Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of
extrinsic and intrinsic factors in rodent populations. Ecology
Appendix S3
1
Sensitivity analysis of vole IBM with
2
phenomenological predator submodule
3
4
1
Sensitivity analysis method ..................................................................................................... 2
5
2
Results of sensitivity analysis .................................................................................................. 6
6
3
Cited Literature ........................................................................................................................ 9
7
8
1
Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of
extrinsic and intrinsic factors in rodent populations. Ecology
Appendix S3
9
1
SENSITIVITY ANALYSIS METHOD
10
We used an improved version of the Morris method (elementary effects method) for sensitivity
11
analysis (Morris 1991, Campolongo et al. 2007, Thiele et al. 2014) in order to identify the most
12
sensitive parameters. For this we varied 22 parameters over five levels with central values being
13
those used for nominal model runs, and additional four levels as following: lower extreme, lower
14
median, upper extreme and upper median (Table S 1). The experimental design consisted of 50
15
trajectories, i.e. we calculated 50 elementary effects using 50*(22 + 1) = 1150 model runs. The
16
sensitivity of the model to each parameter was evaluated by using µ*, the mean of the
17
distribution of absolute values of elementary effects and σ, the standard deviation of the
18
elementary effects values. The bigger the value of µ* associated with a certain parameter, the
19
more sensitive is the model to this parameter. This metric is a measure of the overall impact of a
20
parameter on the output (Campolongo et al. 2007). σ measures the effect of higher order effects,
21
i.e., non-linear and/or interaction effects, i.e. the higher value of σ indicates that this parameter
22
affects the output variable in interaction with other parameters or/and in a non-linear way. As
23
output we used two groups of variables, describing different aspects of population dynamics: i)
24
the effect on overall population dynamics (mean and SD of autumn population size); and ii) the
25
cyclicity of the time series (period and amplitude of cycles, if present). The sensitivity analysis
26
was performed with sensitivity library in R.
27
28
29
2
Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of extrinsic and intrinsic factors in rodent
populations. Ecology
Appendix S3
30
Table S 1. Range of parameter values used for global sensitivity analysis for each of the tested 22 parameters. For each parameter we
31
give its description, source, and process which it acts upon.
Parameter
Survival
𝒔𝒘
𝒔𝒇
𝒔𝒔𝒖𝒎
Rmax
Kwin
Dispersal
𝜶𝒆𝒎
𝜷𝒔𝒆𝒙
𝜷𝒔𝒕𝒂𝒈𝒆
𝜷𝒅𝒆𝒏𝒔
𝜷𝟏
𝜷𝟐
𝜷𝟑
𝒃𝒆𝒎
𝒔𝒅𝒊𝒔𝒑
Parameter description (units)
Range
Source
Lower
extreme
Medium
low
Estimate
Medium
high
Survival rate of weanlings due to male turnover
(week -1)
Survival rate of reproductive females due to male
turnover (week -1)
Baseline survival rate in summer (week -1)
0.22
0.3
0.38
0.48
0.82
0.845
0.87
0.895
0.96
0.97
0.98
0.99
Winter maximum population growth rate (week -1)
Carrying capacity in winter (ind./patch)
0.3
4.5
0.35
5.5
0.4
6.5
0.5
7.5
-1.9653
-1.0044
-0.2561
-1.0919
-1.8892
-1.6062
-0.7793
-0.0174
-0.9691
-1.6954
-1.2470
-0.5542
0.2212
-0.8463
-1.5017
-0.8878
-0.3291
0.4598
-0.7235
-1.3080
0.2338
0.3635
0.4933
0.6231
0.0388
0.1630
0.2873
0.4116
0.5358
–
–
0.814
–
–
0.9
0.92
0.94
0.95
Intercept in the emigration equation
Effect of sex in the emigration equation
Effect of stage in the emigration equation
Effect of density in the emigration equation
Coefficient for sex*stage interaction in the
emigration equation
Coefficient for density*stage interaction in the
emigration equation
Coefficient for density*sex interaction in the
emigration equation
Maximum in the uniform distribution for the
random effect in the emigration equation
Dispersal survival rate (week -1)
Upper
extreme
0.58 Andreassen
and
Gundersen 2006
0.92 Andreassen
and
Gundersen 2006
1 H.P. Andreassen,
unpublished data
0.6 Aars and Ims 2002
8.5 Aars and Ims 2002
-0.5287
-0.1040
0.6985
-0.6007
-1.1142 Andreassen and Ims
2001
0.7528
0.96 H.P. Andreassen,
unpublished data
Reproduction
3
Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of extrinsic and intrinsic factors in rodent
populations. Ecology
Appendix S3
numlitPois
2
2.56
3.12
4.06
Minnumlit
Lambda for Poisson distribution used for the
number of litters produced per female
Minimum number of litters
–
–
1
–
Maxnumlit
Maximum number of litters
–
–
5
–
Weeks
𝜶𝒓𝒆𝒑
𝜷𝒓𝒆𝒑
𝒃𝒓𝒆𝒑
Number of weeks between two reproduction events
Intercept in the reproduction equation
Effect of sociality in the reproduction equation
Maximum in the uniform distribution for the
random effect in the reproduction equation
Maximum number of weanlings
–
0.4922
0.0045
–
–
0.6336
0.0079
–
3
0.7750
0.0113
0.45
–
0.9164
0.0147
–
–
–
6
–
Predator intrinsic growth of increase (yr-1)
2.4
2.6
2.8
3
Q
Predator-prey ratio constant (voles*predator-1)
40
41
42
71
c
Maximum consumption per predator (voles*year1
*predator-1)
150
175
200
250
dlow
Predator mortality rate when prey is scarce (yr-1)
-5
-4.5
-4
-3
Ncrit
Critical prey density for predator reproduction
(voles*ha-1)
Predation half-saturation constant (voles*ha-1)
10
12.5
15
27.5
2
2.5
3
6.5
Lower limit for a predator population density
(predator*ha-1)
–
–
0.005
–
Maxwean
Predation
Smax
D
𝑷𝒎𝒊𝒏
32
33
1
34
2
5 H.P. Andreassen,
unpublished data
– H.P. Andreassen,
unpublished data
– H.P. Andreassen,
unpublished data
– Ims 1997
1.0578 Rémy 2011
0.0181 Rémy 2011
– Rémy 2011
– Ims 1997
3.2 Hanski
and
Korpimaki 1995
100 Turchin and Hanski
1997
300 Hanski
and
Korpimaki 19952
-2
Hanski
and
Korpimaki 19953
40 Turchin and Hanski
19974
10 Turchin and Hanski
1997
– Hanski
and
Korpimaki 1995
Demographic parameters parameterized with the data on Myodes glareolus (unlike the rest of the parameters that are derived for
M. oeconomus)
Maximum consumption per predator had to be adjusted since predation occurs only once a year
4
Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of extrinsic and intrinsic factors in rodent
populations. Ecology
Appendix S3
35
3
Close to the dhigh estimated by Hanski and Korpimaki (1995) to be -5
36
4
Used critical prey density is slightly higher than estimated by Turchin and Hanski (1997): 14 voles*ha-1
5
Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of
extrinsic and intrinsic factors in rodent populations. Ecology
Appendix S3
37
2
RESULTS OF SENSITIVITY ANALYSIS
38
Overall, one out of 22 tested parameters is suggested as the least influential according to most of
39
the used response variables: Ncrit (Figure S 1 and Figure S 2). When we used as output the
40
variables describing the distribution of the autumn population size (mean autumn population size
41
and its standard deviation), the most sensitive parameters were: 𝑠𝑠𝑢𝑚 , Q, numlitPois, and 𝛽𝑟𝑒𝑝
42
(Figure S 1). It does not come as a surprise, since these key survival and reproduction parameters
43
are a priori expected to affect the population size. However, when we used as output the
44
variables describing the cycle (period and amplitude), the most sensitive parameters were all
45
parameters acting on reproduction (𝛼𝑟𝑒𝑝 , 𝛽𝑟𝑒𝑝 , numlitPois), of which 𝛼𝑟𝑒𝑝 and 𝛽𝑟𝑒𝑝 represent
46
sociality effects, dispersal (𝑠𝑑𝑖𝑠𝑝 , 𝛼𝑒𝑚 , 𝛽𝑠𝑒𝑥 ) and predation (𝑄; Figure S 2). Thus, the cycles are
47
affected by social parameter settings (facilitation of reproductive output in presence of several
48
females), survival during dispersal, intercept in the emigration equation and the effect of sex on
49
emigration probability and predator-prey ratio.
50
Importantly, many parameters that were identified as affecting the model output to a large
51
extent were acting in an interactive or non-linear way. Such parameters were: Q, numlitPois, and
52
𝛽𝑟𝑒𝑝 in case of the model output reflecting the distribution of the autumn population size; and
53
𝛼𝑟𝑒𝑝 , 𝛽𝑟𝑒𝑝 , 𝛼𝑒𝑚 , and 𝑠𝑑𝑖𝑠𝑝 in case of the model output describing the population cycles.
54
These findings underline the importance of a combination of social factors, dispersal and
55
predation in formation of the cyclic pattern characteristic of the vole populations in nature.
56
Moreover, the cyclic pattern is a result of the complex interactions among the parameters to
57
which the model output is the most sensitive.
58
6
Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of
extrinsic and intrinsic factors in rodent populations. Ecology
Appendix S3
b)
80
80
a)
Q
K s
1 win disp
d en s
R maxsex
S
3 max
sPtas2ge
f
min
D
s
em
w
60
D s t age
sPfmin
r ep
numlit Pois
re p
20
sw
N crit
N crit
0
-20
0
20
40
60
0
80
20
Meanpop
c)
60
* 40
Meanpop
80
60
50
d)
50
re p
sex
em
SDpop
30
rep
s ex s sum
d
20
rep
em
s sum
K1 win
d2ens
s disp
3
st ag e
D
P
ssS
f max
min
w
10
10
r ep
numlit Pois
R max
d low
20
Klow
win
R
2d e nssmax
disp
3
s f st
S max
s awge
D P min
1
Q
40
40
numlit Pois
30
Q
*
SDpop
s sum
s1exd
em low
S
Rmax
Kens
max
win
s
d
3 disp
2
40
Meanpop
rep
d low
20
40
60
rep
s sum
0
*
Meanpop
Q
numlit Pois
-20
0
N crit
0
0
N crit
20
SDpop
40
0
10
20
* 30
SDpop
40
50
60
59
60
Figure S 1. Sensitivity of two output variables, the mean population size (Meanpop: a,b) and
61
standard deviation of the population size (SDpop: c,d) to the 22 parameters. µ is the mean of the
62
elementary effects values, µ* is the mean of the absolute values of elementary effects; and σ is
63
the standard deviation of the elementary effects. High µ indicates the overall sensitivity of the
64
output to the parameter; low µ simultaneously with high µ* indicates a non-monotonic
65
relationship; high µ with high σ indicates interactive effect of the parameter on the output and
66
high µ with low σ indicates the main effect (no interactions) of the parameter on the output.
7
Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of
extrinsic and intrinsic factors in rodent populations. Ecology
Appendix S3
a)
200
b)
rep
300
em
K win
S2max
r ep
1en s
age
sdQ
Rsstmax
disp
w
s sum
se x
P min
d low
numlit Pois
sf D 3
sex
s ta ge Pois
P min
50
100
100
D d low
sf
Q
s sum
200
S max
1 d ens
K win em
s disp
2
R max s w numlit
Ampl
150
r ep
3
*
Ampl
re p
-50
N crit
0
0
N crit
0
50
0
100
100
*
50
Ampl
2
12
s t age
1
8
P min
s disp
2 s
d low d1en
s
Q
sum
s wD
xe m
Kse
3
Ssmax
winp
f numlit
re
Pois
r ep
6
Q
s sum
d en3s s ex
rep
K win
numlit Pois
d low
srep
f
D
S max s
w
R max
R max
4
Period
6
4
s disp
2
2
*
Period
st a ge
10
d)
em
200
Ampl
c)
P min
150
N crit
0
0
N crit
-4
67
-2
0
Period
2
4
0
2
4
*
Period
6
68
Figure S 2. Sensitivity of two output variables, the amplitude (Ampl: a,b) and period (Period:
69
c,d) to the 22 parameters. µ is the mean of the elementary effects values, µ* is the mean of the
70
absolute values of elementary effects; and σ is the standard deviation of the elementary effects.
71
High µ indicates the overall sensitivity of the output to the parameter; low µ simultaneously with
72
high µ* indicates a non-monotonic relationship; high µ with high σ indicates interactive effect of
73
the parameter on the output and high µ with low σ indicates the main effect (no interactions) of
74
the parameter on the output.
8
Radchuk V., Ims, R.A, Andreassen H.P. From individuals to population cycles: the role of
extrinsic and intrinsic factors in rodent populations. Ecology
Appendix S3
3
CITED LITERATURE
Aars, J., and R. A. Ims. 2002. Intrinsic and climatic determinants of population demography:
The winter dynamics of tundra voles. Ecology 83:3449–3456.
Andreassen, H. P., and G. Gundersen. 2006. Male turnover reduces population growth: An
enclosure experiment on voles. Ecology 87:88–94.
Andreassen, H. P., and R. A. Ims. 2001. Dispersal in patchy vole populations: Role of patch
configuration, density dependence, and demography. Ecology 82:2911–2926.
Campolongo, F., J. Cariboni, and A. Saltelli. 2007. An effective screening design for
sensitivity analysis of large models. Environmental Modelling & Software 22:1509–
1518.
Hanski, I., and E. Korpimaki. 1995. Microtine rodent dynamics in Northern Europe parameterized models for the predator-prey interaction. Ecology 76:840–850.
Ims, R. A. 1997. Determinants of geographic variation in growth and reproductive traits in
the root vole. Ecology 78:461–470.
Morris, M. D. 1991. Factorial sampling plans for preliminary computational experiments.
Technometrics 33:161–174.
Rémy, A. 2011. Linking behaviour with individual traits and environmental conditions, and
the consequences for small rodent populations. University of Oslo.
Thiele, J., W. Kurth, and V. Grimm. 2014. Facilitating Parameter Estimation and Sensitivity
Analysis of Agent-Based Models: A Cookbook Using NetLogo and R. Journal of
Artificial Societies and Social Simulation 17.
Turchin, P., and I. Hanski. 1997. An empirically based model for latitudinal gradient in vole
population dynamics. American Naturalist 149:842–874.
9