1
Supplementary Figures and Tables
1
2
3
4
Table A1. Details of parameterization for the grassland model in the left panels of Fig. 1. Parameters with source marked “Unknown”
were tuned to achieve systems with realistic equilibrium biomasses and N input rate values, and to meet the stability criteria for Case B
described in the main text.
State variable or parameter
Range
Sources and notes
Value
̅
Native biomass [kg C/ha]
[7500, 15000]
(Guo et al. 2008; Lunt and Morgan 1999a;
Varies
𝐵𝑛
Morgan and Lunt 1999)
Exotic annual biomass [kg C/ha]
[1000, 7500]
Lower than native biomass.
Varies
𝐵̅𝑒
̅
Plant-available N [kg N/ha]
Not constrained
Unknown
Varies
𝐴
N input rate [kg N/ha/yr]
𝐼
[0,15]
(Menge et al. 2009)
Varies
Half-saturation constant in biomass/cover
ℎ
[500,10000]
Unknown
3890
functions [kg C/ha]
Litterfall N-loss coefficients [.]
𝛿𝑛
[0.003,0.7]
(Menge et al. 2008)
0.06
𝛿𝑒
[0.003,0.7]
0.06
Plant-available N soil leaching rate [/yr]
𝑘
[0.0,1.0]
Unknown
0.4
Exponent in biomass/cover functions [.]
𝑝
{1,2,3}
Unknown
3
N use efficiencies [kg C/kg N]
𝜔𝑛
[30,40]
(Groves and Whalley 2002)
39.6
𝜔𝑒
[20,30]
27.1
Biomass turnover rates [/yr]
𝜇𝑛
[0.0,0.2]
Tuned with 𝑚𝑖 to give realistic turnover rates
0.14
𝜇𝑒
[0.0,1.0]
(Morgan 1998; Morgan and Lunt 1999).
0.81
N uptake rates [ha/kg C/yr]
𝜈𝑛
[0.02,0.5]
(Menge et al. 2009)
0.0211
𝜈𝑒
[0.02,0.5]
0.0877
Maximum increase in biomass turnover rates
𝑚𝑛
[0.0,0.5]
See 𝜇𝑖 .
0.12
due to light competition [/yr]
𝑚𝑒
[0.0,0.5]
0.32
Light competition coefficients [.]
𝛼𝑒𝑛
[0.0,100.0]
Exotic constrained to be superior light
0.05
𝛼𝑛𝑒
[0.0,100.0]
competitor (𝛼𝑛𝑒 > 𝛼𝑒𝑛 ) (Garden and Bolger
19.00
2001).
Table A2. Details of parameterization of the grasslands model in the right panels of Fig. 1
(note that only the value of 𝑚𝑛 differs from Table A1; these values are also the basis of Figs.
2–4) and in Figs. A1–A2. Sources and notes are as for Table A1.
State variable or parameter
Range
Value
Value
Value
Figs. 1,
Fig. A1 Fig. A2
2–4
̅
Native biomass [kg C/ha]
Varies
Varies
Varies
𝐵𝑛 [7500, 15000]
Exotic annual biomass [kg C/ha]
Varies
Varies
Varies
𝐵̅𝑒 [1000, 7500]
Plant-available N [kg N/ha]
Varies
Varies
Varies
𝐴̅ Not
constrained
N input rate [kg N/ha/yr]
𝐼
[0,15]
Varies
Varies
Varies
Half-saturation constant in
ℎ [500,10000]
3890
5060
7200
biomass/cover functions [kg
C/ha]
Litterfall N-loss coefficients [.]
𝛿𝑛 [0.003,0.7]
0.06
0.56
0.01
𝛿𝑒 [0.003,0.7]
0.06
0.52
0.01
Plant-available N soil leaching rate 𝑘 [0.0,1.0]
0.4
0.035
0.4
[/yr]
Exponent in biomass/cover
𝑝 {1,2,3}
3
2
1
functions [.]
N use efficiencies [kg C/kg N]
𝜔𝑛 [30,40]
39.6
30.7
35.2
𝜔𝑒 [20,30]
27.1
24.6
24.8
Biomass turnover rates [/yr]
𝜇𝑛 [0.0,0.2]
0.14
0.023
0.14
𝜇𝑒 [0.0,1.0]
0.81
0.045
0.88
N uptake rates [ha/kg C/yr]
𝜈𝑛 [0.02,0.5]
0.0211
0.21
0.0875
𝜈𝑒 [0.02,0.5]
0.0877
0.32
0.488
Maximum increase in biomass
𝑚𝑛 [0.0,0.5]
0.17
0.098
0.15
turnover rates due to light
𝑚𝑒 [0.0,0.5]
0.32
0.197
0.11
competition [/yr]
Light competition coefficients [.]
𝛼𝑒𝑛 [0.0,100.0]
0.05
0.096
0.60
𝛼𝑛𝑒 [0.0,100.0]
19.00
0.750
1.30
Fig. A1 As for Fig. 1, but parameterized so that bistability does not occur, all transitions are
gradual and the exotic species can never drive the native species extinct (this
parameterization corresponds to Case C in the main text). The vertical dotted grey line on
each panel represents 𝐼 = 𝐼𝑐𝑟𝑖𝑡,𝑛 . See Table A2 for parameter values.
Fig. A2 As for Fig. 1, but parameterized so that bistability does not occur, all transitions are
gradual and the exotic species takes over at high N inputs (this parameterization
corresponds to Case D2 in the main text). The vertical dotted grey lines represent, from left
to right, 𝐼 = 𝐼𝑐𝑟𝑖𝑡,𝑛 and 𝐼 = 𝐼𝑐𝑟𝑖𝑡,𝑒 . See Table A2 for parameter values.
Fig. A3 Conceptual model of dynamics in south-eastern Australian temperate grasslands
based on empirical understanding of the system. Transitions are shown between native
Themeda-dominated states and exotic-dominated states caused by changes to nutrient
inputs and the disturbance regime (in state name abbreviations, n = native-dominated; e =
exotic-dominated; L, M, and H = low, intermediate and high nutrient levels). Nativedominated states (nL, nM) are maintained by moderate or irregular biomass removal
through fire or grazing (represented by black symbols). Exotic-dominated states on the left
of the diagram (eL’, eM’) occur after ≈3 yr heavy continuous grazing by pastoral animals
(again represented by black symbols), which depletes the underground reserves of
Themeda and thus its ability to regenerate. Exotic-dominated states on the right of the
diagram (eL’’, eM’’) occur after 7–15 yr of no disturbance, which causes Themeda tussocks
eventually to senesce and release nutrients. The exotic-dominated state at bottom-center
(eM) is the hysteretic state that can persist under the same external conditions as state nM
and is induced by temporary disturbances: very high nutrient addition (via state eH; see Fig.
4a), changes to the disturbance regime, or a combination of both.
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