1. science
2. biology
3. zoology
4. endangered species

Online Supplementary Material to: How to predict community responses to
perturbations in the face of imperfect knowledge and network complexity
Helge Aufderheide,1, 2 Lars Rudolf,2 Thilo Gross,2 and Kevin D. Lafferty3
1
Max-Planck-Institute for the physics of complex systems, Dresden, Germany
University of Bristol, Merchant Venturers School of Engineering, Bristol, UK
3
U.S. Geological Survey, Western Ecological Research Center,
Marine Science Institute, University of California, Santa Barbara, CA 93106
2
Contents
I. Review of the generalized modeling approach
1
3
II. Predator-prey example
III. Step-by-step parametrization of the Gatun Lake food web
A. Parametrization
B. Estimation of the impact caused by the introduction of peacock bass
3
4
5
IV. Derivation of influence and sensitivity
6
7
V. Model food webs
VI. Quality assessment of an impact prediction
VII. Figures for a higher food web connectance
8
8
10
References
I.
REVIEW OF THE GENERALIZED MODELING APPROACH
In this section we first review the process of generalized modeling for food webs, which is presented more rigorously
in Ref. 1. Further, we give the explicit expressions of the Jacobian matrix required for the perturbation impact
evaluation in the paper.
The generalized model approach presented in Ref. 1 consists of four steps. First, a generalized model describes
different food web populations X1 , . . . , XN . Depending on the application, the variable Xi can describe species
density, biomass abundance, carbon abundance or another characteristic quantity. Here, we assume that it denotes
the total amount of biomass that can be attributed to a given species. The interactions and dynamics of the different
populations X1 , . . . , XN are described by N differential equations of the form
d
Xi = Gi (X) + Si (Xi ) − Li (X) − Mi (Xi ),
dt
(1)
where Gi , Li , Mi , and Si are unspecified functions describing, respectively, the biomass gain by predation (Gi ), the
biomass loss by predation (Li ), the loss due to natural mortality (Mi ), and the gain by primary production (Si ) of
the focal species. We note that, while the primary production Si and the mortality Mi only depend on the focal
population Xi , the predatory gain Gi and loss Li reflect the structure of the feeding relations in the food. Therefore
Gi and Li may depend on all other populations X. For instance, for a species i with exactly one predator j, the loss
through predation Li depends on the abundance of the species Xi , on the abundance of the predator Xj , and possibly
on the abundances of other species that are also prey to j due to apparent competition. However, Li does not depend
directly on the abundances of the remaining species in the food web. Second, we assume that each species in this
food web has a positive, yet unknown, abundance Xi∗ > 0, i.e. all the species in the system coexist.
Mathematically this means, that Gi , Si , Li , Mi are only reasonable modeling functions if the Jacobian matrix of
the system (1) near X ∗ has only negative eigenvalues (eigenvalues with negative real part) [3]. Generalized models
facilitate finding such interaction functions Gi , Si , Li , Mi through the two remaining steps.
2
Third, we formally establish the Jacobian matrix J for (1) for the unknown value of X ∗ and for unknown functions Gi , Si , Li , Mi . Simply writing the necessary derivatives, the Jacobian matrix does not contain the functions
Gi , Si , Li , Mi directly, but only contains their partial derivatives, such as ∂Gi /∂Xi |∗ , where |∗ means that after taking the derivative by Xj the values of X ∗ should be inserted. For any given function Gi and a given value of X ∗ this
derivative is a simple number. Finding suitable functions Gi , Li , Mi , Si is therefore reduced to finding functions for
which the partial derivatives at the steady state, e.g. ∂Gi /∂Xi |∗ , result in such values that the Jacobian matrix has
only negative eigenvalues. Thus we can reinterpret the partial derivatives as parameters that describe the functional
forms used in the model and study the stability of the Jacobian matrix in terms of their values.
Fourth, because the partial derivatives in the Jacobian matrix are hard to interpret directly and because they may
depend on the (unknown) steady state, we write each of them in terms of intuitive parameters that can be measured
in real-world systems. More precisely, we use the identity
∂Gi G∗i ∂ log Gi = ∗
,
(2)
∂Xi ∗
Xi ∂ log Xi ∗
which is true for Gi (X ∗ ), Xi∗ > 0 (a condition that is met for a system that is in a state of coexistence). This breaks
G∗
each local derivative up into the product of two values. The first value, Xi∗ , is the total per capita loss rate of species
i
log Gi i at the steady state. The second value, ∂∂ log
,
is
a
logarithmic
derivative,
also called elasticity. It measures
Xi
∗
the nonlinearity of Gi at the steady state. For instance, for a power-law, f (x) = Axp , the logarithmic derivative
is ∂ log f /∂ log x = p, independently of A or x. Or, if the gain by predation of a predator i follows the well-known
Holling type-II functional response Gi ∝ Xj /(1 + aXj ), then the logarithmic derivative of Gi with regard to the prey
abundance Xj equals 1/(1 + aXj∗ ), which is one for small prey abundances (linear increase of predation gain
with
∂Gi prey abundance) and zero for very large prey abundances (predator is saturated with prey). Similar to ∂Xi , the
∗
partial derivatives of Li , Si and Mi can be broken up into one per-capita rate of loss or gain and into one elasticity
characterizing its nonlinearity at the steady state. In the following, the turnover rates and elasticities characterizing
the partial derivatives are called generalized parameters.
In summary, for a given food web, choosing a combination of the generalized parameters is equivalent to choosing
specific values for the partial derivatives in the Jacobian matrix, and thus to choosing the properties of the model
functions Gi , Li , Mi , Si near the steady state. Characterizing those combinations of parameters that lead to a Jacobian
matrix with only negative eigenvalues is therefore equivalent to finding food web models that allow a stable steady
state on the given food web topology.
For the generalized model for food webs presented in Ref. 1, the generalized parameters resulting from the described
procedure are given in Tab. I (see main article). Further, the diagonal and off-diagonal entries of the Jacobian matrix
in terms of these generalized parameters are then given by [1]
Jii = αi ρ˜i Φi + ρi (γi χii λii + Ψi )
−σ
˜i µi − σi
X
!!
βki λkj ((γk − 1)χki + 1)
k
Jij = αi ρi γi χij λij
− σi
βji +
X
k
βki λkj ((γk − 1)χk,j )
!!
,
where ρ˜ = 1 − ρ and σ
˜ = 1 − σ.
Intuitively, the terms in the first line of each expression result from biomass intake, either from primary production
Si of the focal species i (ρi = 0 for primary producers) or through the predation gain Gi from species j (ρi = 1, χij 6= 0
if i predates on j). The expressions in the second line of each expression result from to the loss of the focal species
i, either from natural mortality Mi of the focal species i (˜
σi > 0) or through the predation loss Li caused by species
j (σi > 0, βj,i 6= 0 if j predates on i). The sums over k hereby result from the effects of apparent competition if i
and j share a common predator k, because the predation of k changes when more prey is available. A more detailed
discussion is given in Ref. 1.
3
10
1 Nanophytoplankton
2 Filamentous green algae
3 Zooplankton
4 Insects
5 Silverside (Melaniris chagresi)
6 Tetras (Characidae)
7 Sailfin Molly/Mosquito Fish (Poeciliidae)
8 Blackbelt Cichlid (Cichlasoma maculicauda)
9 Bigmouth Sleeper (Gobiomorus dormitor)
10 Tarpon (Tarpon atlanticus)
11 Black Tern (Chlidonias niger)
12 Herons and Kingfishers
B Peacock Bass (Cichla ocellaris)
5–11, and B are species,
1–4 and 13 are aggregated taxa.
B
9
11
12
5
6
7
3
4
8
2
1
FIG. 1: Food Web description of the Gatun Lake ecosystem [4]. Each species or aggregated taxon is marked by a node,
the feeding interactions between the species as edges. Edges from a node to itself mark cannibalistic interactions, commonly
encountered by the fish species feeding on younger individuals of their own species. Numbered species and black (solid) edges
indicated the established ecosystem. Bold edges represent feeding links from a predator’s principle food source. Colored edges
mark the feeding interactions of the introduced peacock bass populations.
II.
PREDATOR-PREY EXAMPLE
In this appendix we provide the explicit calculations of the perturbation impact for the simple producer-grazer
example. In this system, a grazer of biomass density X1 consumes a primary producer of biomass density X2 . The
Jacobian matrix near the steady state is then (see Sec. I)
α1 (ψ1 − µ1 )
α1 γ1
.
J=
−α2 σ2 ψ1 α2 (φ2 − σ2 γ2 − σ
˜ 2 µ2 )
A discussion of stability of this system in terms of the generalized modeling parameters can be found in Refs. 1, 2.
For calculating the impact of a given perturbation for each of the two species, we require the inverse of the Jacobian
1
α2 (φ2 − σ2 γ2 − σ
˜ 2 µ2 )
−α1 γ1
−1
J =
,
α2 σ2 ψ1
α1 (ψ1 − µ1 )
det J
where
det J = α1 α2 ((φ2 − σ2 γ2 − σ
˜2 µ2 )(ψ1 − µ1 ) + σ2 γ1 ψ1 )
denotes the determinant of J.
As a perturbation, we now consider a new grazer that has a direct negative effect on the primary producer, but no
direct effect on the established grazer. The perturbation K is therefore given by K1 = 0, K2 < 0. Now we plug this
vector K in the impact equation to obtain I = −J−1 (0, K2 )T . The impact on the grazer,I1 , and the impact on the
producer, I2 , are therefore
I1 =
III.
α1
γ1 K 2
det J
and I2 =
α1
(µ1 − ψ1 )K2 .
det J
STEP-BY-STEP PARAMETRIZATION OF THE GATUN LAKE FOOD WEB
Here we apply the generalized model approach to parametrize the real-world food web of Gatun Lake in Panama
[4] shown in Fig. 1. The intention of this example is to demonstrate the method described in the paper and give a
more hands-on approach towards such a parametrization. To keep the parametrization process clear, we therefore
choose a very simplistic parametrization. After the parametrization, we analyze the perturbation consequences of an
invasion of this food web by the predatory peacock bass Cichla ocellaris, as described in 4. We note that the process
of parametrization and computation of the perturbation impact on this food web was done using the program code
provided as online supplement to this paper [5, 6].
4
Species
i
1
2
3
4
5
6
7
8
9
10
11
12
B
Name
Nanophytoplankton
Filamentous green algae
Zooplankton
Insects (mosquitoes)
Silverside
Tetras
Sailfin M.&Mosquito F.
Blackbelt Cichlid
Bigmouth Sleeper
Tarpon
Black Tern
Herons and Kingfishers
Peacock bass
Generalized Parameters
Other Parameters
Spec.
Gen. Param.
αi
ρi
σi
χii
βii
Mi (g)
χBi
i
j
χji
βji
203.3
30
30
5
2.509
1.432
3.89
0.847
0.38
0.33
1.39
0.71
0.64
0
0
1
0
1
1
1
1
1
1
1
1
-
0.62
0.50
0.9
0.82
0.80
0.59
0.66
0.49
0.26
0.13
0
0
-
0.25
0.09
0.08
0.20
0.18
0.14
0.08
-
0.32
0.11
0.13
0.30
0.37
0.56
1
-
10
0.0003
0.002
0.003
5.2
49
0.89
400
1716
24000
55
800
1213
0.30
0.17
0.39
0.09
0.05
-
1
2
3
3
4
4
4
5
5
5
6
7
8
9
3
8
5
6
5
6
7
9
10
11
12
12
12
10
1
0.82
0.46
0.46
0.45
0.47
0.80
0.86
0.83
1
0.33
0.46
0.21
0.09
1
1
0.91
0.08
0.74
0.07
0.19
0.31
0.40
0.18
0.87
0.69
0.63
0.50
−6
TABLE I: Modeling Parameters estimated for the Gatun Lake food web. αi , ψi , ρi , σi , χij and βij are generalized parameters
as used throughout the paper. All other generalized parameters are identical for all species, i.e. γ = 0.95, φ = 0.5, λ = 1, µ =
1, ψf ish = 1, ψbird = 1.02, mi = 0.12. Mi is the used typical body mass of individuals from each species, and χBi is the relative
diet composition the peacock bass.
A.
Parametrization
For a given food web, the generalized model presented in Sec. I describes a broad class of different food web
models. Testing a given model in this class through numerical computations requires setting all parameters
α, γ, µ, φ, ψ, β, χ, σ, ρ to specific values, i.e. we have to parametrize the model. In our example parametrization,
the parameters α, γ, µ, φ, ψ are chosen independently from the underlying food web topology, while the remaining
parameters χ, β, σ, ρ depend on the strength of the feeding relations between the different species.
The parameters used in our model are set using the following rules. The resulting parameters are given explicitly
in Tab. I.
• αi = (0.00485 Mi)−1/4 . The species turnover rates are scaled allometrically with their body mass Mi , unless
more precise explicit empirical knowledge is available. The pre-factor 0.00485 was determined by the best fit of
available turnover data.
• µi = 1. The loss of biomass through natural mortality of each species increases linearly with the species’
abundance.
• ψi = 1 for fish and 1.02 for birds. The predation gain increases linearly with the abundance of fish. For birds
the predation gain increases slightly faster because increasing bird abundance decreases the fish refugium from
predation.
• φi = 0.5. The biomass gain of primary producer species’ in the food web increases sublinearly with the primary
producer’s abundance, because of space or nutrient constraints.
• γi = 0.95. The biomass gain through predation of a predator species increases slightly slower than the abundance
of its prey. Intuitively, this is due to a saturation in prey of the predator as in the well-known Holling type-II
functional response.
• λij = 1. All prey switching coefficients are equal. The value 1 indicates, that a predator has no dietary
preferences and switches passively between different prey species.
• ρi = 0 for primary producers and ρi = 1 for predators. Each species gains biomass either by primary production
or by predation, but not both.
• χji reflects the relative contribution of species i to species j’s diet. To parametrize this, we use allometric
relationships between consumer and prey body sizes. More precisely, we estimate for each predator the preferred
5
Species
No.
1
2
3
4
5
6
7
8
9
10
11
12
Name
Nanophytoplankton
Filamentous green algae
Zooplankton
Insects (Mosquitoes)
Silverside
Tetras
Sailfin Molly &Mosquito Fish
Blackbelt Cichlid
Bigmouth Sleeper
Tarpon
Black Tern
Herons and Kingfishers
Observed
Impact
change [4]
IB /κ
decrease
?
increase
increase
-50%
-95%
-100%
+50%
-90%
decrease
decrease
decrease
-0.06
+0.06
+0.05
+0.04
-0.6
-0.3
+0.24
-0.06
-0.16
-1.25
+3
-0.1
TABLE II: Comparison of the observed long-term changes in the species abundances after the introduction of the peacock bass
[4] with the assessed impact of the perturbation corresponding to this introduction. For the impact IB represents the response
to the population of peacock bass.
prey body mass and a typical prey body mass range from Table 1 in Ref. [7], individually considering allometric
scaling for birds, fishes, and invertebrate predators. Then, we consider that for each predator with a preferred
prey body mass Mpref the feeding on prey with a body mass Mprey is distributed normally with log(Mprey /Mpref ),
where the width of the normal distribution is estimated from the typical prey body mass range of each consumer.
Thus we obtain for each predator-prey relation in the food web the preference
P eki of the predator k to consume
the prey i. To obtain χki , we normalize these contributions, i.e. χki = eki / j ekj .
• σi , indicating biomass loss due to predation relative to the total loss, strongly
P with predation pressure
P increases
mi is a
on a species. Using the feeding preferences established for χ, we set σi = k eki /( k eki + mi ), whereP
measure for the importance of the loss through mortality relative to the feeding pressure measured by k eki .
• βji represents the relative loss of a species i to its different predators j. However, in general this predation loss
of a species to different predators is not independent from the predation gain of the respective predators. In
other words, the parameters β are dependent on χ. To resolve this dependency consistently, we assume that
when a predator kills a prey, it consumes all its biomass, such that biomass is conserved throughout the process.
We denote the total biomass loss rate of a species i as Bi . For top predators, this loss is due only to natural
mortality and we set Bi = 1. For all P
other species the conservation of biomass implies that the loss Bi equals
the intake of its predators, i.e. Bi =
χji Bj /σj , where j runs over all of i’s predators. This system of linear
equations can be solved for all the species biomass outflows B. Then the parameters β are the contributions of
each predator j to the total loss of i, i.e. βji = (χji /Bi ) Bj /σj .
This configuration of parameters is included as an example in our online code [5].
Once we have established the parametrization, we can write the Jacobian matrix J using the expressions given in
Sec. I. Stability of any particular steady state (corresponding to a particular set of generalized parameters) is ensured
by checking that all eigenvalues of the Jacobian have negative real parts [3]. For the Gatun Lake Food Web with
the parametrization above, the eigenvalues with the largest real parts are the pair of complex conjugate eigenvalues
λ± ≈ −0.001 ± 0.02i. Therefore, the chosen parametrization indeed corresponds to a steady state of coexisting species.
B.
Estimation of the impact caused by the introduction of peacock bass
Now, we compute the impact of a perturbation to the Gatun Lake food web. More specifically, we estimate the
initial effect of the invasion of the lake by a predatory fish, such as the peacock bass, as described in Ref. 4.
We consider the increase of predation pressure on species that are prey to the peacock bass as the main direct
perturbation of the established ecosystem. To quantify this perturbation for the different established species i we now
estimate the effect of the additional population of peacock bass YB . The perturbation of species i due to increased
6
predation by B is
KiB = − ∂Li /∂YB |∗ = −αB χBi κi ,
(3)
where we used (2) to split the derivative into the turnover rate αB χBi (αB is the turnover rate of the peacock bass
and χBi the relative contribution of i to B’s diet), and an elasticity κi = ∂ log Li /∂ log YB . For simplicity, we assume
in the following that κ is identical for all species i and therefore drop the corresponding index.
The peacock bass feeds on the secondary consumer fishes (tetras, silverside, and sailfin molly/mosquito fish, blackbelt cichlid), and on the bigmouth sleeper [4]. To obtain the relative contributions of each of these species to its diet,
χai , we employ the allometric scaling rules introduced to estimate the diets of species in the food web. This leads to
the values indicated in Tab. I.
Plugging J and the perturbation for the peacock bass K into I = −J−1 K we obtain the ultimate impact of the
perturbation shown in Fig. 1 of the paper. This computation can be easily done using our supplementary online code
[6]. The resulting response (Fig. 2m) is given explicitly in Tab. II alongside the observed changes [4].
In addition to the perturbation caused by the peacock bass (Fig. 2m), we also consider the perturbation of the
system due to a specialist predator feeding on only one species i, i.e. the diet of the predator (P) is characterized by
χP,i = 1. We observe, that the response to these perturbations (Fig. 2a-l) agrees well with the predicted sensitivity
and influence (Fig. 2n,o). If an influential species is perturbed, generally all species show a strong response (e.g.
Fig. 2k), but when a species with low influence is perturbed, generally only few species are strongly affected (e.g.
Fig. 2b). Further, sensitive species generally respond strongly to perturbations elsewhere (e.g. species 6), while species
with low sensitivity (e.g. species 5) show generally smaller responses.
IV.
DERIVATION OF INFLUENCE AND SENSITIVITY
Here, we derive the expressions for the influence and sensitivity of the food web species that are motivated in the
paper with intuitive arguments. For this, we decompose the Jacobian matrix into the dynamical modes of the system.
Then we use the impact-definition to identify sensitive species as those for which the expected impact of randomly
excited modes is largest, and to identify influential species as those for which the expected excitation of dynamical
modes is largest when they are perturbed.
The inverse Jacobian matrix J−1 , required for the impact calculation, has the same right eigenvectors v (k) and left
eigenvectors w(k) as the Jacobian matrix. Each pair v (k) , w(k) correspond to the eigenvalue 1/λk , where λk is the
eigenvalue of J. Therefore, for a perturbation K, the impact equation I = −J−1 K can be written as the sum over
the eigenvectors and eigenvalues of J as
I=
X
v (k)
k
w(k) · K
,
−λk
where k runs over all eigenvalues and where · denotes the scalar product.
We see that for a given mode k the contribution to the impact on a given node depends on three factors. First, the
product w(k) · K determines which dynamical modes are excited by the perturbation. Second, v (k) determines which
entries in I are affected by this dynamical mode. And finally, the excitability 1/λk indicates how strongly the mode
will be excited.
For the sensitivity of a species, we do not want to explicitly refer to any particular perturbation K. In absence
of additional information, we therefore consider the case of a perturbation affecting every dynamical mode k with
identical probability. The sensitivity Sei can thus be measured as the effects on node i,
!
X v (k) i Sei = log
,
−λk
k
where v (k) i denotes the absolute value of v (k) on node i, and where we used the logarithm to bring the numerical
values into a more manageable range.
For the influence of a species, we evaluate the impact of a perturbation affecting only this specific species K. In
absence of additional information, the influence is measured by the resulting excitation of all the dynamical modes
!
X w(k) i .
Ini = log
−λk
k
7
FIG. 2: Predicted response of the Gatun Lake food web [4] to different perturbations. Shown are species (circles) and predatorprey relationships (arrows, in the direction of the biomass flow). In each panel a-l) a different population is affected by a
hypothetical specialized predator (P), such that the abundance of this species decreases. Panel m) shows the perturbation
caused by the peacock bass (B). The color code denotes the response of each species to the different perturbations. In panels
n) and o) the gray-scale denotes the sensitivity and the influence of each species respectively. Color online.
V.
MODEL FOOD WEBS
For the numerical experiments in our paper, we generate model food webs as suggested in Ref. 8. This process
consists of two steps, topology generation and parametrization, which can be reproduced using our supplementary
online code [5].
To generate the food web topology, we use the niche web algorithm [9]. This algorithm establishes a topology by
assigning to each species a so-called niche value, denoting a species fitness or body size. Further it assigns to each
species a range of niche values, called feeding range. If a species has a niche value that lies inside the feeding range
of another species, then it is prey to this species.
Formally, the niche value ni of species i is drawn from a uniform distribution between 0 and 1. Further, the
8
feeding range of viable prey niche values is given by ci − ri /2 . . . ci + ri /2, where ri denotes its size and ci its center.
ri is obtained for each species by independently drawing a beta distribution variable (between 0 and 1) and then
multiplying with the species’ niche value ni . The center value ci of this range, is then drawn uniformly between ri /2
and ni . We note, that this makes cannibalistic links possible, because ci − ri /2 . . . ci + ri /2, contains the niche value
ni , if ci > ni − ri /2.
In our simulations, we use only niche model food webs meeting two requirements. First, topologies have to be
connected, such that the resulting food web does indeed contain the desired number of species and does not separate
into multiple smaller food webs. Second, the connectance of a generated food web cannot deviate more than 5% from
the desired value.
After the topology of a food web is established, we parametrize the generated food web topology using the generalized
model reviewed in Sec. I [1] . In other words, we sample a specific food web model for the given topology by drawing
a set of specific values for the generalized parameters. For this sampling, species turnover rates αi are set using
allometric scaling with body mass, where the niche values are used as an indicator for their body size. All other
parameters are sampled uniformly and
P independently
Pfrom the ranges indicated in Tab. I in the main paper, for the
parameters χ and β, the conditions i χij = 1 and j βij = 1 are enforced by subsequent normalization.
Once the food web is parametrized with a specific set of parameters, we can compute the Jacobian matrix and thus
the stability of the chosen configuration. In our numerical experiments, we retain only configurations, that allow for
a stable coexistence of species, i.e. configurations for which the largest Jacobian matrix eigenvalue has a negative real
part.
VI.
QUALITY ASSESSMENT OF AN IMPACT PREDICTION
To assess the quality of an impact estimation, we compare the true impact I computed using the true Jacobian
matrix with the estimated impact I˜ computed using the estimated Jacobian matrix. The quality Q of the impact
estimation is evaluated as the cosine of the angle between the true and the estimated impact vector. Mathematically,
Q = cos I 6 I˜ =
I · I˜
,
˜
||I|| ||I||
where · denotes the scalar product and ||I|| is the norm of the vector I.
˜ i.e. if the estimation is exactly correct, then Q = I · I/||I||2 = 1. Further, if I = −I,
˜ i.e. if
For instance, if I = I,
the estimation is exactly the opposite of the true impact, then Q = −I · I/||I||2 = −1.
VII.
FIGURES FOR A HIGHER FOOD WEB CONNECTANCE
Throughout the paper, we show graphs and results for for model food webs with a connectance of C = 0.04. Because
of the great importance of the connectance for food web properties, for example for the stability of a food webs [8],
we show here the same results based on model food webs with twice the connectance, i.e. for C = 0.08. The detailed
description of the shown quantities and the interpretations of the graphs are given in the main paper. All of the
following graphs have the same parameters as in the paper.
The impact as a function of importance and sensitivity is virtually identical to the results for C = 0.04 (Fig. 3).
Close inspection reveals a slight increase in the overall values of importance and sensitivity.
The increase in prediction quality with iterative parameter estimation is qualitatively identical to the results for
C = 0.04 (Fig. 4), but the overall quality of the prediction decreases.
The decrease of prediction quality with an error in one species property is qualitatively identical to the results for
C = 0.04 (Fig. 5). The decrease with parameters referring to dietary preferences χ, β is more pronounced.
The correlations of sensitivity and influence with several species properties (Fig. 6) are similar to those found for
C = 0.04. A significant increase is observed for the negative correlation between sensitivity and species vulnerability.
In summary, we observe that doubling the connectance generally has only a quantitative, but not qualitative effect
on our results.
9
10
0.8
4
0.7
3
3
8
2
2
1
6
1
4
0
0
2
-1
0
0
−1
-1
−2
1.0
0.8
0.6
α
µ
φ
ψ
0.4
0.2
0.0
0
0.5
0.4
0.3
0.2
0.1
γ
χ
β
6
8
10
2
4
12
Sensitivity of error-carrying node
14
0
20
40
60
80
100
% of species measured with high precision
FIG. 4: Focusing on sensitive and important species can reduce the measurement effort when species are measured successively with higher precision. See Fig. 3 in the main paper
for details. Parameters: C = 0.08.
Quality of impact prediction
Quality of impact prediction
FIG. 3:
The average absolute impact on a species with
given sensitivity if a species of given influence is perturbed.
See Fig. 2 in the main paper for details. Parameters: N = 50,
C = 0.08.
N=10
N=30
N=50
randomly
max. Sensitivity
max. Influence
max. Sens.× Inf.
0.6
0.0
2 4 6 8 10 12 14
Sensitivity (observed sp.)
Error in:
Quality of the prediction
4
5
log10 |Impact|
12
-1
Influence (perturbed sp.)
14
1.0
0.8
0.6
Error in:
α
µ
φ
ψ
0.4
0.2
0.0
0
2
γ
χ
β
6
8
10
4
12
Influence of error-carrying node
14
Correlation Coefficient R
FIG. 5: Average quality of an impact prediction in the presence of measurement errors. The average quality of an impact
prediction if one node with the specified sensitivity (left) or influence (right) in a food web is subject to a measurement error.
See Fig. 4 in the main paper for details. Parameters: N = 50, C = 0.08.
0.4
0.2
0.0
−0.2
−0.4
Sensitivity
Influence
p.
. im
pol
To 10 p.
s =ol. im
p
To 2
s=
e
gre
y
De
ilit
rab
lne
Vu
ty
ali
ner
Ge ary
m
r
Pri duce
prohic
p
Tro el
lev ver
rno
Tu te
ra
FIG. 6: Correlations of species’ properties with their sensitivity and influence. See Fig. 5 in the main paper for details.
Parameters: N = 50, C = 0.08.
10
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