1
Figure S1. Spatial autocorrelation among sampling points in the herbivore exclosures. Moran’s
2
I (based on the first principal component calculated from the following soil variables: total N and
3
C, mineralizable N and C, Mehlich-P and acid phosphatase) as a function of lag distance across
4
trees inside the Hlangwine (granite) and N’washitshumbe (basalt) exclosures in Kruger National
5
Park. Filled symbols would indicate a significant autocorrelation (note that there were none)
6
based on permutation tests using 1000 iterations and a progressive Bonferroni correction
7
beginning with the first lag.
1.0
Hlangwine
N'washitsumbe
0.8
Moran's I
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
0
8
200 400 600 800 10001200140016001800
Distance (m)
1
Figure S2. Partial soil moisture (S) time series for two sensors associated with canopy (U) or
2
open (B) positions at the same tree. We note differences in initial peak moisture following
3
rainfall, and differences in subsequent decay, with the U position showing faster losses
4
immediately following a precipitation event and the N position showing faster losses later on,
5
after S has fallen below field capacity (~ 0.34).
U
B
6
2
1
Table S1. Tests of main effects and interactions of tree canopies (CAN) and herbivore exclusion (HERB) on soil variables on granite,
2
based on mixed effects models with site and tree within site as random effects.
Basis
Fixed effect
Mass
HERB
Bulk Density
F
P-value
27.5 0.0001
(g kg-1)
CAN
52.5
<.0001
61.0
<.0001
10.6
0.0028
2.8
0.1
19.2
0.0002
20.4
0.0001
2.8
0.1
HERB × CAN
0.3
0.6
0.1
0.8
0.1
0.8
1.2
0.3
3.3
0.080
3.3
0.079
0.3
0.6
-
-
3.9
0.068
0.0
1.0
1.6
0.2
6.9
0.019
7.2
0.017
n/a
n/a
CAN
-
-
33.3
<.0001
7.1
0.0125
0.2
0.6
6.3
0.019
7.7
0.010
n/a
n/a
HERB × CAN
-
-
0.9
0.3
0.0
1.0
1.8
0.2
2.3
0.1
2.5
0.1
n/a
n/a
Area (g m-2) HERB
3
F
0.0
Cmin
P-value
0.9
F
0.2
Nmin
Mehlich-P
P-value F P-value
0.6
0.2
0.6
Total N
F
P-value
1.7
0.2
Total C
C:N
F
P-value F P-value
2.3
0.1
3.5
0.08
1
Table S2. Tests of effects of tree canopies (CAN) and herbivore exclusion (HERB) on soil
2
variables on basalt, based on mixed effects models with site (for effects of fencing) or tree (for
3
effects of canopies) as random effects.
Fixed effect Variable
Mass basis (g kg-1) Area basis (g m-2)
F
CAN
4
F
-
P-value
Bulk Density
31.6 <.0001
Cmin
23.2
0.0003
13.2
0.003
Nmin
15.5
0.0015
17.3
0.001
6.1
0.027
6.7
0.02
Total N
16.2
0.0013
10.6
0.006
Total C
13.4
0.0025
6.7
0.02
C:N
3.6
0.08
n/a
n/a
Bulk Density
3.5
0.083 -
Cmin
4.4
0.055
1.2
0.3
Nmin
1.2
0.3
1.6
0.2
Mehlich-P
1.2
0.3
1.6
0.2
Total N
0.1
0.8
0.3
0.6
Total C
0.3
0.6
0.2
0.6
C:N
0.2
0.6
n/a
n/a
Mehlich-P
HERB
P-value
-
-
1
Table S3. Tests of effects of tree canopies (CAN) and herbivore exclusion (HERB) on soil
2
temperature on granite
Mean (°C)
Factor
d.f. Estimate
SE
F
Range (°C)
P-value Estimate
SE
F
P-value
Intercept
5
25.18
0.52 5500.7
<.0001
7.81
1.23 84.6
0.000
CAN
5
-1.94
0.50
15.0
0.012
-4.46
1.33 10.8
0.022
HERB
5
-0.01
0.51
0.0
0.985
1.61
1.28
0.265
1.6
3
5
1
Table S4. Tests of effects of tree canopies (CAN) and herbivore exclusion (HERB) on soil
2
temperature on basalt
Factor
CAN
HERB
Dependent variable Estimate
SE
d.f. t-value P-value
Mean
-2.28
0.65
3
-3.52
0.039
Range
-1.09
3.97
3
-0.27
0.802
Mean
1.50
0.55
3
2.73
0.072
Range
2.70
1.29
3
2.10
0.127
3
6
1
Appendix S1. Description of soil moisture dynamics state-space model. State-space models can
2
be divided into two components, a process model and an observation model. We describe each of
3
these components below.
4
SOIL MOISTURE PROCESS MODEL
5
We used a relatively simple soil moisture dynamics model, modified from Rodriguez-Iturbe et
6
al. (1999):
7
nz
dS
 RT  L
dt
[1]
8
Here, n is soil porosity, z is the depth of the soil profile under consideration, S is soil saturation,
9
R is rainfall, T represents vegetation effects that modify soil moisture inputs from R, and can
10
either reduce (in the case of interception) or enhance (in the case of stem flow or runoff) the
11
supply of water from precipitation events, and L represents losses to evapotranspiration and
12
drainage to deeper layers, with the relative contribution of each being a function of S (see
13
below). The use of WinBUGS precluded the use of continuous time, so we discretized the model,
14
using a daily time step:
15
S jk ,t 1  S jk ,t 
Rt T jk ,t  L jk ,t
nz
[2]
16
Here, the t subscript specifies time (in days) and the subscripts j and k specify sites and trees
17
within sites, respectively. The daily time step was necessary to make solving the model tractable,
18
even though we had rainfall and soil moisture data resolved over 5-min intervals. The potential
19
difficulty here is that a daily time step fails to capture the fast dynamics of wetting events,
20
followed by the rapid initial decay in soil moisture as a result of leakage during the time window
21
when soil saturation exceeds field capacity. We solved this issue by subdividing our soil
22
moisture data into two metrics: maximum (obtainable from our high temporal resolution data)
7
1
and mean daily S (SMAX and SMEAN, respectively). We assume that the maximum value is
2
achieved immediately following a wetting event, and that the ensuing dry-down period brings
3
soil moisture to a value that is to a first approximation captured by the daily mean. In the process
4
model, we further subdivided each of these variables into deterministic components SMAX det
jk ,t
5
and SMEAN det
jk ,t (the values we would expect in the absence of environmental stochasticity) and
6
true
their ‘true’ values SMAX true
jk ,t and SMEAN jk ,t , which add process error (representing
7
environmental stochasticity) to the baseline deterministic values. We assumed that the dynamics
8
of SMAX are driven by rainfall inputs as follows:
9
true
SMAX det
jk ,t  SMEAN jk ,t 1 
Rt T jk ,t
nz
 b j  c jk
10
Here, SMAX det
jk ,t increases after a rainfall event Rt from a baseline value dictated by the mean
11
value from the previous time step ( SMEAN true
jk ,t ), and the input term Tjk,t is given by a linear
12
function of the fixed effects CAN and HERB:
13
T jk ,t   0   1CAN jk   2 HERB jk
[3]
[4]
14
Also in eq. 3, bj is a site random effect, and cjk is a tree within site random effect. In reality, a
15
further random effect of sensor within tree (the U and B sensor) could be modelled, with
16
individual sensors showing consistently under- or over-estimating soil moisture, for example.
17
We tested a model specifying such effects but found a tradeoff between the sensor and tree
18
effects, suggesting a so-called identifiability issue (Gelfand & Sahu 1999), so chose to simplify
19
the model. A further source of non-independence is potentially introduced at the sensor level by
20
the autocorrelated nature of the data (given that individual measurements represent a time series
21
for each sensor). The state space model addresses this issue by assuming that changes in soil
8
1
moisture represent a Markov process, where the state of SMAX and SMEAN are conditionally
2
dependent on their states in the preceding time period (Clark & Bjornstad 2004). We assumed
3
that bj values were normally distributed with mean 0 and standard deviation σb, and that bjk
4
values were normally distributed with mean bj and standard deviation σc.
5
6
From eq.3, we obtain SMAX true
jk ,t by adding normally distributed process error σproc to
SMAX det
jk ,t :

det
SMAX true
jk ,t ~ Normal SMAX jk ,t ,  proc
7

[5]
8
SMAX true represents peak soil moisture following a wetting event. The rate of decline from this
9
peak depends on the value of SMAX true in relation to two important inflection points: the field
10
capacity (SF) and the soil saturation value (S*) that leads to reduced transpiration (Rodriguez-
11
Iturbe et al. 1999). The loss term Ljk,t (from eq. 2) takes on one of three values:
12
L jk ,t

SMAX true
jk ,t  S F
K
 jk ,t
1 SF


E jk ,t

SMAX true
jk ,t
 E jk ,t

S*
if
S F  SMAX true
jk ,t  1
if
S *  SMAX true
jk ,t  S F
if
0  SMAX true
jk ,t  S *
[6]
13
In eq. 6, Kjk,t and Ejk,t represent saturated conductivity and evapotranspiration, respectively. From
14
eq. 6, when S is above field capacity, L is dominated by K, which diminishes to 0 as S
15
approaches SF. Between S* and SF, L is constant and given purely by evapotranspiration E.
16
Below S*, plants begin to experience water stress and E declines linearly between S* and 0
17
(Rodriguez-Iturbe et al. 1999). Ordinarily, K and E are treated as fixed parameters (Rodriguez-
18
Iturbe et al. 1999), but here we allow them to be functions of our covariates:
19
Kjk,t = exp(α0 + α1×CANjk + α2×HERBjk)
[7]
20
Ejk,t = exp(β0 + β 1×CANjk + β 2×HERBjk)
[8]
9
1
The exponential functions prevent K and E from taking on negative values during model fitting.
2
SMEAN det
jk ,t is then given by:
true
SMEAN det
jk ,t  SMAX jk ,t 
3
L jk ,t
nz
 b j  c jk
[9]
4
Note that R and L (and its components K and E) are all expressed in units of mm d-1, and division
5
by nz in the equations above converts these variables into units of soil saturation S. As was the
6
case for SMAX (eq. 3), we include random effects for site and tree within site to the update
7
function for SMEAN in eq. 9. We tested using separate random effects for SMAX and SMEAN,
8
but found were similar values for both variables (suggesting that site effects on maximum and
9
mean soil moisture were similar), so we simplified the model by including only two overall
10
true
random effects. Finally, we applied a process error to SMEAN det
jk ,t to obtain SMEAN jk ,t :

det
SMEAN true
jk ,t ~ Normal SMEAN jk ,t ,  proc
11

[10]
12
OBSERVATION MODEL
13
We coupled the process model to our data by adding observation error (e.g., due to spikes in
14
sensor readings due to equipment failure) as follows:


15
true
2
SMAX obs
jk ,t ~ Normal SMAX jk ,t ,  obs
16
true
2
SMEAN obs
jk ,t ~ Normal SMEAN jk ,t ,  obs

[11]

[12]
17
obs
Where SMAX obs
jk ,t and SMEAN jk ,t are observed daily values of maximum and mean soil
18
saturation across the j sites (j = 1 to 4) and k trees within each site (k = 1,2) on either granite or
19
basalt. At the granite site, our daily rainfall data had an obvious gap between days 153 and 189
20
where no precipitation was recorded but soil moisture sensors indicated rainfall events. To
21
address this issue, we treated rainfall for this period as a random variable, assuming a compound
10
1
distribution. We assumed that the occurrence of a rainfall event ηt on day t followed a Bernoulli
2
distribution with probability p:
ηt ~ Bernoulli(p)
3
4
[13]
Given a rainfall event (ηt = 1), we assumed that Rt followed a Poisson distribution with mean μ:
Rt ~ Poisson(μ)
5
[14]
6
The imputed values had clearly defined posterior distributions and correctly identified the days
7
where soil moisture spikes indicated rainfall events.
8
PRIORS
9
We assumed flat Normal priors (with mean 0 and variance 104) for all fixed effect coefficients
10
(α’s, β’s and γ’s). We assumed flat Gamma priors (with shape and scale parameters equal to 10-2)
11
for the precisions (our model implementation in WinBUGS expresses variance terms as
12
precicisons, equal to the inverse of the variance) of the observation and process error terms σobs
13
and σproc, and for the site and tree within site hyperpriors σb and σc. We assumed a uniform prior
14
in the interval [0,1] for p (eq. 13) and a gamma prior (with shape and scale 10-2) for μ.
15
MODEL PARAMETERS
16
We used values from the literature for the soil conditions typical of the granite and basalt sites
17
(Guswa, Celia & Rodriguez-Iturbe 2002; Holdo 2013). On granite, we used the following
18
parameter values: SF = 0.29, nz = 0.42, and S* = 0.1. On basalt, we assumed the following: SF =
19
0.50, nz = 0.45, and S* = 0.2. We initialized the model with values for SMEAN from day 1 of our
20
data set. Some sensors began logging data on day 2; in these cases we used mean values (across
21
all sensors with day 1 data) as initial values.
11
1
MODEL IMPLEMENTATION
2
We used WinBUGS 1.4 (Spiegelhalter et al. 2003), to obtain estimates of the posterior
3
distributions of the model parameters. We ran three versions of the model in duplicate: the model
4
as described above for the granite exclosure, and two separate models to test for canopy or
5
herbivore effects on basalt (with eqs. 4, 7 and 8 modified to contain the relevant fixed effects).
6
We ran two versions of each model, one in which we calculated S with site-specific bulk density
7
data, and one in which the S variables were calculated using mean bulk density for the entire
8
exclosure (see main text). We ran each model for 105 iterations, discarding the first half of these
9
as “burn-in.” We used three chains (with different initial values for each parameter) for each
10
model and checked for model convergence both visually (by establishing that chains were well-
11
mixed and had converged on a stable mean value) and quantitatively by plotting for each node
12
the convergence diagnostic R̂ , which has an expected value of 1 (Sturtz, Ligges & Gelman
13
2005). We checked that our sampling interval did not lead to autocorrelation between successive
14
realizations of each variable. We used the 95% credible intervals of the posterior means for the
15
fixed effects coefficients to determine if CAN or HERB contributed meaningfully to explain
16
patterns in the data.
17
ECOLOGICAL INTERPRETATION OF MODEL COEFFICIENTS
18
In this exercise we applied a relative simple model of soil moisture dynamics to an inference
19
problem, with the objective of identifying the specific mechanisms whereby tree canopies and
20
herbivore exclusion might affect soil moisture. Because our use of the model is inductive rather
21
than deductive, it is important to note that it may be difficult to pin down the exact mechanisms
22
at work in the various phases of the model. The model provides a good fit to the data, and we are
23
able to infer effects of e.g., tree canopies on moisture inputs following rainfall events, but we
12
1
note that whether these effects are the result of the tree canopies themselves or other effects
2
associated with canopies is difficult to determine. Similarly, it is difficult to fully tease apart
3
uptake and drainage differences, so rather than using the terms in the original model formulation
4
for some of the key processes involved, we prefer to use more descriptive terms that capture the
5
patterns observed in the data without making definitive claims about the underlying processes. In
6
particular, we refer to the linear function that drives soil moisture decay immediately following
7
rainfall events (when drainage typically dominates) as “fast output” and refer to function that
8
drives decay when soils are drier (typically dominated by evapotranspiration) as “slow output”.
9
MODEL TESTING
10
To confirm that the model is able to recover the parameter values for the underlying process
11
model, as well as the process and observation error terms, we created a simulated data set, driven
12
by the actual observed rainfall time series from our data. We assigned values to coefficients (α’s,
13
β’s and γ’s) that were similar to those estimated for the granite site. We constrained SMAXobs and
14
SMEANobs to be greater than 0 (the normally-distributed process and observation errors otherwise
15
allows negative values when soil moisture is near 0). For simplicity, we ignored random effects
16
to focus on the process model, and generated pseudo-data for four soil moisture “sensors” with a
17
factorial combination of canopy and fencing effects. We then applied a modified version of our
18
Bayesian state-space model (minus the tree and site effects) to this simulated data set, allowing it
19
to run for 105 iterations (more than enough to achieve convergence). Table S7 compares the true
20
and model-estimated parameter values, and shows that the model performs well (estimated
21
parameter means are consistent with true values, and in almost all cases the true values fall
22
within the credible intervals of the estimated values).
23
Below, we list the WinBUGS code corresponding to the granite analysis:
13
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
# Soil moisture dynamics state-space model
# The following data set is loaded into WinBUGS:
# DAYS and N (scalar variables, DAYS=205, N=16 [4 sites, 2 trees per site, two canopy
# positions per tree = 16 sensors])
# CAN, HERB, SITE and TREE (matrices with DAYS rows and N columns,
# taking values 0 or 1)
# S.obs.mean and S.obs.max (matrices with DAYS rows and N columns
# containing soil moisture data)
# S0 (a vector of N initial values)
# R (a vector of length DAYS containing rainfall data)
# SF, Sstar, and nz (model parameters)
model {
# rainfall model for imputation of missing values
for(t in 1:DAYS){
R[t] ~ dpois(m[t])
m[t] <- eta[t]*mu;
eta[t] ~ dbern(p)
}
# Obtain initial values (t=1)
for(j in 1:N){ # cycle through N sensors
# The logit and reverse logit keep T positive
logit.T[1,j] <- gamma0 + gamma1*CAN[1,j] + gamma2*HERB[1,j]
T[1,j] <- exp(logit.T[1,j])/(1 + exp(logit.T[1,j]))
S.det.max[1,j] <- S0[j] + R[1]*T[1,j]/nz # Initial conditions
# mx is an intermediate variable that makes the script easier to run
mx[1,j] <- S.det.max[1,j] + b[SITE[1,j]] + c[site[1,j],TREE[1,j]]
S.true.max[1,j] ~ dnorm(mx[1,j],tau.proc) # Process model
S.obs.max[1,j] ~ dnorm(S.true.max[1,j],tau.obs) # Observed peak daily S
# Identify where S is in relation to S* and Sfc
# Create an index with value 1,2 or 3 corresponding to eq. 6
idx1[1,j] <- trunc(S.true.max[1,j]-SF+1)+trunc(S.true.max[1,j]-Sstar+1)+1
log(E[1,j]) <- beta0 + beta1*CAN[1,j] + beta2*HERB[1,j]
# rho is a temporary variable
rho[1,(j-1)*3+1] <- E[1,j]*S.true.max[1,j]/Sstar
rho[1,(j-1)*3+2] <- E[1,j]
log(K[1,j]) <- alpha0 + alpha1*CAN[1,j] + alpha2*HERB[1,j]
rho[1,(j-1)*3+3] <- K[1,j]*(S.true.max[1,j]-SF)/(1-SF)
idx2[1,j] <- (j-1)*3+idx1[1,j]
L[1,j] <- rho[1,idx2[1,j]]
S.det.mean[1,j] <- max(0, S.true.max[1,j] - L[1,j]/nz) # Keep values between 0 and 1
14
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
# mn is a temporary variable
mn[1,j] <- S.det.mean[1,j] + b[SITE[1,j]] + c[SITE[1,j],TREE[1,j]]
S.true.mean[1,j] ~ dnorm(mn[1,j],tau.proc)
S.obs.mean[1,j] ~ dnorm(S.true.mean[1,j],tau.obs)
}
for(j in 1:N){ # cycle through N sensors
for(t in 2:DAYS){ # cycle through DAYS
logit.T[t,j] <- gamma0 + gamma1*CAN[t,j] + gamma2*HERB[t,j]
T[t,j] <- exp(logit.T[t,j])/(1 + exp(logit.T[t,j]))
S.det.max[t,j] <- min(1, S.true.mean[t-1,j] + R[t]*T[t,j]/nz) # Water input from rainfall
mx[t,j] <- S.det.max[t,j] + b[SITE[t,j]] + c[SITE[t,j],TREE[t,j]]
S.true.max[t,j] ~ dnorm(mx[t,j],tau.proc) # Process model
S.obs.max[t,j] ~ dnorm(S.true.max[t,j],tau.obs) # Observed peak daily S
idx1[t,j] <- trunc(S.true.max[t,j]-SF+1)+trunc(S.true.max[t,j]-Sstar+1)+1
log(E[t,j]) <- beta0 + beta1*CAN[t,j] + beta2*CAN[t,j]
rho[t,(j-1)*3+1] <- E[t,j]*S.true.max[t,j]/Sstar
rho[t,(j-1)*3+2] <- E[t,j]
log(K[t,j]) <- alpha0 + alpha1*CAN[t,j] + alpha2*HERB[t,j]
rho[t,(j-1)*3+3] <- K[t,j]*(S.true.max[t,j]-SF)/(1-SF)
idx2[t,j] <- (j-1)*3+idx1[t,j]
L[t,j] <- rho[t,idx2[t,j]]
S.det.mean[t,j] <- max(0, S.true.max[t,j] - L[t,j]/nz) # Water losses from ET and
drainage
mn[t,j] <- S.det.mean[t,j] + b[SITE[t,j]] + c[SITE[t,j],TREE[t,j]]
S.true.mean[t,j] ~ dnorm(mn[t,j],tau.proc) # Process model
S.obs.mean[t,j] ~ dnorm(S.true.mean[t,j],tau.obs) # Observed mean daily S
}
}
# Priors
p ~ dunif(0,1)
mu ~ dgamma(0.01,0.01)
beta0 ~ dnorm(0,0.0001)
beta1 ~ dnorm(0,0.0001)
beta2 ~ dnorm(0,0.0001)
gamma0 ~ dnorm(0,0.0001)
gamma1 ~ dnorm(0,0.0001)
gamma2 ~ dnorm(0,0.0001)
alpha0 ~ dnorm(0,0.0001)
alpha1 ~ dnorm(0,0.0001)
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
alpha2 ~ dnorm(0,0.0001)
tau.obs ~ dgamma(0.01,0.01)
tau.proc ~ dgamma(0.01,0.01)
# Obtain standard deviations from precisions
sigma.obs <- 1/sqrt(tau.obs)
sigma.proc <- 1/sqrt(tau.proc)
# Priors for site effect
for(s in 1:4){
b[s] ~ dnorm(0,tau.site)
# Priors for tree effect within site
for(k in 1:2){
c[s,k] ~ dnorm(b[s],tau.tree)
}
}
# Hyper-priors
tau.site ~ dgamma(0.01,0.01)
sigma.site <- 1/sqrt(tau.site)
tau.tree ~ dgamma(0.01,0.01)
sigma.tree <- 1/sqrt(tau.tree)
}
References
Clark, J.S. & Bjornstad, O.N. (2004) Population time series: Process variability, observation
errors, missing values, lags, and hidden states. Ecology, 85, 3140-3150.
Gelfand, A.E. & Sahu, K. (1999) Identifiability, improper priors, and Gibbs sampling for
generalized linear models. Journal of the American Statistical Association, 94, 247-253.
Guswa, A.J., Celia, M.A. & Rodriguez-Iturbe, I. (2002) Models of soil moisture dynamics in
ecohydrology: a comparative study. Water Resources Research, 38, 1-15.
Holdo, R.M. (2013) Revisiting the Two-Layer Hypothesis: Coexistence of Alternative
Functional Rooting Strategies in Savannas. PLoS ONE, 8, e69625.
16
1
Rodriguez-Iturbe, I., Porporato, A., Ridolfi, L., Isham, V. & Cox, D.R. (1999) Probabilistic
2
Modelling of Water Balance at a Point: The Role of Climate, Soil and Vegetation.
3
Proceedings: Mathematical, Physical and Engineering Sciences, 455, 3789-3805.
4
Spiegelhalter, D., Thomas, A., Best, N. & Lunn, D. (2003) WinBUGS User Manual.
5
Sturtz, S., Ligges, U. & Gelman, A. (2005) R2WinBUGS: A Package for Running WinBUGS
6
from R. Journal of Statistical Software, 12, 1-16.
7
17
1
Table S5. Parameter estimates for hierarchical Bayesian state-space model of soil moisture
2
dynamics, using site-specific bulk density data.
Basalt CAN effects†
Granite
Process or
Factor
Par
Mean
CI
Mean
CI
Basalt HERB effects†
Mean
CI
quantity
Fast output
Slow output
Input
2.5%
97.5%
2.5%
97.5%
Intercept
α0
2.94
2.86
3.01
0.96
0.56
1.23
CAN
α1
0.42
0.35
0.50
0.15
-0.38
0.53
HERB
α2
-0.48
-0.56
-0.40
Intercept
β0
-0.47
-0.92
-0.16
-25.6
-166.3
-0.3
CAN
β1
0.01
-0.27
0.27
-12.5
-154.0
75.8
HERB
β2
0.18
-0.29
0.68
Intercept
γ0
-1.24
-1.31
-1.17
-0.45
-0.54
-0.36
CAN
γ1
0.27
0.19
0.35
-0.66
-0.78
-0.55
HERB
γ2
-0.08
-0.16
0.00
0.095
0.046
0.209
0.083
0.042
0.162
11.74
10.81
12.65
0.28
0.22
0.35
0.036
0.034
0.034
0.032
Site random effect
2.5%
97.5%
0.78
0.50
1.03
-0.64
-1.02
-0.30
-89.5
-235.0
-3.7
-36.8
-195.6
114.7
-0.43
-0.51
-0.34
0.26
0.05
0.44
0.087
0.041
0.189
0.089
0.042
0.222
0.037
0.031
0.030
0.033
0.032
0.031
0.034
0.035
0.031
0.030
0.033
0.030
0.028
0.032
σsite
SD
Tree random
σtree
effect SD
Mean daily
μ
imputed rainfall
Imputed rainfall
ρ
hyper-parameter
Observation error
σobs
SD
Process error SD
σproc
18
1
Note: positive coefficients for CAN and HERB indicate positive effects of herbivores (O>I) and
2
tree canopies (U>B) on a given process, respectively, compared to baseline values (given by
3
intercepts). Coefficients shown in bold differ from zero.
4
†CAN and HERB effects were examined separately on basalt due to the absence of U (under
5
canopy) sites outside the fence at this site.
19
1
Table S6. Parameter estimates for hierarchical Bayesian state-space model of soil moisture
2
dynamics, using mean bulk density data.
Basalt CAN effects†
Granite
Process or
Factor
Par
Mean
CI
Mean
CI
Basalt HERB effects†
Mean
CI
quantity
Fast output
Slow output
Input
2.5%
97.5%
2.5%
97.5%
Intercept
α0
3.02
2.94
3.10
0.68
0.36
0.92
C
α1
0.21
0.15
0.29
-0.21
-0.71
0.21
F
α2
-0.46
-0.54
-0.38
Intercept
β0
-0.34
-0.73
-0.06
-88.51
-223.78
-6.89
CAN
β1
-0.02
-0.27
0.22
-37.60
-205.38
111.33
HERB
β2
0.17
-0.24
0.63
Intercept
γ0
-1.25
-1.33
-1.17
0.27
-0.06
0.46
CAN
γ1
0.43
0.35
0.52
-0.45
-0.69
-0.14
HERB
γ2
-0.16
-0.24
-0.07
0.094
0.044
0.220
0.080
0.042
0.167
11.70
10.86
12.57
0.28
0.22
0.35
0.036
0.035
0.038
Site random
2.5%
97.5%
0.73
0.47
0.94
-0.69
-1.15
-0.33
-89.95
-224.97
-5.08
-37.25
-201.77
110.30
0.08
-0.17
0.38
-0.27
-0.60
0.04
0.087
0.042
0.191
0.089
0.042
0.222
0.035
0.033
0.037
0.034
0.032
0.036
σsite
effect SD
Tree random
σtree
effect SD
Mean daily
imputed
μ
rainfall
Imputed
rainfall
ρ
hyperparameter
Observation
σobs
20
error SD
Process error
0.035
0.033
0.037
0.035
0.033
0.037
0.031
0.029
0.033
σproc
SD
1
Note: positive coefficients for CAN and HERB indicate positive effects of herbivores (O>I) and
2
tree canopies (U>B) on a given process, respectively, compared to baseline values (given by
3
intercepts). Coefficients shown in bold differ from zero.
4
†CAN and HERB effects were examined separately on basalt due to the absence of U (under
5
canopy) sites outside the fence at this site.
21
1
Table S7. True values of simulated model parameters and mean and upper and lower 95%
2
credible intervals of estimated values derived from the Bayesian state-space soil moisture
3
dynamics model.
Parameter True value
Estimated value
Mean
CI
2.5%
97.5%
α0
3.000
2.971
2.874
3.061
α1
0.200
0.282
0.182
0.386
α2
-0.500
-0.597 -0.702 -0.496
β0
-0.340
-0.384 -0.548 -0.237
γ0
-1.200
-1.178 -1.254 -1.106
γ1
0.400
0.426
γ2
-0.200
-0.263 -0.343 -0.182
σobs
0.020
0.019
0.018
0.021
σproc
0.020
0.020
0.019
0.021
0.345
0.509
4
22
1
Figure S3. Soil bulk density as a function of total C (a) inside and (b) outside exclosure fence at
2
the granite site (Hlangwine). P-values are based on linear mixed model regressions conducted in
3
R package nlme with TREE as a random effect.
(a)
P < 0.0005
1.6
1.4
1.3
1.2
1.1
1.0
1.2
4
-3
1.5
1.6
2.0
2.4
-1
Total C (g kg )
2.8
P = 0.4
1.9
Bulk density (g cm )
-3
Bulk density (g cm )
1.7
(b)
1.8
1.7
1.6
1.5
1.4
1.3
1.4
1.8
2.2
2.6
3.0
-1
Total C (g kg )
23
1
Figure S4. Canopy effects on short-term soil moisture inputs on (a) granite and (b) basalt.
2
ΔGWC = GWCinput for U - GWCinput for B locations per tree, averaged across all trees for a given
3
day, where GWCinput = GWCmax - GWCmean (maximum and mean Gravimetric Water Content)
4
for every sensor.
(a)
(b)
0.06
0.03
R2
0.05
= 0.57
0.01
0.03
GWC
GWC
0.04
0.02
0.00
-0.01
0.01
-0.02
0.00
-0.03
-0.01
-0.04
-0.02
-0.05
0
5
R2 = 0.09
0.02
20
40
60
Daily rainfall (mm)
80
0
20
40
60
80
Daily rainfall (mm)
24
1
Table S8. ANOVA tables for regressions of ΔGWC (see Fig. S4 and main text) against daily
2
rainfall at the granite and basalt sites.
Site
Variable
Estimate
SE
Granite Intercept -1.25E-04 3.47E-04
Basalt†
3
†
t-value
-0.36
P-value
0.7
Rainfall
5.24E-04 3.18E-05
16.5 << 0.0001
Intercept
1.34E-04 3.70E-04
0.36
Rainfall
-3.87E-04 8.46E-05
Rainfall2
4.40E-06 1.08E-06
0.7
-4.58 << 0.0001
4.08
< 0.0001
The introduction of a quadratic term improved the fit at the basalt (but not granite) site.
4
25
1
Table S9. Species and crown diameters of trees used in the soil moisture dynamics and
2
temperature studies at the granite and basalt sites in Kruger National Park.
Site
Label† Species
Exclosure
Crown
position
diameter (m)‡
Granite T3
Terminalia sericea
I
8.9
Granite T7
Terminalia sericea
O
10.7
Granite T12
Diospyros mespiliformis
I
9.4
Granite T16
Sclerocarya birrea
O
6.4
Granite T19
Terminalia sericea
I
12.3
Granite T23
Sclerocarya birrea
O
10.5
Granite T28
Terminalia sericea
O
9.5
Granite T30
Terminalia sericea
I
10.1
Basalt
T36
Ozoroa engleri
I
12.3
Basalt
T39
Acacia nigrescens
I
7.4
Basalt
T44
Colophospermum mopane
I
7.7
Basalt
T49
Colophospermum mopane
I
9.1
Basalt
T52
Combretum imberbe
O
n/a
Basalt
T55
Combretum imberbe
O
n/a
Basalt
T60
Colophospermum mopane
O
n/a
Basalt
T65
Colophospermum mopane
O
n/a
3
†
See Fig. 1 map in main text for tree location
4
‡
Mean value calculated from the major and minor axes of an ellipse
26