a second-order impact model for forest fire regimes
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Theoretical Population Biology 70 (2006) 174–182
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A second-order impact model for forest fire regimes
Stefano Maggia, Sergio Rinaldia,b,�
aDipartimento di Elettronica e Informazione, Politecnico di Milano, Milano, ItalybAdaptive Dynamics Network, International Institute for Applied Systems Analysis, 2361 Laxenburg, Austria
Received 29 March 2005
Available online 24 May 2006
Abstract
We present a very simple ‘‘impact’’ model for the description of forest fires and show that it can mimic the known characteristics of
wild fire regimes in savannas, boreal forests, and Mediterranean forests. Moreover, the distribution of burned biomasses in model
generated fires resemble those of burned areas in numerous large forests around the world. The model has also the merits of being the
first second-order model for forest fires and the first example of the use of impact models in the study of ecosystems.
r 2006 Elsevier Inc. All rights reserved.
Keywords: Forest fires; Savannas; Boreal forests; Mediterranean forests; Impact models; Chaos
1. Introduction
In savannas, as well as in boreal and in Mediterraneanforests, wildfires are recurrent, but with remarkablydifferent characteristics. Fires in savannas are almostperiodic surface fires with return times ranging from 1 to2 yr in moist areas (Goldammer, 1983) to 5–10 yr in aridareas (Rutherford, 1981). Fires in northern boreal forestsare also quite regular, but they prevalently involve crowns(Kasischke et al., 1995) and occur every 50–200 yr (Roweand Scotter, 1973; Zackrisson, 1977; Engelmark, 1984;Payette, 1989). By contrast, in Mediterranean areas, mixed(crown and surface) fires are almost the rule and occur inan apparently random fashion, with highly variable returntimes (Kruger, 1983; Davis and Burrows, 1994).
While it is true that natural forest fires originate fromrandom events (mostly lightening) and are influenced bymeteorological conditions (Bessie and Johnson, 1995), it isalso true that fires can develop only if there is enough drymatter on the ground and if plants are sufficientlyabundant in at least one of the various vegetational layers
e front matter r 2006 Elsevier Inc. All rights reserved.
b.2006.01.007
ing author. Dipartimento di Elettronica e Informazione,
ilano, Via Ponzio 34/5, 20133, Milano, Italy.
99 3412.
esses: [email protected] (S. Maggi),
limi.it (S. Rinaldi).
of the forest (for a relatively detailed discussion of this issuesee Casagrandi and Rinaldi, 1999 and references therein).This suggests the idea that long-term predictions of forestfires can be roughly performed with deterministic modelsdescribing the growth processes, while more precise short-term predictions can only be performed through stochasticmodels (conceptually comparable with those used inweather forecast).Here, we propose a simple deterministic model for the
long-term prediction of forest fires in which the vegeta-tional growth is described by standard ordinary differentialequations, while fire episodes are modeled as instantaneousevents. The fire develops when there is enough fuel on theground, and, under suitable assumptions, this occurs whenthe mix of biomasses of the various layers reaches pre-specified values. The consequence of a fire is therefore aninstantaneous reduction of the biomasses which is heur-istically described by a simple rule in state space. Modelswith discontinuities of this kind are called ‘‘impact models’’and have been first used in mechanics (see Brogliato, 1999and references therein) to describe the dynamics ofmechanical systems characterized by impacts amongvarious masses. They are quite special and can be used toexplain a number of rather subtle phenomena like the‘‘Zeno chattering’’ (e.g., the diminishing return times of theimpacts of a ping-pong bouncing ball) that other modelscannot explain. Impact models represent the most naıve
ARTICLE IN PRESSS. Maggi, S. Rinaldi / Theoretical Population Biology 70 (2006) 174–182 175
approach for the description of systems characterized bydynamic phenomena occurring at very diversified timescales (in our case the slow building up of biomass and itsfast destruction through fire). They should not be confusedwith an apparently similar but substantially different classof models, namely that of periodically pulsed systemswhere the discontinuity in state space is generated by aperiodic exogenous shock on the system. Many are theexamples of this second class of models in biology: thecontrol of continuously stirred-tank reactors (Funasakiand Kot, 1993), the study of pulsed chemotherapy(Lakmeche and Arino, 2001) and vaccination (Shulgin etal., 1998) and a number of contributions dealing with theeffects of periodic harvesting or immigration (Ives et al.,2000; Liu and Chen, 2003; Chau, 2000; Grant et al., 1997;Geritz and Kisdi, 2004; Reluga, 2004). However, in thosemodels the return time of the discontinuous event isconstant and a priori fixed, while in our impact model thefire return times are neither constant nor pre-specified butare endogenously created by the interactions among thevarious layers of the forest.
The impact model we propose in this paper has only twodifferential equations (i.e. it is a so-called second-ordermodel), one for the lower and one for the upper layer of theforest. We, therefore, exclude from our study single-layeredshrublands. The simplest deterministic model availableuntil now for the study of the dynamics of the fire returntimes was a standard (i.e. nonimpact) fourth-order model(Casagrandi and Rinaldi, 1999) in which the four statevariables are the burning and nonburning biomasses of thelower and upper layers of the forest. It is important to keepin mind that the impact model we propose in this papershould not be intended as an approximation to thatfourth-order model. However, it can mimic the qualitativefeatures of the periodic fire regimes of savannas andboreal forest, as well as the chaotic fire regimes ofMediterranean forests suggested by the fourth-ordermodel. As far as we know, this is also the first time thatthe general idea behind impact models is applied inecology, although forestry and agricultural practicescorrespond very closely to the same idea: harvest whenthe population reaches a specified state. For this reason, itwould be surprising if related models have not been appliedin that context. In any case, there are certainly many otherpotential applications, since population dynamics are veryoften the result of slow dynamical processes interrupted byshort devastating events.
2. The model
A continuous-time impact model is described by a set ofn ordinary differential equations
_xðtÞ ¼ f ðxðtÞÞ, (1)
which hold at any point in state space except on a (n� 1)dimensional manifold X�, where the impact occurs. Whenthe state x reaches the manifold X� at point x�, an
instantaneous transition described by a map
xþ ¼ jðx�Þ x� 2 X� (2)
occurs. The set
Xþ ¼ jðX�Þ
is the set of the states of the system immediately after theimpact. For this reason, the sets X� and Xþ are called, ingeneral, pre- and post-impact manifolds. In the specificapplication considered in this paper, they simply representthe pre- and post-fire conditions of the forest and aretherefore called pre- and post-fire manifolds. Obviously,first-order impact models are of no interest because if n ¼ 1the manifolds X� and Xþ are just two points and theirmost complex behavior is just a cycle passing through X�
and Xþ. This is why impact models are usually presentedfor nX2.The model we propose is a crude simplification of the
real world. Species diversity, age structure, spatial hetero-geneity, and plant physiology are not taken into accountsince we look only at total biomasses (see Shugart, 1984,Chapter 6 for a discussion). However, in order todistinguish fires in different layers, we assume that theforest is composed of two layers: a lower vegetational layer(from now on called ‘‘bush’’) that, depending on the forest,is composed of bryophytes, herbs, shrubs, or any mix ofthese plants, and an upper vegetational layer (from now oncalled ‘‘tree’’), in general composed of plants of variousspecies. The corresponding biomasses are denoted by B
(bush) and T (tree). The equations of growth (1)characterizing our model are
_B ¼ rBB 1�B
KB
� �� aBT ,
_T ¼ rT T 1�T
KT
� �. ð3Þ
This means that, in the absence of fire, trees growlogistically toward the carrying capacity KT , while plantsof the lower layer do not tend toward their carryingcapacity KB because tree canopy reduces light availability.A detailed discussion of the validity and limitations of Eqs.(3) can be found in Casagrandi and Rinaldi (1999), whererealistic values for the five vegetational parametersðrB; rT ;KB;KT ; aÞ are also suggested.As for the fire, we know (see, for example, Viegas, 1998)
that the ignition phase is possible if there is enough deadbiomass on the ground (leaves, twigs, branches, moss,herbs, etc.). Since the biochemical processes regulating themineralization of dead biomass are relatively fast withrespect to plant growth (Esser et al., 1982; Seastedt, 1988),we can reasonably assume, on the time-scale at which wedescribe bush and tree growth, that the rate of mineraliza-tion (proportional to the amount of dead biomass) equalsthe inflow rate of new necromass into the ground layer(proportional to bush and tree biomass). As a result, wecan consider B and T as appropriate indicators of theabundance of fuel on the ground. Of course, also the water
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�T S +
C +
S −S −
C−C −X −
X +X −
X +
T /
KT
T /
KT
1 1
�T
�B 10
0 100
2−
1−
0
B / KB
�B
λT�T
�T
�T
λT�T
λB�B�B
B / KB
�BλB�B
S+1+
2+
C+
(a) (b)
Fig. 1. Model behavior. (a) The pre- and post-fire manifoldsX� andXþ; the dotted lines with double arrows are the instantaneous transitions fromX� to
Xþ due to a fire (see Eq. (2)); horizontal (vertical) lines correspond to surface (crown) fires in which trees (bushes) are not involved; oblique lines starting
from the segment C�S� of X� correspond to mixed fires; (b) state portrait of the model; continuous lines with a single arrow represent the growing phase
of the forest and are described by Eq. (3).
S. Maggi, S. Rinaldi / Theoretical Population Biology 70 (2006) 174–182176
content of the fuel particles matters, and, indeed, modelsfor short-term fire prediction often include a number ofrelevant factors related with soil moisture. However, suchfactors vary at high frequency and can therefore beneglected if the target is long-term fire prediction (in otherwords, if we like to predict within how many years (ordecades) we are going to experience a new fire we canobviously forget high-frequency phenomena like weeklyweather variability).
After this premise, we can specify (see Fig. 1(a)) the pre-and post-fire manifolds X� and Xþ and the map (2)interpreting the impact of the fire.
Let us first focus on the pre-fire conditions by noticingthat the function TðBÞ identifying the manifold X� ispiecewise linear and nonincreasing and that the set belowthe manifold X� is convex. The first property is obviousbecause less fuel originated from trees (i.e. less trees) isnecessary for fire ignition if more fuel originated frombushes is available on the ground.
The second property simply says that if x0 ¼ ðB0;T 0Þ andx00 ¼ ðB00;T 00Þ are two states of the forest at which fireignition is not possible (i.e. two points below the manifoldX�Þ no mix of these two states (i.e. no points of thesegment connecting x0 with x00) can give rise to fire ignition.A formal support to these two properties, which are hereassumed to hold, is available in the Appendix.
Intuition suggests that X� should be a smooth manifoldwhile our choice (see Fig. 1) has been in favor of a lessrealistic but simpler piecewise linear manifold X�. Thereason for this choice is that our manifoldX� allows one tosharply identify surface fires (vertical segment of X�),crown fires (horizontal segment of X�) and mixed fires(central segment of X�). By definition, surface fires do notinvolve the upper layer, so that the post-fire conditions areon the vertical segment characterized by Bþ ¼ lBrBKB ¼
lBB�. In other words, rB is the proportion of the lower
layer carrying capacity KB at which surface fires occur andlB is the proportion of the lower layer biomass thatsurvives to surface fires. Similarly, fires in the upper layerare characterized by a vertical instantaneous transitionfrom T� ¼ rT KT to Tþ ¼ lT T�. The most extremesurface fire is represented by the transition S� ! Sþ,while the most extreme crown fire is represented by thetransition C� ! Cþ. The assumption that mixed firesinitiate on the segment C�S� implies, by continuity, thatpost-fire conditions are on a curve connecting points Cþ
and Sþ. Of course, mixed fires initiating close to point C�
should end up close to point Cþ. Since we have beenunable to find simple suggestions on the shape of themanifold Xþ from point Cþ to point Sþ, we have assumedfor simplicity that mixed-fires terminate on the segmentCþSþ and preserve the relative distances Z and ð1� ZÞfrom the two extreme points. In formulas, the extremepoints of the pre-fire segment C�S� are
C� ¼ ðsBKB;rT KT Þ; S� ¼ ðrBKB; sT KT Þ
and each mixed fire starts from a point ðB�;T�Þ belongingto the segment C�S�, i.e.
B� ¼ ZsBKB þ ð1� ZÞrBKB,
T� ¼ ZrT KT þ ð1� ZÞsT KT , ð4Þ
where 0pZp1 (Z ¼ 0 and 1 correspond to points S� andC�, respectively). The extreme points of the post-firesegment CþSþ are
Cþ ¼ ðsBKB; lTrT KT Þ; Sþ ¼ ðlBrBKB;sT KT Þ
and the post-fire conditions are
Bþ ¼ ZsBKB þ ð1� ZÞlBrBKB,
Tþ ¼ ZlTrT KT þ ð1� ZÞsT KT . ð5Þ
Thus, the map xþ ¼ jðx�Þ for mixed fires is nothing butthe transformation of point ðB�;T�Þ into point ðBþ;TþÞ.
ARTICLE IN PRESSS. Maggi, S. Rinaldi / Theoretical Population Biology 70 (2006) 174–182 177
This completes the description of the model. Notice thatFig. 1 has been drawn for the case sT4lTrR and sB4lBrB
but that Eqs. (4), (5) hold for all possible cases.The sequence of the fires can be easily obtained from the
model, as shown in Fig. 1(b). Starting from a given initialcondition, say point 0 in Fig. 1(b), one integrates thedifferential equations (3) until the solution hits the pre-firemanifold X� at point ðB�;T�Þ (see point 1� in Fig. 1(b)).From any one of the two equations (4) one can derive thevalue of Z associated with point 1� and then use Eqs. (5)for computing point 1þ. Then, the procedure is iteratedand a series of fires 2� ! 2þ; 3� ! 3þ; . . . is obtained.
As pointed out in Fig. 1(b), the trajectory of the system isthe concatenation of slow transitions (continuous lines)corresponding to growing phases, and fast (actuallyinstantaneous) transitions (dotted lines) corresponding tofires. It is worth noticing that fast transitions can intersectwith slow and fast transitions. This is why the model can bechaotic even if it is only a second-order model.
3. Results
In this section we show that our second-order model canmimic, for suitable values of its parameters, the character-istic fire regimes of savannas, boreal forests, and Medi-terranean forests. For each kind of forest we present theresult of a typical simulation and compare it with the resultobtained with the more complex fourth-order model(Casagrandi and Rinaldi, 1999). Simulations were per-formed with fourth-order Runge–Kutta–Fehlberg methodwith fifth-order error estimate. Impacts of the trajectorywith the pre-fire manifold X� were detected with an
0 10 20 30 40 50 60 70 800
0.1
0.2
0.3
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0.5
0 10 20 30 40 50 60 70 800
0.1
0.2
0.3
0.4
0.5
biom
ass
[10K
g/m
2 ]bi
omas
s [1
0Kg/
m2 ]
time [y]
time [y]
T
B
T
B
(a)
(c)
Fig. 2. Examples of fire regimes in savannas. First row: time series (a) and st
rB ¼ 1:5, rT ¼ 1, KB ¼ 0:4, KT ¼ 0:5, a ¼ 1:5, rB ¼ 0:49, rT ¼ 4, sB ¼ 0:1, sT
(d) obtained with the fourth-order model (see Casagrandi and Rinaldi, 1999 f
adaptive step integration procedure with absolute toleranceof 10�9. Then, we also show that the distributions of theburned biomasses in model generated fires have structuralproperties quite similar to those emerging from statisticalanalysis of the areas burned by fires in numerous largeforests around the world.
3.1. Savannas
A typical series of fires in savannas obtained throughsimulation of the second-order model is shown in Fig. 2(a).The fires occur every 8 yr, in good agreement withRutherford (1981). The cycle, shown in Fig. 2(b), is setup by the lower layer, and in fact the fire is essentially asurface fire devastating the herbs (the post-fire bushbiomass is only 10% of the pre-fire bush biomass), whiletree biomass remains almost constant, as observed byHopkins (1965).The second row of Fig. 2 shows very similar results
obtained with the fourth-order model.
3.2. Boreal forests
A typical fire regime of a boreal forest obtained with ourimpact model is shown in Fig. 3(a) and (b). In agreementwith many data and studies on boreal forests at highlatitudes, the fires are essentially crown fires and occurevery 100 yr (Yarie, 1981). After a fire, the biomass of thelower layer increases for 20–30 yr (while conifers grow veryslowly) and then decreases and reaches the pre-fire level(manifold X� in Fig. 3(b)) within 60–100 yr as predicted byViereck (1983).
90 100 0 0.05 0.1 0.15 0.2
0.49
0.492
0.494
0.496
0.498
0.5
0 0.05 0.1 0.15 0.2
0.49
0.492
0.494
0.496
0.498
0.5
90 100B
B
x−
TT
(b)
(d)
ate portrait (b) obtained with the second-order model (parameter values:
¼ 0:985, lB ¼ 0:1, lT ¼ 0:2). Second row: time series (c) and state portrait
or parameter values).
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0.5
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0.5
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x−
time [y]
T
B
T
time [y]
T
B
T
B
B
biom
ass
[10K
g/m
2 ]bi
omas
s [1
0Kg/
m2 ]
(a) (b)
(d)(c)
Fig. 3. Examples of fire regimes in boreal forests. First row: time series (a) and state portrait (b) obtained with the second-order model (parameter values:
rB ¼ 0:3, rT ¼ 0:067, KB ¼ 0:1, KT ¼ 3, a ¼ 0:045, rB ¼ 10, rT ¼ 0:98, sB ¼ 0:2, sT ¼ 0:1, lB ¼ lT ¼ 0:01). Second row: time series (c) and state portrait
(d) obtained with the fourth-order model (see Casagrandi and Rinaldi, 1999 for parameter values).
S. Maggi, S. Rinaldi / Theoretical Population Biology 70 (2006) 174–182178
A very similar fire regime obtained with the fourth-ordermodel is shown in Fig. 3(c) and (d). Obviously, the twomodels does not explain special surface fires (observed inboreal forests), due to litter accumulation on the ground(Kilgore and Taylor, 1979).
3.3. Mediterranean forests
Using suitable parameter values our impact model pointsout chaotic fire regimes. The first row of Fig. 4 shows thestrange attractor (b) and a 200 yr long time series (a)extracted from the strange attractor. Fires are recurrent butnot periodic: the fire return times vary from 10 to 50 yr, ingood agreement with a number of studies on two layersforests (Hanes, 1971; Le Houerou, 1974; Keeley, 1977;Schlesinger and Gill, 1978; Horne, 1981). Moreover, someof the fires are surface fires (indicated by S in Fig. 4(a)),while others are mixed fires (indicated by M in Fig. 4(a)).Again, quite similar results can be obtained with thefourth-order model (see second row of Fig. 4).
3.4. Comparison with field data
The comparison of the fire regimes proposed by oursecond-order model with real fire regimes is very difficult, ifnot impossible. A first difficulty is that the model describesthe behavior of a natural forest, while the great majority ofavailable data refer to forests where fire-fighting effortswere systematically performed. A second and moreimportant obstacle is that data do not refer to burnedbiomasses but rather to burned areas which are largely
influenced by meteorological conditions (Bessie andJohnson, 1995). Under these circumstances, we can atmost hope that some relevant features of forest fires(detectable from statistical analyses of field data) havesome sort of analogy in the model behavior. The mostrelevant of such features is, undoubtedly, the power-law
distribution of the burned areas. The power law has beenfirst suggested on the basis of a rather naıve interpretationof forest fires in terms of probabilistic cellular automata(Bak et al., 1990; Drossel and Schwabl, 1992) and thensupported by some statistical analyses of field data(Malamud et al., 1998; Ricotta et al., 1999). This lawwould imply that the log–log plot of the cumulativedistribution of the burned areas is a straightline. However,more recently, Ricotta et al. (2001), Reed and McKelvey(2002) and Telesca et al. (2005) have shown throughstatistical analysis of extensive fire records concerningMediterranean areas in Italy, Spain, Corsica and Greeceand six regions in North America (Sierra Nevada, NezPerce, Clearwater, Yosemite, N. E. Alberta, North WestTerritories) that the log–log plot of the cumulativedistribution is not a straightline but can be approximatedby three straight segments, as shown in the first row ofFig. 5. It is, therefore, very interesting to note (see Fig. 5(c)and (d)) that quite similar log–log plots can be obtained bycomputing the cumulative distributions of burned bio-masses in model generated fires.Moreover, a detailed analysis of the fires generated by
the model shows that the fires associated to flat segments ofthe cumulative distributions are mainly surface fires whilethose associated to the steepest part of the cumulative
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0 20 40 60 80 100 120 140 160 180 2000
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0 0.2 0.4 0.6 0.8 10
0.2
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time [y]
time [y]
T
0
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T
B
0 0.2 0.4 0.6 0.8 1B
T
B
TB
S
S S
SM S M S M S
M M M M M M
biom
ass
[10K
g/m
2 ]bi
omas
s [1
0Kg/
m2 ]
(a) (b)
(d)(a)
Fig. 4. Examples of fire regimes in Mediterranean forests. First row: time series (a) and state portrait (b) obtained with the second-order model (parameter
values: rB ¼ 3=8, rT ¼ 1=16, KB ¼ KT ¼ 1, a ¼ 129=800, rB ¼ 0:85, rT ¼ 14=15, sB ¼ 0:6, sT ¼ 0:35, lB ¼ lT ¼ 10�4). Second row: time series (c) and
state portrait (d) obtained with the fourth-order model (see Casagrandi and Rinaldi, 1999 for parameter values). The arrowsM and S point out mixed and
surface fires.
S. Maggi, S. Rinaldi / Theoretical Population Biology 70 (2006) 174–182 179
distributions are crown fires. This suggests that theproperty empirically pointed out by Ricotta et al. (2001),Reed and McKelvey (2002) and Telesca et al. (2005) couldsimply be related with forest morphology, i.e. with theexistence of various layers.
4. Discussion
We have presented a second-order model capable ofmimicking all the main forest fire regimes. This is a quiteappreciable result since many problems in ecology have,since long, their paradigmatic second-order model (e.g.Lotka–Volterra models for competition and predation,Leslie model for two stage populations, Streeter–Phelpsmodel for biodegradable pollution in rivers and lakes, SIRmodel for epidemics, etc.), forest fires still did not have it.
To our knowledge, the model is also the first impactmodel proposed in ecology. Other impact models couldpossibly be used for other ecosystems in which fast anddevastating events recursively occur. Promising applica-tions are plankton blooms in shallow lakes and insect–pestoutbreaks in forests.
Our analysis has shown that the proposed model israther flexible and can be adapted to savannas, borealforests and Mediterranean forests. Moreover, the statisticalcharacteristics of the intensity of the fires generated by themodel resemble those emerging from the statistical analysisof large data sets of various forests in Mediterraneancountries and in North America.
The analysis of the bifurcations of the model is of greatinterest, since it would produce in a systematic way thewhole catalogue of fire regimes described by the model.However, the solution of this problem is far from beingtrivial, since the theory of bifurcations of impact models isstill incomplete and, on the top of this, our manifolds X�
and Xþ are not smooth. A first attempt in this direction(Dercole and Maggi, 2004) has pointed out that the modelis sensitive to the parameters lB and lT which are theproportions of surface and tree biomass that survive tosurface and crown fires, respectively. In particular, theanalysis has shown that the transition from chaotic fireregimes (typical of Mediterranean forests) to cyclic fireregimes (typical of savannas and boreal forests) can berather sharp and interpretable as a so-called bordercollision bifurcation.Some problems concerning the model could be further
explored. The analysis of one or more stochastic versionsof the model would be interesting, for example, by lettingthe set X� of pre-fire conditions depend upon meteor-ological conditions. This would certainly amplify thechaoticity of the deterministic component of the system,thus giving to weather variability the role it deserves (Bessieand Johnson, 1995). The model could also be used todetermine, at least qualitatively, the most important effectsthat climate change and different management policies (e.g.thinning, grazing, remote monitoring, etc.) have on firefrequencies and intensities. Another issue of practicalinterest, that could be explored with our impact model, is
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101
102
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cum
ulat
ive
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(num
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of f
ires
)
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ulat
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(num
ber
of f
ires
)
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ires
)
cum
ulat
ive
dist
ribu
tion
(pro
port
ion
of f
ires
)
(c) (d)
(a) (b)
Fig. 5. Examples of fire statistics. First row: cumulative distributions (number of fires) of burned areas in (a) Alicante region, 1973–1996 (obtained from
Ricotta et al., 2001) and (b) Gargano region, 1997–2003 (b) (obtained from Telesca et al., 2005). Second row: cumulative distributions (proportion of fires)
of burned biomasses obtained with the second-order model for two parameter settings ((c): rB ¼ 0:3, rT ¼ 0:067, KB ¼ 0:7, KT ¼ 3, a ¼ 0:03, rB ¼ 0:8605,rT ¼ 0:98, sB ¼ 0:465, sT ¼ 0:4, lB ¼ lT ¼ 10�4; (d): as in (c) except KB ¼ 1, a ¼ 0:0375, rB ¼ 0:84, sT ¼ 0:405).
S. Maggi, S. Rinaldi / Theoretical Population Biology 70 (2006) 174–182180
the description of fire propagation in spatially extendedforests.
Acknowledgments
The authors are grateful to Renato Casagrandi for hishelpful suggestions and for his criticisms on a first draft ofthe paper and to one of the anonymous reviewers whoallowed us to improve the quality of the paper. Financialsupport was provided by MIUR under project FIRB2001-RBNE01CW3M.
Appendix
The fourth-order model described in Casagrandi andRinaldi (1999) is the following
_B ¼ rBB 1�B
KB
� �� aBT � bB
B
Bþ hBB
F B
� gB
B
Bþ hBT
F T , ðA:1Þ
_T ¼ rT T 1�T
KT
� �� bT
T
T þ hTT
F T � gT
T
T þ hTB
FB,
(A.2)
_FB ¼ bB
B
Bþ hBB
F B þ gB
B
Bþ hBT
F T � dBFB, (A.3)
_FT ¼ bT
T
T þ hTT
F T þ gT
T
T þ hTB
FB � dT F T , (A.4)
where B and T (FB and F T ) are the nonburning (burning)biomasses of the lower and upper layers of the forest, andall lower-case letters, as well as KB and KT , are constantparameters. If there is no fire, i.e. if F B ¼ FT ¼ 0, themodel describes the growing phase of the forest andreduces to Eqs. (3).If we are interested in determining the pre-fire manifold
X�, we must simply determine the pairs ðB;TÞ at which agrowing forest settles the conditions for fire ignition andpropagation. For this we can analyze Eqs. (A.3), (A.4)which describe the dynamics of the fire. If there is no fire,
ARTICLE IN PRESSS. Maggi, S. Rinaldi / Theoretical Population Biology 70 (2006) 174–182 181
system (A.3), (A.4) with B and T frozen at their currentvalues, is at the trivial equilibrium FB ¼ F T ¼ 0. If thistrivial equilibrium is stable, a small accidental fire (i.e. asmall positive perturbation of FB and F T ) will not have thechance to propagate. By contrast, the same smallaccidental fire will propagate if the trivial equilibrium isunstable. Thus, the pre-fire manifold X� can be interpretedas the set of values ðB;TÞ at which the trivial equilibrium ofsystem (A.3), (A.4) becomes unstable. But system (A.3),(A.4) with B and T frozen is a linear system with Jacobianmatrix given by
J ¼
bB
B
Bþ hBB
� dB gB
B
Bþ hBT
gT
T
T þ hTB
bT
T
T þ hTT
� dT
2664
3775.
For relatively small values of B and T, the diagonalelements of J are negative (see Casagrandi and Rinaldi,1999) since dB and dT are high. Indeed, dB and dT are therates of decay of the fires when there is nothing to burn (i.e._FB ¼ �dBF B if B ¼ T ¼ 0, and similarly for crown fires).Thus, the transition to instability of the Jacobian matrix issimply revealed by the annihilation of its determinant, i.e.
bB
B
Bþ hBB
� dB
� �bT
T
T þ hTT
� dT
� �
� gBgT
B
Bþ hBT
T
T þ hTB
¼ 0. ðA:5Þ
This condition implicitly defines a function TðBÞ whichcorresponds to the pre-fire manifold X�. Taking the firstand second derivatives with respect to B of Eq. (A.5) withT ¼ TðBÞ, one obtains two relationships involving T 0ðBÞ
and T 00ðBÞ, which can be used to prove that
T 0ðBÞo0 T 00ðBÞo0
i.e. that the function TðBÞ is decreasing and the setTpTðBÞ is convex. The proof is quite simple ifhBB ¼ hTT ¼ hBT ¼ hTB ¼ h, which is actually the caseconsidered in all simulations of Mediterranean forestspresented in Casagrandi and Rinaldi (1999). In fact, in sucha case, Eq. (A.5) can be solved with respect to T and gives
T ¼aB� b
cB� d, (A.6)
where
a ¼ ðdT þ hÞðbB � dBÞ,
b ¼ ðdT þ hÞdBh,
c ¼ ðbT � dT ÞðbB � dBÞ � gBgT ,
d ¼ ðbT � dT Þ.
Assuming that each layer can burn provided its biomass isinfinitely large, from Eqs. (A.3) and (A.4) we obtain
bB4dB bT4dT
which imply that the four parameters a, b, c, d are positive,since gB and gT (the inter-layers fire attack rates) are small
(Casagrandi and Rinaldi, 1999). Moreover,
bc
ad¼ dBh�
gBgT
ðbB � dBÞðbT � dT ÞodBh
so that
bcoad
since dBh is smaller than 1 (see Table 2 in Casagrandi andRinaldi, 1999). Thus, from Eq. (A.6) it follows that thefunction TðBÞ is positive in the interval
0oBob
a,
while its first and second derivatives
T 0ðBÞ ¼bc� ad
ðcB� dÞ2T 00ðBÞ ¼
2ðad � bcÞc
ðcB� dÞ3
are negative in the same interval.In conclusion, the fourth-order model of Casagrandi and
Rinaldi (1999) suggests that the pre-fire manifold X� is asmooth manifold decreasing with respect to B and that theset below the manifold X� is a convex set.
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