SOM - Jelle Lever

The sudden collapse of pollinator communities
— Supplementary material —
J. Jelle Lever1,2∗, Egbert H. van Nes1 , Marten Scheffer1 , and Jordi Bascompte2
1
Department of Aquatic Ecology and Water Quality Management, Wageningen
University, P.O. Box 47, NL-6700 AA, Wageningen, The Netherlands
2
Integrative Ecology Group, Estación Biológica de Doñana, CSIC
Calle Américo Vespucio s/n, 41092 Sevilla, Spain
∗ To
whom correspondence should be addressed. E-mail: [email protected], Phone: +31 647238146, Fax: +31 317419000
1
A
A
A
SUPPLEMENTARY MATERIAL 1
P
(a) dA < rA
P
(b) dA > rA
P
(c) dA >> rA
Figure S1: Nullclines of two mutualistically interacting species. Filled dots indicate stable equilibria,
open dots indicate unstable equilibria. Fundamentally different configurations exist when (a) the driver
of pollinator decline, dA , is smaller than intrinsic growth rate rA , (b) when the driver of pollinator
decline, dA , is bigger than intrinsic growth rate rA and, (c) when the driver of pollinator decline, dA ,
is substantially larger than intrinsic growth rate rA . By increasing the driver of pollinator decline, dA ,
we change from a regime with one stable state, presented in a, to the regime with two alternative stable
states presented in b, until eventually a tipping point is reached where pollinators collapse to extinction.
For a further analysis of models with two mutualistically interacting species see May (1978), Dean (1983),
and Wright (1989).
2
SUPPLEMENTARY MATERIAL 2
Random network structure
Nested network structure
3.5
2
Abundance of Plants
3
1.5
2.5
2
1
1.5
1
0.5
0.5
0
0
0
0.2
0.4
0.6
0.8
1
0
Driver of pollinator decline (d )
0.2
0.4
0.6
0.8
1
Driver of pollinator decline (d )
A
A
Figure S2: Collapse of plant populations when increasing the mortality dA of pollinators. Results are
shown for a random (left) and a nested (right, N=0.6) network. Parameter settings are the same as in
figure 2.
Random network structure
Nested network structure
3.5
2
Abundance of Plants
3
1.5
2.5
2
1
1.5
1
0.5
0.5
0
0
0
0.2
0.4
0.6
0.8
1
0
Driver of pollinator decline (d )
0.2
0.4
0.6
0.8
1
Driver of pollinator decline (d )
A
A
Figure S3: Re-establishment of plant populations when decreasing the mortality of pollinators dA .
Results are shown for a random (left) and a nested (right, N=0.6) network. Parameter settings are the
same as in figure 3.
3
SUPPLEMENTARY MATERIAL 3
We tested the extent to which our results depend on the specific number of species, connectance or
fraction of forbidden links chosen (see figure S4, S5, S6 and S7).
Furthermore, we show in figure S8 and S9 what our results look like if we do not allow any species to
have less than 2 partners during any step of the algorithm we used to generate nested networks.
We only found qualitative differences in the behaviour of our model.
4
Fraction of networks
Random network structure
Increasingly nested structure
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1 point of collapse
3 points of collapse
2 points of collapse >3 points of collapse
fraction of feasible networks
0
1 point of collapse
3 points of collapse
2 points of collapse >3 points of collapse
fraction of feasible networks
0
0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.35
0.2
0.25
0.3
Connectance
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Nestedness
Figure S4: Results when using the same parameter settings as in figure 4, only now the community
consists out of 35 plant and 35 pollinator species. As in figure 4, the coloured bars represent the fractions
of feasible networks in which a certain number of collapses is found. The fraction of networks in which
feasible solutions are found is indicated with the green diamonds.
Random network structure
Increasingly nested structure
Driver of pollinator decline (dA)
1.1
Final collapse
First recovery
1
Final collapse
First recovery
1.2
1.1
0.9
1
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.1
0.15
0.2
0.25
0.3
0.2
0.35
Connectance
0.3
0.4
0.5
0.6
0.7
Nestedness
Figure S5: Points of collapse (circles) when the driver of pollinator decline, dA , is increased, and points
of recovery (triangles) when the driver of pollinator decline, dA , is decreased. In case of multiple collapses
and/or recoveries, the final point of collapse and the first point of recovery was plotted. Parameter settings
are as in figure S4.
5
Fraction of networks
Random network structure
Increasingly nested structure
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1 point of collapse
3 points of collapse
2 points of collapse >3 points of collapse
fraction of feasible networks
0
1 point of collapse
3 points of collapse
2 points of collapse >3 points of collapse
fraction of feasible networks
0
0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45
0.3
0.35
0.4
Connectance
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Nestedness
Figure S6: Results when using the same parameter settings as in figure 4, only now competition
between species is a bit stronger, Cij ∼ U (0.025, 0.075), and in communities with increasingly nested
network topologies (right panel), the connectance is fixed to 0.25, and the fraction of forbidden links is
fixed to 0.25. As in figure 4, the coloured bars represent the fractions of feasible networks in which a
certain number of collapses is found. The fraction of networks in which feasible solutions are found is
indicated with the green diamonds.
Random network structure
Increasingly nested structure
Driver of pollinator decline (dA)
0.9
1
Final collapse
First recovery
Final collapse
First recovery
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.2
0.25
0.3
0.35
0.4
0.45
0.3
Connectance
0.4
0.5
0.6
0.7
0.8
Nestedness
Figure S7: Points of collapse (circles) when the driver of pollinator decline, dA , is increased, and points
of recovery (triangles) when the driver of pollinator decline, dA , is decreased. In case of multiple collapses
and/or recoveries, the final point of collapse and the first point of recovery was plotted. Parameter settings
are as in figure S6.
6
Fraction of networks
Random network structure
Increasingly nested structure
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1 point of collapse
3 points of collapse
2 points of collapse >3 points of collapse
fraction of feasible networks
0
1 point of collapse
3 points of collapse
2 points of collapse >3 points of collapse
fraction of feasible networks
0
0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45
0.3
0.35
0.4
Connectance
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Nestedness
Figure S8: Results when using the same parameter settings as in figure S6, only now each species has
at least two interactions. As in figure S6, the coloured bars represent the fractions of feasible networks
in which a certain number of collapses is found. The fraction of networks in which feasible solutions are
found is indicated with the green diamonds.
Random network structure
Increasingly nested structure
Driver of pollinator decline (dA)
0.9
1
Final collapse
First recovery
Final collapse
First recovery
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.2
0.25
0.3
0.35
0.4
0.45
0.3
Connectance
0.4
0.5
0.6
0.7
0.8
Nestedness
Figure S9: Points of collapse (circles) when the driver of pollinator decline, dA , is increased, and points
of recovery (triangles) when the driver of pollinator decline, dA , is decreased. In case of multiple collapses
and/or recoveries, the final point of collapse and the first point of recovery was plotted. Parameter settings
are as in figure S8.
7
REFERENCES SUPPLEMENTARY MATERIAL
Dean, A. M. (1983). A simple model of mutualism. Am. Nat., 121, 3, pp. 409−417.
May, R. (1978). Studies in Mathematical Ecology. II Populations and Communities, chapter
Mathematical aspects of the dynamics of animal populations., pp. 317−366. Washington, DC:
Mathematical Association of America.
Wright, D. H. (1989). A simple, stable model of mutualism incorporating handling time. Am. Nat.,
134, 4, pp. 664−667.
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