
0
0.0001 d1
background per capita mussel mortality rate
d1
total per capita mortality rate of a mussel in position x, y (Eq A7)
 xy
cell pred1 d1
predator attack rate at position x, y (Eq A4)
0
0.0 cell pred1 d1
minimum predator attack rate within lattice gradients
1
1.0 cell pred1 d1
maximum predator attack rate within lattice gradients
c
0.04 mm1
decrease in predator attack rate with mean mussel size
d
1 (unitless)
predation neighborhood radius
 xy (t )
Predator Movement
pred cell1
predator density
0.01 pred cell1 d1
predator immigration rate
(t )
pred
number of predators immigrating into the lattice (Eq A12)
EP (t )
d1
per capita predator emigration rate (Eq A8)
5.0 mm pred1 d2
predator emigration constant per unit prey consumed
mm
total mussel biomass consumed during previous time interval
(unitless)
probability that predator emigrates during interval  (Eq A13)
pred
number of predators emigrating from the lattice (Eq A14)
P (t )
I
e0
Cˆ (t )
eP
 (t )
†
Time steps were varied using an optimization procedure.
Figure S1. Three dimensional plots of recruitment data (points) from a study site on Bird
Rock off the coast of Santa Catalina island fitted with a distance-weighted least squares
surface (grid). Level refers to tidal height and the sites are along a gradient of low (2) to
high (5) wave exposure.
Appendix II. Details of Methods.
II a. Photo-mosaic methods
Our aim was to provide benthic ecologists with a simple method that could be
implemented with inexpensive off-the-shelf components.
Photo-mosaics were assembled from component images taken from the nadir
perspective (see Blakeway et al. 2004 for details). Two focal lengths were used
depending on the size of mussels and complexity of cover. For lower boundaries
comprised of predominantly large mussels, a digital camera (Nikon CoolPix 5000TM
Nikon Corportation Tokyo, Japan) with a 7.1 mm lens was held approximately 1.95
meters above the center of the field by mounting it on a horizontal armature atop a
handheld pole maintained vertical with a spirit level. The photographer took overlapping
component images at regular intervals along taught transect lines. Component images had
a resolution of < 3 mm. For upper boundaries the camera was mounted 0.75 m above the
center of the field, and component images taken at intervals, with a resolution of
approximately 1 mm. Since the graduations of transect line could be read along the mid
line of each component image, and when necessary additional scales were set in the
corners of component images, the scaling for the cells in the sample grids could be held
constant despite changes in magnification resulting from topographic irregularities and
parallax.
For the within-site mosaics, transect lines were laid parallel to the horizon,
separated from one another by 1 m intervals on the perpendicular axis and spanning
approximately 10 m, and then the component frames were taken along each transect. For
the among-sites mosaics, transect lines were laid out to sample a short segment of upper
and lower boundaries at each site. Upper and lower boundary mosaics were aligned
vertically through the approximate center of the mussel bed at each site. For the upper
mosaic, a sequence of component images was shot along the vertical line and spanning
the apparent upper boundary of the mussel bed. For the lower mosaic, a sequence of 1-2
horizontal rows of frames were taken along the lower boundary. The different transect
orientations were chosen according to apparent differences in boundary intensity: the
lower boundary appeared abrupt, and usually could be bracketed within the top and
bottom margins of one row of frames (1 x 3 m), whereas the upper boundaries sometimes
appeared diffuse, requiring a vertical column of frames (3 x 1 m).
Because the slope of the shore was gradual (means for all sites < 15o), the 3-D
surface could be projected onto the 2-D images without loss of small features. Numbered
markers were placed at regular intervals within the frames, and their exact locations
relative to a datum point above mean lower low water (MLLW) were estimated with a
Total Station surveyor. Using these survey coordinates, the 2-D mosaics were registered
to a 3-D elevation surface interpolated as a Triangular Irregular Network (TIN) in a GIS
database (ArcView GISTM). Thus, a specific feature could be related to its location in the
landscape.
II b. Ground-truthing the photo-mosaics
On high shore levels of a few sites, M. californianus overlapped with sparse populations
of its similar congener Mytilus trossulus Gould. The covers were composed of small
individuals, and they sometimes were obscured by a patchy algal overstory. To assure
that M. californianus could be detected and discriminated from M. trossulus, the
following steps were taken. When present, the sparse overstory algae were snipped off
before taking the images. To determine error rates for identifying M. californianus, a 400
cm2 quadrat was placed in the center of each of 10 component frames. Using a focal
length of 1.5 m, a digital image was taken, then every M. californianus in the quadrat was
daubed with fluorescent chalk, and the frame re-shot. In the laboratory, scoring of the
images was done as a double blind design. The images of the unmarked quadrats were
overlain with a sample grid, the apparent M. californianus scored for each cell, and the
scoring of the blind grids then compared with the corresponding images of the chalked
quadrats. Forty-four percent of the 483 cells comprising the 10 grids were occupied by M.
californianus. Cells scored as occupied by the lab observer but unoccupied by the field
observer (false positives) constituted 1.4% of the total cells scored. Cells scored as
unoccupied by the lab observer but occupied by the field observer (false negatives)
constituted 1.0% of the total cells scored.
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