Supplementary Information:
Predator decline leads to decreased stability in a coastal fish community
Gregory Lee Britten 1,2 * Michael Dowd 2 Cóilín Minto 3 Francesco Ferretti 4 Ferdinando
Boero 5 Heike K. Lotze 1
1
Department of Biology, Dalhousie University, Halifax, Nova Scotia, Canada
2
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada
3
Marine and Freshwater Research Centre, Galway-Mayo Institute of Technology, Galway,
Ireland
4
Hopkins Marine Station, Stanford University, Pacific Grove, California, USA
5
DiSteBA, Universita' del Salento, Lecce, Italy; CNR-ISMAR, Italy
* gregbritten@dal.ca
Appendix A
Methods
Formula S1. Formula for 𝐕∞
The MAR(1) model has a stationary distribution which is multivariate normal with mean vector
𝛍∞ and covariance matrix 𝐕∞ given by
𝛍∞ = (𝐈 − 𝐁)−𝟏 ,
(S1)
𝐕∞ = 𝐁𝐕∞ 𝐁′ + 𝚺,
where 𝐈 is the identity matrix, 𝐁 is the autoregressive matrix and 𝚺 is the environmental
covariance matrix. An explicit formula for 𝐕∞ is given by
Vec(𝐕∞ ) = (𝐈 − 𝐁⨂𝐁) −1 Vec(𝚺),
(6)
where the Vec operator stacks the columns of a matrix atop one another with the first column on
top and last column on bottom, and ⨂ is the Kronecker product.
Table S1. All species caught in the Camogli trap along with their trophic group denomination,
generation time (in years) and fecundity (# eggs). Generation time and fecundity estimates were
extracted from FishBase (Froese & Pauly, 2011) as available (symbol – represents unavailable
data). The * symbol represents species with relatively uncertain trophic denomination.
Scientific Name
Common Name
Alosa agone
Boops boops
Clupea harengus*
Diplodus puntazzo*
Diplodus vulgaris
Hirundichthys rondeletii
Luvarus imperialis*
Oblada melanura*
Pagellus acarne
Sardina pilchardus
Sarpa salpa
Scomber colias
Scomber scombrus
Scomberesox saurus saurus
Sparus aurata
Spicara maena
Spicara smaris
Trachurus trachurus*
Auxis rochei rochei
Belone belone
Cetorhinus maximus*
Coryphaena hippurus
Dasyatis pastinaca
Dentex dentex
Euthynnus alletteratus
Katsuwonus pelamis
Loligo vulgaris
Lophius piscatorius
Merluccius merluccius
Micromesistius poutassou*
Mobula mobular
Mola mola
Mustelus mustelus
Pomatomus saltatrix
Raja clavata
Sepia officinalis
Todarodes sagittatus
Umbrina cirrosa
Agone
Bogue
Atlantic herring
Sharpsnout seabream
Common two-banded
seabream
Black
wing flyingfish
Luvar
Saddled seabream
Axillary seabream
European pilchard
Salema
Atlantic chub mackerel
Atlantic mackerel
Atlantic saury
Gilthead seabream
Blotched picarel
Picarel
Atlantic horse mackerel
Bullet tuna
Garfish
Basking shark
Common dolphinfish
Common stingray
Common dentex
Little tunny
Skipjack tuna
European squid
Angler
European hake
Blue whiting
Devil fish
Ocean sunfish
Smooth-hound
Bluefish
Thornback ray
Common cuttlefish
European flying squid
Shi drum
Tropic
Group
Guild
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Generation
Time
4
4
5.6
2.4
2
0.7
4
3.8
4
3.6
0.8
3.6
1.4
2.4
6.6
6.1
7.7
4.6
3.1
1.5
8.9
5.1
2.1
8.6
6.5
2.4
4.8
8.0
4.8
10.7
3.4
Fecundity
195,000
395,000
213,448
156,525
417,421
104,508
5,916
1
299,684
5
395,221
400,000
1,000,000
3,741,657
30,000
300,000,00
80
1,000,000
94
-
Zeus faber
Alopias vulpinus
Echinorhinus brucus
Galeorhinus galeus
Isurus oxyrinchus
Lichia amia*
Prionace glauca
Sarda sarda*
Sphyrna zygaena
Thunnus alalunga*
Thunnus thynnus
Xiphias gladius
John dory
Thresher shark
Bramble shark
Tope shark
Shortfin mako shark
Leerfish
Blue shark
Atlantic bonito
Smooth hammerhead
shark
Albacore
Atlantic bluefin tuna
Swordfish
2
3
3
3
3
3
3
3
3
3
3
3
3.4
12.4
8.6
15
9.0
7.6
4.2
15.6
8.0
13.2
8.6
3
18
8
23
2,449,490
10,000,000
5,385,165
Table S2. R code for calculating the three stability metrics
# Code to loop through years, estimate B matrix, stability, and
perform bootstrapping
# Note ‘data’ is a Nx3 data matrix
for(i in 1:length(years))
{
y <- log(data[data$year==years[i],][2:4])
T <- nrow(y)
X <- as.matrix(y[1:T-1,])
Y <- as.matrix(y[2:T,])
# Estimate AR(1) coefficients
B<-solve((t(X)%*%X))%*%(t(X)%*%Y)
B1s[[i]] <- B
# Calculate the residual
Y_hat <- X%*%B
e <- Y - Y_hat
# Bootstrap for uncertainty
for(j in 1:500)
{
# Resample residuals
e_boot <- e[sample(1:nrow(e), replace=TRUE),]
T <- nrow(e_boot)
x0 <- y[1,]
Y_boot <- matrix(NA, T, 3)
Y_boot[1,] <- as.numeric(x0)
# Add resampled residuals back
for(k in 2:T)
Y_boot[k,] <- Y_boot[k-1,]%*%B + e_boot[k,]
X <- as.matrix(Y_boot[1:(T-1),])
Y <- as.matrix(Y_boot[2:(T),])
B_boot <- pseudoinverse((t(X)%*%X))%*%(t(X)%*%Y)
# Use pre-defined stability function to calculate metrics
return_time1[i,j] <- return.func(B_boot,T,3)
eigen1[i,j] <- dominant.eigen(B_boot)
sp_var1[i,j] <- species.var.func(B_boot,T,3)
}
}
# Pre-defined stability functions
#################
#-Resistance-####
#################
resistance <- function(B,T,p)
{
I2 <- diag(p^2)
coef <-(1/(T-p-1))
res <- Y-(X%*%B)
col <- t(res)%*%res
cov <- coef*col
varvec <- solve(I2-(kronecker(B,B)))%*%as.vector(cov)
var <- matrix(varvec,ncol=3,byrow=F)
pro <- det(var-cov)/det(var)
return(pro)
}
####################
# - Resilience - ###
####################
resilience <- function(B)
{
eigen <- eigen(B)
maxeigen <- max(abs((eigen$values)))
index <- which.max(abs(Re(eigen$values)))
maxvec <- eigen$vectors[,index]
return(maxeigen)
}
#####################
# - Reactivity - ####
#####################
reactivity <- function(B,T,p)
{
coef <- (1/(T-p-1))
res <- (Y-(X%*%B))
col <- t(res)%*%res
cov <- coef*col
I <- diag(p)
I <- diag(p^2)
varvec <- solve(I2-(kronecker(B,B)))%*%as.vector(cov)
var < -matrix(varvec,ncol=3,byrow=F)
vartrace <- sum(diag(var))
covtrace <- sum(diag(cov))
return <- -(covtrace/vartrace)
return(return)
}
Figure S1. Kernel density estimates for the distribution of species generation times (in years),
according to trophic denomination. Note that the generation times vary several-fold across all
trophic levels.
Figure S2. Single year example of the raw catch data (Panel A) and the log-transformed,
demeaned CPUE data (log(KG)/day; Panel B), as analyzed in the paper.
Figure S3. The quantile-quantile (QQ) plot of residuals for the autoregressive fit, separated by
trophic level time series. The statistical assumption of normality appears supported.
Figure S4. The multivariate auto-correlation function (ACF) for the set of estimated autoregressive model residuals (n=7577, p=3), note HP = high-predators; MP = meso-predators; and
LT = low-trophic. Results indicate no significant autocorrelation (diagonal plots) or crosscorrelation (off-diagonal plots).
Figure S5. Four additional ecological and environmental variables tested as potential drivers of
stability: A) Relative fishing effort series formed by combining two records of # of registered
fishing boats in the Ligurian Sea area (1950-1964) and total fishing vessel horsepower (19651972; ISTAT 1972); B) Sea surface temperature (SST) within the Ligurian Sea from 1950-1972;
and inset E) SST over 1900-2000 (University of Maryland, 2014); C) Species richness R (# of
species caught per year); D) Diversity of the catch, measured by Shannon’s index ∑𝑅𝑖=1 𝑝𝑖 ln 𝑝𝑖 ,
where 𝑅 is the number of species.
Figure S6. Estimated time series of species richness (R) and species diversity (H) by trophic
group.
Figure S7. Selected species-specific time series of mean annual CPUE. Sarpa salpa, Oblada
melanura, Todarodes sagittatus and Loligo vulgaris are among the most frequently caught low
trophic species which all increase several fold in abundance as predatory sharks (including
Sphyrna zygaena, Prionace glauca, Isurus oxyrinchus, Galeorhinus galeus, Echinorhinus brucu,
and Alopias vulpinus) decrease dramatically and are largely absent from the trap by 1974.
Figure S8. Results from the leave-one-out stability analysis. Stability metrics were calculated
three times by leaving a trophic level out of the CPUE data, estimating the stability of the
bivariate community, and correlating it against the excluded trophic level. Panels A-C give the
resilience metric; panels D-F the reactivity metric; and G-I the resistance metric (see text for
definitions). The sample correlation (r) is given in each plot. HP = high-predators; MP = mespredators; and LT = low trophic.
Figure S9: Results from a multiple regression analysis across three individual stability metrics
(Resilience, Reactivity, and Resistance). Covariates included species richness (total # of species per year),
species diversity (Shannon index value), fishing effort (combined record of total number of regional
fishing boats and horsepower), and SST (sea surface temperature of the Ligurian Sea). Results are shown
for the full-model including all variables; however, only Effort was retained in AIC model selection for
each metric.
Figure S10. Estimated time series for the coefficient of variation (CV) of the CPUE data. CV
was calculated per year, over the 25 years. Dashed lines give the CV for each individual trophic
group and solid black line gives the CV for total CPUE. All trends are statistically insignificant
(P > 0.1).
Figure S11. Landings (A) and proportional catch (B) data for the Mediterranean Sea over the period
1950-2010 (Source: Sea Around Us database, http://www.seaaroundus.org/lme/26.aspx). The time
between dashed lines represents the study period (1950-1974) analyzed in the paper.