1
Supporting Information Notes S1, Figures S1-S3
2
3
Notes S1
4
1. Model description
5
The climate-vegetation-natural fire (CVNF) model is composed of a simple set of
6
modified Lotka-Volterra competition equations:



dg
g
a 
  g g 1  *  c ga *   
g  d g firg g ,


dt  
g
a  IV 


I 

V
III
II


(1)



da
a 
  a a1  *   d a fira a,


dt 
a  

I 
V
II 

(2)
7
to represent the time evolution of the fractional coverage area of grasses (g) and trees
8
(a).
9
Basically, two aspects differ from the classical set of Lotka-Volterra competition
10
equations: 1) plant growth parameters (a, g) depend on environmental conditions
11
instead of being defined as constants (Nordstrom et al., 2005); 2) additional sink terms
12
due to lightning-triggered fire effects on vegetation (dgfirgg, dafiraa) are included.
13
The first term (I) is a source term which is driven by the plant growth parameters
14
defined as:
i 
fi
i
, i  g , a,
(3)
15
where i is the time-scale of establishment of each vegetation type, and fi is a climate-
16
dependent function. i is a constant obtained from an equation proposed by Hughes et
17
al. (2006) (from the TRIFFID model equations (Cox, 2001)) to determine the time scale
18
for vegetation establishment considering fractional coverage areas and not only
19
individual elements of a vegetation population. The expression to calculate
20
described as:
 V f1  1 
 i  i ln  1 ,
 V 1
 s

is
(4)
21
where Ti is the time of establishment of one element of the population i; Vf and Vs are
22
the final and initial fractional coverage of population i, respectively. Typical growth
23
length for individual tropical forest trees is about 50 years to reach a mature phase, i.e.,
24
Ta = 50 years (Chambers et al., 1998). On the other hand, grasses take a much shorter
25
time to completely develop themselves from seed to adult phases. Although field
26
measurements are not conclusive about seed-to-adult growth length of grasses within
27
South American tropical savannas, only the aerial portions of typical herbaceous species
28
of Brazilian savannas take about 2 years to recover after a fire event (Batmanian &
29
Haridasan, 1985). Thus, we assume that the complete seed-to-adult cycle would take at
30
least doubled growth time (Martins et al., 1999), and set Tg ~ 4 years. Then, using
31
equation (4), tree and grass populations would take about 340 and 27 years (a = 340
32
and g = 27) respectively to grow from Vs = 0.01 (almost null) to Vf = 0.9. We do not
33
include different periods for vegetation recovery after environmental disturbances,
34
especially fires. This would be more realistic since that grass vegetation would take a
35
shorter time to regrow as only its aerial portions would be damaged (root chains would
36
be intact). Besides, evergreen trees considered in the model are not as resilient as
37
grasses, and could either be partially damaged and followed by secondary-forest
38
succession, or completely die off and need the same seed-to-adult cycle period to
39
regrow.
40
Functions
41
2005):
are modeled as simple piecewise linear functions (based on Arora & Boer,

 HI  HI l ,i
f i  max min 

 HI u ,i  HI l ,i
 
,1,0
 
(5)
42
where HI is a humidity index and HIu,i (HIl,i) is the upper (lower) bound above (below)
43
which fi is equal to 1 (zero). The humidity index (Hulme et al., 1992), HI, is derived
44
from the aridity or dryness index, AI (Budyko, 1974), and is defined as a combination of
45
monthly precipitation (P) and temperature (T) values by the equation:
HI 
1
P

AI PET
(6)
46
where PET = PET(T) is the potential evapotranspiration given by Thornthwaite’s
47
equation (Thornthwaite, 1948). HI is computed on a monthly basis from precipitation
48
and temperature datasets. Therefore, plant growth parameters (i, i = g, a) are
49
larger/smaller under wetter/drier humidity conditions.
50
Grasses (g) and trees (a) grow depending on different values of HIl,i and HIu,i. These
51
values are defined based on the broad-sense classification of climate regimes suggested
52
by Ponce et al. (2000). For the purposes of this work, we focus on the humid climate
53
delimitation 0.375 ≤ AI < 0.75, which we assume to be related to the development of
54
tropical forest vegetation. A simple algebra turns the above expression into 1.3 ≤ HI <
55
2.6. Therefore, using rounded values, we define HIl,a = 1.0 as the starting point for tree
56
growth, and HIu,a = 3.0 the value from which climate conditions are not yet limiting
57
factors for tree development (Fig. S1a). On the other hand, thresholds for grasses are
58
empirically set based on tree values. As they need much less favorable soil moisture
59
conditions to develop in comparison to trees, they start growing from HIl,g = 0.1, and are
60
no longer controlled by HI when HIu,g ≥ 1.1, when grasses grow freely only restricted by
61
competition factors (represented by the competition coefficient, cga, of equation (1),
62
which will be discussed later) (Fig. S1a).
63
Synthetic time series of monthly precipitation and temperature are used. Precipitation
64
values (P) are generated (based on D’Andrea et al., 2006; Dolman et al., unpublished)
65
using the following equation:

 2t
 
P, T (t )  max   P ,T     P ,T  AP ,T cos
  P ,T ,0.
 
 

(7)
66
The first term on the RHS of equation (7), P,T, represents long-term mean annual
67
precipitation/temperature; the second term, P,T, represents a prescribed random
68
variation from the climatological values and is composed by a random number () with
69
a normal standard distribution (~N(0,1)) and the standard deviation of the mean annual
70
value (P,T); the third term represents the annual cycle (), is the period of 12 months;
71
AP,T is the amplitude; and P,T is the phase), in which precipitation seasonality is
72
embedded and thus the variation of the dry season length. AP,T = P,T – min(P,T) is
73
defined using the minimum value from monthly precipitation/temperature climatology;
74
and P,T = 2 m(P,T),max/  takes into account month m when the maximum value from
75
monthly precipitation/temperature climatology occurs. Precipitation climatology is
76
evaluated by the Climate Research Unit’s (CRU) gridded (0.5o x 0.5o) precipitation time
77
series (1901-2002) (New et al., 1999; Mitchell et al., 2003). Fig. S2a shows the annual
78
cycle of the generated precipitation in the middle of the simulation period (around the
79
500th year). Precipitation gradients in the wet season and longer dry seasons are clearly
80
seen comparing precipitation distribution for closed forest (thicker lines) and savannas
81
(thinner lines). The precipitation regime in the 66W longitude is somewhat different
82
from the other three locations with the dry season taking place earlier in the year.
83
Temperature values (T) are generated analogously. For instance, in the middle of the
84
same simulation period (around the 500th year), temperature decreases reach up to 2oC
85
from 66W to 46W and there is a slight change in the annual cycle of the temperature
86
within the savanna location (44W) (Fig. S2b).
87
The formulation described above may be suitable to simulate future climate scenarios
88
by modifying the mean annual precipitation/temperature (P,T), the inter-annual
89
variability (P,T), or even the amplitude or the peaks of the annual cycle (AP,T or P,T).
90
Besides birth representation, sink terms II and III of intra-specific (quadratic terms,
91
(ggg) / g* or (aaa) / a*) and inter-specific (crossed terms, (gcgaga) / a*)
92
competitions are also driven by the plant growth parameter (i, i = g, a) (equations (1)
93
and (2)). Constants g* and a* correspond respectively to the carrying capacity of grasses
94
and trees, i.e., the maximum value that the vegetation population may reach. As usual,
95
both constant values are set to 1 for general purposes. The crossed term should represent
96
ecological advantages from one vegetation type over another depending on magnitude
97
cga. This parameter in term III represents in equation (1) the negative shading effect that
98
favors tree growth and development instead of grasses within an area (Cox, 2001).
99
Because there are no general advantages of grasses over trees, inter-specific competition
100
(term III) is negligible (cga ~ 0) in equation (2).
101
Term IV in equation (1) is the natural grass mortality, and is assumed to occur at a
102
constant rate  = 0.1 (Hughes et al., 2006). On the other hand, according to 14C isotope
103
measurements, trees in central portions of the Amazon forest may be very old, with ages
104
ranging from 200 to 1,400 years depending on tree species (Chambers et al., 1998).
105
Assume, for simplicity, an average age of 800 years. As it is relatively close to the total
106
simulation period, natural tree mortality is not computed in the CVNF model.
107
As fire mortality is computed in a separate term (term V, see below), only climate could
108
disturb tree coverage and promote death of younger trees in our simplified scheme. Tree
109
type, however, is associated to a generic evergreen-forest species, and thus it is very
110
resistant to climate seasonality due to its deep root zone (Nepstad et al., 2004). Note
111
that because natural tree death is not computed in this model, this term does not
112
influence tree population dynamics (term IV is null in equation (2)).
113
The vegetation sink term due to fire occurrence depends on the maximum population
114
death resulting from fire, di, and on fire effects firi ,i = g, a. This means that even with
115
extreme fire intensities, only a maximum fraction di of population i shall die. Constant
116
parameter di is set as 0.3 for grasses (Gardner, 2006) and 0.1 for trees as proposed by
117
Balch et al. (2008) for the woody biomass burnt within a forest-savanna transition in
118
southeastern Amazon.
119
Fire effect (firi ,i = g, a) is an output of a simplified fire sub-model divided into 3 stages:
120
triggering, intensity and effects. It is far beyond the purpose of this model to simulate
121
fire propagation, displacement and other ecological features and impacts of fire. Instead,
122
we looked for a simplified representation of the main characteristics of fire ecology.
123
124
1.1.
The fire sub-model
125
This section presents the main contributions of this work, concerning the
126
parameterization of lightning-triggered fires and respective effects on grass and tree
127
vegetation that will affect the dynamics of the system (1)-(2).
128
Natural fires depend on three major conditions to start: fuel availability and
129
flammability, and the ignition source (Whelan, 1995). In this context, the first stage is
130
associated to favorable conditions of fire triggering taking into account the three aspects
131
mentioned above. Ecologically, fuel availability is mostly due to the herbaceous layer
132
(Bond, 2008). Therefore, fuel is accumulated when grasses die (at constant rate  = 0.1
133
from equation (1)) and form litter pools (L). Whenever L reaches the threshold value,
134
Lmin = 0.45, the availability condition is satisfied and fire may start. Flammability is
135
defined based on a conceptual assumption that litter pools may be split into 2 vertical
136
layers. This two-layered litter model in which the upper (bottom) layer is highly
137
influenced by air humidity (soil moisture) is inspired by controlled fire field
138
experiments in the Amazon (Balch et al., 2008). In order to start a fire, only the top
139
layer needs to be sufficiently dry; therefore, the flammability threshold is set according
140
to the humidity index (HI) of equation (6). Whenever HI ≤ HImax = 1.2, fire may start.
141
Both optimum thresholds (Lmin, HImax) are set empirically through computational
142
experiments with the CVFN model.
143
Ignition is caused by lightning flashes. To represent the occurrence of these flashes, we
144
propose the monthly lightning (R) time series generation, similarly to the procedure
145
outlined for precipitation and temperature in equation (7):

 2t
 
R(t )  0.25  max   R     R  AR cos
  R ,0.
 
 

(8)
146
Therefore, a random component is explicitly incorporated in the lightning time series
147
generation to represent the highly stochastic nature of lightning strikes. Statistics (R,
148
R, mR,max, AR, R) used for the time series generation is extracted from a combined 7-
149
year (1995-2002) dataset of two lightning-detection satellite sensors (LIS/OTD) from
150
the Global Hydrology Resource Center - NASA (http://thunder.nsstc.nasa.gov).
151
This dataset includes the entire lightning activity over the tropics, in either cloud-to-
152
cloud or cloud-to-ground flashes. As only cloud-to-ground flashes can initiate fires, we
153
use a global value of 25% (0.25) to represent the lightning activity of interest (Pride &
154
Rind, 1993). Fig. S2c shows the annual cycle of the lightning activity generated by this
155
methodology. It is remarkable that there is more lightning activity over closed forests
156
than over savannas for most part of the year. However, forests do not frequently burn
157
because even if fire is triggered, the high soil moisture limits its intensity (see equation
158
(10)).
159
Moreover, not all cloud-to-ground flashes initiate a fire. There are few estimates of the
160
relation between flashes and fire initiation in boreal forests (for instance, see Larjavaara
161
et al., 2005, for Finland). An estimate for southeast Brazil is proposed based on
162
observational data from the Brazilian Lightning Detection Network – BrasilDat (Pinto
163
Jr. et al., 2006). As the negative cloud-to-ground flashes with long continuing currents
164
(Saba et al., 2006) cause the most serious damage associated to thermal effects, i.e. fires
165
among others, Correia & Saba (2008) measure the presence of that current type in
166
several thunderstorms. Authors suggest that about 28% of the negative flashes have
167
long continuing currents and may promote fire episodes. In addition, this estimate can
168
be extended to neighboring areas of Brazil without loss of generality. In general, 90% of
169
the total could-to-ground lightning flashes are negative (Saba et al., 2006). The
170
combination of both percentages (28% and 90%) results in an estimate of 25%.
171
Based on Arora & Boer (2005), we suggest a probability function of fire (
172
the equation (5), expressed as:

 R  Rl  
PR  max min 
,1,0 ,
R

R
u
l

 

), similar to
(9)
173
depending on the number of monthly flashes (R) and empirical lower and upper bounds
174
defined as Rl = 25 and Ru = 65. Although empirically defined, these bound values are
175
inspired on the assessment of 25% described above (Correia & Saba, 2008). Let Rf be
176
the number of lightning flashes that could cause fire ignition, such that Rf = 0.25R.
177
Applying this relation to the Rl and Ru values results in approximately Rf = 6 and Rf =
178
16. Thus, in a very simplified scheme, probability PR is assumed to be practically null
179
for less than 6 flashes, as this would mean 1 flash that could ignite a fire for every 5
180
days. Then, PR is assumed to enhance linearly as Rf (R) increases up to 16 (65), above
181
which PR = 1 (Fig. S1b). Moreover, R ≥ 65 or Rf = 16 means that at least 1 flash for
182
every 2 days in the month may promote fire ignition, which is quite a high prospect.
183
Thus, PR for this case is assumed to be maximum.
184
Ignition is likely to occur if PR ≥ PR,min = 0.55 (Fig. S1b). We take the 46W longitude
185
point in the middle of the simulation period (after approximately 500 years). At this
186
specific location, Fig. S1d shows an example of the exact moment when the three
187
conditions are met in the model. Fire occurs in the transition between the dry and the
188
wet seasons in November, when there is enough dried litter (L ≥ 0.45 and HI ≤ 1.2) on
189
the ground, and the probability of fire ignition is above the threshold value (0.55).
190
After fire starts, fire effects depend on fire intensity and vegetation types. Thus, the
191
second stage is to define a fire-intensity function. In this study, the latter varies
192
according to the amount (L) and the dryness of the litter available to burn, which is
193
associated to a soil moisture index (w). It is defined as a function of the annual mean
194
precipitation value, P in mm month-1:

 w  P  
w  max min  u
,1,0,
w

w
u
l

 

(10)
195
where wl = 110 mm month-1 and wu = 200 mm month-1 are the empirical lower and
196
upper bounds of precipitation values. These constants limit the point below which w is
197
not a limiting factor, and above which w is null (totally limiting), respectively (Fig.
198
S1c).
199
The fire intensity dependence of litter is modeled after Anderies et al. (2002) according
200
to:
LbL
h( L, k L , bL )  bL
,
k L  LbL
(11)
201
where constants kL and bL are empirically set to 0.7 and 2.0, respectively, and are chosen
202
to represent the increase in fire intensity due to the higher amount of litter (Fig. S3).
203
Fire intensity is thus defined as the combination of two limiting factors:
I  I (w, L)  w  h( L, k L , bL ) .
(12)
204
As fire effects are different on grasses and trees, we make assumptions to represent
205
these differences (see Fig. S3). Grass vegetation cover burns as soon as fire starts and
206
regrows faster than trees (equation (3)). Trees, on the other hand, are more resistant to
207
low fire intensities; however, whenever fire intensity is strong enough, trees burn
208
significantly, and need more time to recover.
209
The formulation to represent this relation is similar to equation (11), except that it does
210
not depend on litter amount, but on fire intensity I (Anderies et al., 2002):
I bi
firi  h( I , k i , bi )  bi
, i  g, a ;
k i  I bi
211
212
with kg =0.3, bg = 2.0, ka = 0.7, ba = 8.0 (Fig. S3).
(13)
213
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302
Figs S1–S3
303
304
Fig. S1 – (a) fi, i = g, a as a function of humidity index; continuous (dotted-continuous)
305
line refers to the plant growth function of grasses (trees); (b) fire probability (PR) as a
306
function of lightning number and the respective threshold (0.55); (c) soil moisture index
307
(w) as a function of annual precipitation; (d) illustration of a fire event in longitude 46W
308
after about 500 years of simulation – continuous, dashed and dotted lines refer to the 3
309
fire limiting factors: the humidity index (HI), the probability of fire ignition (PR) and the
310
available litter (L), respectively, with HImin ≤ 1.2, PR,min ≥ 0.55 and Lmin ≥ 0.45.
311
312
Fig. S2 – Annual cycle of the generated (a) precipitation, (b) temperature and (c)
313
lightning flashes for four locations: closed forest with two different annual precipitation
314
regimes (65W and 58W), forest-savanna transition (46W) and savanna core (44W).
315
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317
Fig. S3 – Variations of function h used in the model, depending on parameters
and b.
318
319
Fig. S1
(a)
(b)
(c)
(d)
320
321
322
Fig. S2
(a)
(b)
(c)
323
324
325
326
Fig. S3