Journal of Theoretical Biology 355 (2014) 68–76

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Eco-hydrology driven fire regime in savanna Nadia Ursino n Department ICEA, Universita di Padova, via Loredan 20, I 35131 Padova, Italy

H I G H L I G H T S








A new eco-hydrological predator–prey model for savanna wildfire is presented. Fire frequency arises as an ecosystem property from savanna composition. When grass fuel is scarce, fires are more frequent and less destructive. Dry and mesic savannas are characterized by different fire regimes. The vulnerability of trees to grass fire regulates the shift between the fire regimes.

art ic l e i nf o

a b s t r a c t

Article history: Received 26 August 2013 Received in revised form 27 March 2014 Accepted 1 April 2014 Available online 12 April 2014

Fire is an important evolutionary force and ecosystem consumer that shapes savanna composition. However, ecologists have not comprehensively explained the functioning and maintenance of flammable savannas. A new minimal model accounting for the interdependence between soil saturation, biomass growth, fuel availability and fire has been used to predict the increasing tree density and fire frequency along a Mean Annual Rainfall (MAR) gradient for a typical savanna. Cyclic fire recurrence is reproduced using a predator prey approach in which fire is the “predator” and vegetation is the “prey”. For the first time, fire frequency is not defined a priori but rather arises from the composition of vegetation, which determines fuel availability and water limitation. Soil aridity affects fuel production and fuel composition, thus indirectly affecting the ecosystem vulnerability to fire and fire frequency. The model demonstrates that two distinct eco-hydrological states correspond to different fire frequencies: (i) at low MAR, grass is abundant and the impact of fire on the environment is enhanced by the large fuel availability, (ii) at higher MAR, tree density progressively increases and provides less fuel for fire, leading to more frequent and less destructive fires, and (iii) the threshold MAR that determines the transition between the two states and the fire frequency at high MAR are affected by the vulnerability of trees to grass fire. The eco-hydrology-driven predator–prey model originally predicts that the transition between dry and wet savanna is characterized by a shift in wildfire frequency driven by major differences in soil moisture available for plants and savanna structure. The shift and the role of fire in conserving savanna ecosystems could not have been predicted if fire was considered as an external forcing rather than an intrinsic property of the ecosystem. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Predator–prey Mean Annual Rainfall (MAR) gradient Tree density Fuel abundance Tree vulnerability

1. Introduction The savanna question Sarmiento (1984) posed concerning the reason trees and grasses coexist in savanna stimulated the formulation of several hypotheses and modeling concepts; however, this ecological issue remains partially unresolved. Recent remotely sensed estimate of tree cover in Africa evidenced an increase of

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tree cover with MAR up to the threshold MAR E1000 mm yr  1 when tree closure is possible and bimodality of tree cover appears (Staver et al., 2011b). Whether the bimodality of tree density is globally widespread, and associated with a sharp transition between alternative stable states is unclear. Global patterns of tree density (Hirota et al., 2011) as well as fire manipulation experiments (Higgins et al., 2007) may be used to infer the resilience of tropical forest and savanna to critical transitions. Indeed, whether savannas are intrinsically unstable systems maintained by disturbances such as fire and grazing or whether savannas are stable systems that persist despite these disturbances

N. Ursino / Journal of Theoretical Biology 355 (2014) 68–76

is unclear (Sankaran et al., 2004). Fire, grazing and rainfall may represent chance events for one or the other species (Archer et al., 1995; Bond and Keeley, 2005; Hoffmann et al., 2004; Scholes et al., 2002). Possible determinants of savanna structure include (i) competition and niche differentiation (e.g., root niche separation) with respect to limiting resources (e.g., water) (Walter, 1971; Walker et al., 1981; Walker and Noy-Meir, 1982; Eagleson and Segara, 1985; Fernandez-Illescas and Rodriguez-Iturbe, 2004; van Langevelde et al., 2003) and (ii) demographic bottlenecks to tree seedling germination and establishment (Menaut et al., 1990; Hochberg et al., 1994; Jeltsch et al., 1996, 2000; Higgins et al., 2000; van Wijk and Rodriguez-Iturbe, 2002; Gignoux et al., 2009). Resource based controls are modeled here with soil saturation dependent growth rates and tree carrying capacity, in order to predict the extent to which the niche differentiation between grass and trees may impact fire dynamics. Fire could be a major cause of these bottlenecks in frequently burned savannas (Archer et al., 1995). However, fire ignites if enough fuel is present and if the fuel is dry (Liedloff and Cook, 2007). Recent catastrophic fire events have “ignited” the debate on the interconnection between climate and fire regime (e.g., Moritz, 2012). The frequency and the influence of fire are affected by climate, species composition and fuel availability. Introduced grasses can alter the fire regime (Gill et al., 2009), and fire suppression may cause species losses and the switch from savanna to forest ecosystems (Peterson and Reich, 2001). Several minimal models analyze the impact of fire, grazing or climate on the stability of savanna, but none, as far as I know, accounts for the influence of climate on fuel production and, consequently, fire regime. Among others, Staver and Levin (2012) incorporated a positive grass–fire feedback into a simple space implicit model and demonstrated that bistability of tree cover is determined by fire feedbacks at intermediate rainfall, Beckage et al. (2009) developed a fire disturbance model incorporating firevegetation feedbacks and demonstrated that positive and negative feedbacks result in a stable savanna state. Beckage et al. (2011) used a stochastic modeling framework to demonstrate that a strong feedback between grass density and fire probability may stabilize savanna. Baudena et al. (2010) demonstrated that stochastic disturbances such as fire widen the parameter range over which tree–grass coexistence is expected. Accatino et al. (2010) showed that the co-existence of trees and grasses could be controlled by fire and rainfall scarcity. In disturbance models (Accatino et al., 2010; Baudena et al., 2010; Beckage et al., 2009, 2011), fire is an external source of additional vegetation mortality. Using a substantially different approach, Casagrandi and Rinaldi (1999) assumed that fire could ignite via either lightening or human action in any year, provided that sufficient fuel (mainly grass) is present. Because the frequency of fire is not defined a priori, the ecosystem itself dictates the frequency and intensity of fire. Fire is a crucial determinant of tree cover at intermediate MAR (Staver et al., 2011a; Staver and Levin, 2012). Positive feedbacks between grass abundance and fire spread, and between tree establishment and rainfall, and a negative feedback between tree establishment and fire, ensure that at very low and high MAR tree cover is stable and determined by climate. Previous models accounting for the grass–fire feedback (Casagrandi and Rinaldi, 1999; Staver et al., 2011a; Staver and Levin, 2012) link the fire regime to the ecosystem characterization but neglect the influence of climate on growth of vegetation, fuel production and the environmental conditions for fire ignition that will be evaluated in this study. Fire has been defined as a “global herbivore” (Bond and Keeley, 2005). In the following sections, the cyclic savanna fire regime is predicted by a predator–prey model, in which burning biomass is the predator and standing biomass is the prey. This new modeling

69

concept is formulated to clarify if the observed increase in fire frequency along a MAR gradient (Sankaran et al., 2007) can be attributed to the interrelation between fuel production and fireinduced mortality of grasses and trees and the extent to which this interrelation is affected by niche competition. The classical predator–prey model, based on a two-species system (Lotka, 1920; Volterra, 1926), considers one predator and one prey in a static environment in which the food for prey is unlimited and exhibits only two types of behavior: static point and limit cycle. The model described in the following section is a two-predator–two-prey model, in which grass and trees are the prey and the burning grass and trees (the predators) feed on both vegetation species. The balance of fuel is not taken explicitly into account, but fuel (the dry biomass) is considered to be proportional to the living biomass. The two-predator–two-prey model can exhibit richer dynamics than the classical Lotka–Volterra model and includes chaos (Casagrandi and Rinaldi, 1999). To account for the eco-hydrological interaction between climate, soil and vegetation, the model includes an ordinary balance equation for soil moisture in addition to the balance equations for two predators and two prey. The soil moisture balance equation drives the system of two-predator–two-prey to different behaviors as the hydrological forcing changes, including the static point (no fire regime), the limit cycle (with more or less frequent fire occurrence) or chaotic behavior (corresponding to a more complex structure of the fire occurrence), as demonstrated by Ursino and Rulli (2011) for a Mediterranean forest. The interdependence between moisture, biomass and fire is introduced here for the first time in a minimal modeling framework to predict wildfire frequency in savanna. The minimal model is described in Section 2. The novel modeling framework may be used to evaluate the importance of fire versus resources in shaping the tree–grass composition of savannas, as well as to forecast the resulting wildfire regime in climate change scenarios. This study aims at testing new modeling hypotheses rather than reproducing site-specific behavior, although I also compared literature data with the model outcomes. First, the model equilibrium in the absence of fire was analyzed (Section 3). Second, the impact of soil moisture on the vegetation growth and thus on savanna composition was investigated (Section 4). Third, the model outcome was compared with real data (average tree density and fire frequency) along a MAR gradient (Section 5). The results are discussed in Section 6.

2. Model description A conceptual implicit-space model describes the dynamics of trees, grass and soil moisture. Fuel load is proportional to vegetation biomass density. Fire ignites any time the fuel load is above a certain threshold then burns a fraction of the trees and grass that cover the soil, before becoming extinguished when the flammable fuel is depleted. The reduction in burning biomass density allows standing biomass to build up again until the following fire ignites after several growing seasons. Scarce moisture availability limits the biomass growth and may be ascribed either to low MAR or to large excess rainfall and low effective rainfall entering the soil, when intense and infrequent rainfall events occur. Here, effective rainfall is modeled as proportional to MAR. Neither the interannual variability of dry season length and rainfall abundance nor the daily variability of rainfall, soil moisture, temperature and wind is taken into account, even though they influence plant growth. Thus, the soil moisture variability is strongly interconnected with the fire regime and the changes of biomass density that fire determines. Because of its simplicity, the model allows the analytical calculation of the steady state in the absence of fire and

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the numerical estimation of tree density and fire frequency of less than 1 yr  1. The model consists of five ordinary differential equations formulated for the soil saturation S ½  , the biomass density of trees Bt, the biomass density of grass Bg, the burning biomass of trees Rt and that of grass Rg [kg m  2]. The burning biomass of trees and grass attacks the inter-specific and intra-specific living biomass. Burning biomass grows if enough fuel (prey) is present and otherwise decays at a much faster time scale than the characteristic time scale of biomass growth ∂S 1 ¼  ½I ETðS; Bt ; Bg Þ  LðSÞ ∂t nH

ð1Þ

∂Bt ¼ Gt ðS; Bt Þ  F t ðBt ; Rt ; Ru Þ ∂t

ð2Þ

because of the morphological differentiation of their root apparatus (root partitioning), because of their physiology, because prolonged severe drought periods result in a high tree seed mortality that limits tree establishment, whereas grasses survive dormant in the seed bank, ready to sprout at the beginning of the rainy season (Higgins et al., 2000) or because many consecutive dry years lower soil moisture to the extent that trees die. The model accounts for these resource based controls with the soil saturation dependent growth rates and carrying capacity and predicts the extent to which the niche differentiation between grass and trees may impact fire frequency. The phenological differences between grass and trees are idealized:   Bt Gt ðS; Bt Þ ¼ r t Sαt Bt 1  ð8Þ kt S

∂Bg ¼ Gg ðS; Bg ; Bt Þ  F g ðBg ; Rt ; Ru Þ ∂t

ð3Þ

  Bg  γ Bt Bg Gg ðS; Bg ; Bt Þ ¼ r g Sαg Bg 1  kg

∂Rt ¼ F t ðBt ; Rt ; Ru Þ  Dt ðRt Þ ∂t

ð4Þ

∂Rg ¼ F g ðBg ; Rt ; Ru Þ  Dg ðRg Þ ∂t

ð5Þ

In Eq. (1) H identifies the depth of the soil control volume where plant roots are active, n is the soil porosity, I is the annual average effective rainfall entering the control volume, ET is the soil saturation-dependent annual rate of evapotranspiration and L is the leakage out of the control volume that contributes to the water balance when soil moisture content is closed to saturation. I, ET, and L resemble processes characterized by variation at daily and seasonal time scales that must be properly upscaled to evaluate I, ET, and L as functions of annual average eco-hydrological parameters. To reduce the complexity of the model, the variability of evapotranspiration and leakage of soil moisture is drastically simplified by assuming that ET ¼ ET t

Bg Bt S þ ET g S kt kg

ð6Þ

where ET t  S and ET g  S are the maximum yearly evapotranspiration rates achievable by the trees and grass, respectively, according to the standing environmental conditions when the density of each species is equal to the carrying capacity (that is kg for grass and kt for the trees when S approaches 1). In arid savanna where the tree density is water limited, the effect of resource limitation on the tree cover is modeled by assuming that the carrying capacity of trees (kt S) increases linearly with soil saturation. The water stress limits the evapotranspiration, as the average soil saturation S decreases, according to the linear relation (6) between evapotranspiration and soil saturation (Kim et al., 1996). Leakage L is modeled as follows: L ¼ KSα

ð7Þ

where K and α are the soil characteristic parameters (Ursino, 2005). The dynamics of the biomass density of the two vegetation species, described by Eqs. (2) and (3), resembles the individual processes of growth, extinction and replacement of living biomass (expressed by the functions Gt and Gg), and the additional mortality due to fire (expressed by the functions Ft and Fg). The dynamics of the burning biomass density of trees Rt and of the burning biomass density of grass Rg are the sum of fire ignition and fire extinction (expressed by the functions Ft and Fg and Dt and Dg), and are much more rapid than the dynamics of living biomass. The biomass growth rates Gt and Gg are functions of biomass density, the maximum species' carrying capacity and soil saturation. Trees and grasses competing for water resources may coexist

ð9Þ

The effective parameters αt and αg have been introduced because the daily variability of the climatic forcing is not modeled explicitly, neither the soil moisture dynamics is resolved over depth. Pulses of rainfall often cause limited infiltration depth, whereas in deeper soil layers water remains available for longer times. Unfrequent rainfall events may indeed favor plants' reproduction and germination to maintain species diversity (Chesson et al., 2004), and allow vegetation persistence in an arid land (Baudena et al., 2007). The exponents αt and αg are vegetation characteristic parameters that account for the ability of each species to resprout after a prolonged drought or to survive despite persistent water scarcity, profiting off of small or infrequent rainfall events. On an average annual basis, to account for the ability of grass to resprout after drought a low value has been attributed to αg. The term γ Bt Bg accounts for the fact that trees are superior competitors for light according to Casagrandi and Rinaldi (1999). The burning biomass of each species attacks the flammable biomass of both species, igniting fire according to the following functions: F t ðBt ; Rt ; Rg Þ ¼ ðβt Rt þ γ t Rg Þ 

Bt Bt þ h t

F g ðBg ; Rg ; Rt Þ ¼ ðβg Rg þ γ g Rt Þ 

Bg Bg þhg

ð10Þ ð11Þ

where βt and βg are the intra-specific fire attack rates of the burning biomass of trees and grass, respectively, γt and γg are the inter-specific fire attack rates, and ht and hg are the half saturation constants associated with tree and grass density, respectively, because burning biomass growth is limited by fuel availability and fuel is assumed to be proportional to standing biomass density. Burning trees and grass become extinct at the given rates, δt and δg, according to Eqs. (4) and (5), where Dt ðRt Þ ¼ δt Rt

ð12Þ

Dg ðRg Þ ¼ δg Rg

ð13Þ

The model parameters (Table 1) are taken from the literature (Fernandez-Illescas and Rodriguez-Iturbe, 2004; Accatino et al., 2010; Ursino, 2013; Casagrandi and Rinaldi, 1999). The major contribution of grass to fuel production is expressed by the condition βg =δg 4 βt =δt , implying that trees are less susceptible to fire than grass. This assumption accounts for the fact that large individuals rarely suffer mortality (Higgins et al., 2000, 2007). As a consequence, grass fires extinguish when the grass layer is almost completely destroyed, and conversely, tree fires extinguish much earlier and never completely destroy the

N. Ursino / Journal of Theoretical Biology 355 (2014) 68–76

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Table 1 Model parameters. Parameter

Parameter value

Unit

Definition

Water balance equation parameters ETt ETg H n

400 250 800 0.4

mm yr  1 mm yr  1 mm –

Tree reference evapotranspiration Grass reference evapotranspiration Soil depth Soil porosity

K α

400 3

mm yr  1 –

Leakage coefficients

Biomass growth parameters rt αt rg αg kt kg

0.8 0, 1, 5 2 0, 1, 0.1 5 4

yr  1 – yr  1 – kg m  2 kg m2

Tree growth rate Tree water use efficiency exponent Grass growth rate Grass water use efficiency exponent Tree maximum carrying capacity Grass carrying capacity

γ

0.2

kg m  2 yr  1

Interspecific competition rate

Fire dynamics parameters βt βg γt γg ht hg δt δg

50 80 1, 2, 5 1 0.075 0.06 53 72

yr  1 yr  1 yr  1 yr  1 kg m  2 kg m  2 yr  1 yr  1

Intra-specific tree fire attack rate Intra-specific grass fire attack rate Inter-specific tree fire attack rate Inter-specific grass fire attack rate Half saturation burning tree constant Half saturation burning grass constant Burning tree extinction rate Burning grass extinction rate

tree layer that acts as a barrier (Archibald et al., 2009). Thus, when a grass fire occurs, grass is mainly dry and flammable, whereas when a tree fire occurs, trees are just partially flammable. By attributing different values to the parameters of the water stress functions Sαt and Sαg , different model concepts may be addressed. If αt ¼ αg , the growth rate of grass and trees is similarly affected by water scarcity. When αt ¼ αg ¼ 0, the link between ecology and hydrology is restricted to the linear variability of the tree carrying capacity with S. The case αt ¼ αg ¼ 1 in the absence of fire is characterized by a saturation independent density of grass in the savanna solution (see the next section), where grass abundance is one of the major determinants of fire frequency. Particular attention is given here to the case αt ¼ 5; αg ¼ 0:1. When αt 4 αg , the model accounts for the fact that drought limits the growth of both species (O'Connor, 1985; Sankaran et al., 2005), but some niche competition for water between grass and trees may provide an additional chance for grass to establish, and reach a sufficiently high biomass density and affect the fire regime when water is scarce. When γ t ¼ γ g 5 βt ; βg , less relevance is given to the interspecific development of fire than to the intra-specific fire attack. This condition minimizes the impact of grass fire on tree density, e.g., by affecting the demographic bottleneck to tree establishment. To evaluate the impact of inter-specific development of grass fire on fire frequency, the base case study, with γt ¼1 yr  1 and the cases where γt ¼2 yr  1 and γt ¼ 5 yr  1, was examined.

3. Equilibrium in the absence of fire In the absence of fire Rt ¼ Rg ¼ 0, the system of differential equations (1)–(5) reduces to the three equations (1)–(3). Since I is constant, then the dependent variables Bt, Bg, and S converge to a constant value in the absence of fire. In addition to the trivial bare soil solution: Bt ¼ Bg ¼ 0, the model has a savanna type solution: Bt ¼ kt S and Bg ¼ kg ð1  γ kt Sð1  αg Þ =r g Þ; a forest solution Bt ¼ kt S and Bg ¼0; and a grass solution Bt ¼0 and Bg ¼ kg. If αg ¼ 1 the grass density in the savanna solution does not

1.0

Bt Bg , kt k g

0.8

0.6

0.4

0.2 100

200

300 400 I [mm/yr]

500

600

Fig. 1. Steady state solution in the absence of fire for αg ¼ 0:1 and αt ¼ 5. Squares: savanna solution; circles: forest solution. Open squares: dimensionless biomass density of grass Bg =kg ; solid squares: dimensionless biomass density of trees Bt =kt .

depend on S. The savanna solution exists for S o ðγ kt =r g Þ1=ðαg  1Þ . According to the model parametrization specified in Table 1, γ kt =r g ¼ 0:5. Since αg o 1, then ðγ kt =r g Þ1=ðαg  1Þ 4 1 is always higher than S. This means that the model always predicts savanna. In Fig. 1 the savanna and the forest solutions are plotted as a function of I, for αg ¼ 0:1, αt ¼ 5 and other parameters specified in Table 1. A linear stability analysis concerning the steady states or the equivalent phase-plane analysis (Jordan and Smith, 1999) provides necessary and sufficient conditions for the linear stability of the steady state solutions. Based on the results of the linear stability analysis, in the absence of fire, the savanna solution is stable, according to the assertion that arid and semiarid savannas may be considered stable systems in which water constrains

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N. Ursino / Journal of Theoretical Biology 355 (2014) 68–76

woody cover and allows tree–grass coexistence (Sankaran et al., 2005). The grass solution and the forest solution are unstable.

4. Numerical solution: soil moisture and biomass patterns Adding the dynamics of the burning layers, each layer is interpreted as a prey–predator assemblage, and the system becomes an oscillator even in the absence of external hydrologic forcing. The set of equations (1)–(5) has been integrated numerically by implementing a MATLAB 7.10 code. Three different scenarios have been simulated by setting I at 100, 400, and 600 mm yr  1. It is reasonable to assume that MAR is positively correlated with I; thus, the model outcome, obtained with different values of I, suggests how climate can influence the fire regime. By considering the model results obtained with different αt and αg values, different modeling assumptions concerning plant adaptations to drought may be evaluated. Eqs. (1)–(5) have been integrated numerically for 0 r t r 150 yr with a time step Δt ¼ 2  10  3 yr, starting with the initial conditions S ¼0.2, Bt ¼ 0:9  kt S, Bg ¼ 0:9  kg , Rt ¼ 0:1  kt , Rg ¼ 0:1  kg and the model outcome in the last 50 yr of simulation time has

I

100 mm yr

1

I

been shown in dimensionless form in Figs. 2–4, where Bt =kt , Bg =kg and S are plotted functions of time. The three dimensionless variables range from 0 to 1, and their time variability is due to occurrence of fire that causes a sudden reduction of Bt =kt and Bg =kg . When αt ¼ αg ¼ 0, the growth rate of trees and grass is not limited by the soil moisture availability. The saturation dependent tree carrying capacity affects tree density and indirectly the grass density that determines the fuel availability, dictating the frequency of fire along the MAR gradient (Fig. 2). By assuming αt ¼ αg ¼ 1 (Fig. 3), meaning that the soil moisture availability limits the growth of both species, the results show that grass growth is limited by soil moisture availability at low I and by trees (that are stronger competitors for light) at high I. As a consequence, the fuel availability is always low and fire is less destructive than in the previous case with αt ¼ αg ¼ 0. The assumption that some niche differentiation due to the preferential adaptability of grass to the environmental conditions, originated by a low net rainfall income, may affect the fire regime was tested by assuming that αt ¼ 5 and αg ¼ 0:1. With this parameter choice, the water-stress function for grass, Sαg , is concave to emphasize the grass water-stress tolerance. Conversely,

400 mm yr

1

I

600 mm yr

1

1.0

0.8

0.6

0.4

0.2

0

10

20

30

40

50 0

t [yr]

10

20

30

40

50 0

10

20

t [yr]

I

30

40

50

t [yr]

Fig. 2. Patterns of biomass and soil saturation as a function of time, αt ¼ αg ¼ 0 (the growth rate of trees and grass is not limited by the soil moisture availability). Dots: soil saturation; continuous line: dimensionless biomass density of trees Bt =kt ; dashed line: dimensionless biomass density of grass Bg =kg . Left to right I¼ 100, 400, 600 mm yr  1.

1.0

0.8

0.6

0.4

0.2

0

10

20

30

t [yr]

40

50 0

10

I

20

30

t [yr]

40

50 0

10

20

30

40

50

t [yr]

Fig. 3. Patterns of biomass and soil saturation as a function of time, αt ¼ αg ¼ 1 (the soil moisture availability limits the growth of both species). Dots: soil saturation; continuous line: dimensionless biomass density of trees Bt =kt ; dashed line: dimensionless biomass density of grass Bg =kg . Left to right I¼ 100, 400, 600 mm yr  1.

N. Ursino / Journal of Theoretical Biology 355 (2014) 68–76

73

1.0

0.8

0.6

0.4

0.2

0

10

20

30

40

50 0

10

20

30

50 0

40

10

20

30

40

50

t [yr]

t [yr]

t [yr]

I Fig. 4. Patterns of biomass and soil saturation as a function of time, αt ¼ 5 and αg ¼ 0:1 (niche differentiation: grass are better adapted than trees to the environmental conditions, originated by a low net rainfall income). Dots: soil saturation; continuous line: dimensionless biomass density of trees Bt =kt ; dashed line: dimensionless biomass density of grass Bg =kg . Left to right I¼ 100, 400, 600 mm yr  1.

the water-stress function for trees is assumed to be convex to account for the fact that at low I, which leads to a low S, the tree growth rate is limited by reduced soil moisture availability. When I ¼100 mm yr  1, tree growth is slow and (Fig. 4, left) grass provides a lot of fuel for fire before trees reach a high biomass density and out-compete grass. The fire regime is characterized by a low fire frequency (3 fire events in 40 years), and each fire event completely destroys the grass layer. The fire regime changes for I ¼400 and I ¼600 mm yr  1 (Fig. 4, center and right), and the impact of fire on the environment changes as well. The frequency of fire increases but the amount of biomass that each fire event destroys decreases. Trees provide less fuel for fire than grass and are moderately attacked by grass fires. Consequently, after each fire event, the biomass density of trees is just partially reduced, trees limit grass growth and less fuel for fire is available. The resulting fire frequency when I ¼600 mm yr  1 is 0.2 yr  1 (10 fire events in 50 years).

1.0

Bt Bg , kt k g

0.8

0.6

0.4

0.2

0.0 100

5. Numerical solution: fire frequency and average tree density along a MAR gradient Sankaran et al. (2005, 2007) collected data from 854 sites across Africa, each one characterized by a different management strategy, species composition, and climate. The data have been used to demonstrate that the maximum woody cover in arid and semiarid savanna receiving MAR below 650 mm yr  1 increases linearly with MAR (Sankaran et al., 2005, 2007). Data on fire frequency along the MAR gradient are also provided by Sankaran et al. (2005, 2007), but as far as I know, they have not been used to speculate on the general trend of fire frequency along a MAR gradient. The model predicts the dependence of fire frequency on the eco-hydrological standing conditions (MAR, potential evapotranspiration, vegetation growth rates, fuel threshold amount that leads to fire ignition, etc.), based on the following basic assumptions: (i) the maximum tree cover is limited by the soil saturation, (ii) increasing average soil saturation favors tree growth and grass production, (iii) trees tend to out-compete grass for light, but (iv) grass grows faster than trees at low soil saturation, profiting of any pulse of rainfall, and providing fuel for fire under conditions of water scarcity. Setting γ ¼0 and neglecting this interspecific competition leads to slightly higher estimated grass density and

200

300 400 I [mm/yr]

500

600

Fig. 5. Steady state solution with and without fires (symbols and lines respectively). Open squares: dimensionless biomass density of grass Bg =kg in the no-fire savanna solution; solid squares: corresponding dimensionless biomass density of trees Bt =kt . Time average values of dimensionless tree and grass density for different values of I, αt ¼ 5 and αg ¼ 0:1. Continuous line: dimensionless biomass density of trees Bt =kt ; dashed line: dimensionless biomass density of grass Bg =kg .

frequency of fire but substantially unchanged main conclusions (not shown). Finally, (v) fire developing in either the grass or the tree layer attacks flammable biomass in both layers. In the absence of fire the model predicts stable savanna. The model (Eqs. (1)–(5)) has been integrated repeatedly for different I, αt ¼ 5 and αg ¼ 0:1, for 0 rt r 300 yr. Ignoring the model outcome in the first 100 yr of simulation time, the modeled time series of the dimensionless dependent variables Bt =kt , Bg =kg and S, in the following 200 yr of simulation time have been used to estimate (i) their average value Bt =kt , Bg =kg and S and (ii) the frequency of fire f. Fire frequency has been estimated numerically as the maximum of the power spectra of Gg =kg ðtÞ in 100 rt r 300 yr. When 100 o I o 600 mm yr  1 , the model predicts a progressive change in ecosystem composition (Fig. 5), and lower biomass density of both species than in the no fire scenario (confront symbols and lines). For I 4350 mm yr  1 Bg =kg does not depend

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on S any further, and Bt =kt slowly approaches its maximum. Correspondingly, the fire frequency f switches from f E0.07 yr  1 to f E0.2 yr  1(as shown in Fig. 4). These results should not be attributed exclusively to the fact that the tree carrying capacity increases along the MAR gradient. As a result of the fire dynamics, the model predicts that the tree density progressively increases and fire becomes more frequent and less destructive along the MAR gradient even though the tree carrying capacity is set at a constant value (i.e. does not depend on S), provided that there is a niche differentiation between grass and trees as represented by the saturation dependent growth rates (not shown). In Fig. 6 the model outcome has been plotted versus MAR together with the data collected by Sankaran et al. (2004). The calculated tree carrying capacity (corresponding to the tree cover in the no fire savanna solution) and Bt both divided by kt are shown in the left panel. The difference between the two curves (the dashed and the continuous lines in Fig. 6, left) correspond to the reduction of tree coverage due to fire. A linear relation between I and MAR, accounting for the partitioning of rainfall into infiltration and runoff, was calibrated in order to fit the calculated to the observed maximum tree coverage. The linear relation is I ¼0,6 MAR, and it results in agreement with literature data (Franz et al., 2012). The corresponding calculated fire frequency f is shown in the right panel. Despite the local scale heterogeneity of the vegetation composition, the tree cover changes continuously with rainfall (Favier et al., 2012) and the model captures this response of tree cover to moisture and fire (Fig. 6, left). Accatino et al. (2010) found a similar result with prescribed fire frequency. The new result obtained here is the calculated fire frequency along the MAR gradient. It represents the new insight of the two-predator–two-prey model into the wildfire regime of savannas. The model fit is good at very low MAR (arid and semiarid savanna), the climatic condition that causes fire occurrence to be driven by grass availability and climate. But the maximum modeled fire frequency (f  0:2 yr  1) is much lower that the observed one at high MAR (Fig. 6, right), suggesting that some relevant processes may have been underestimated. Buccini and Hanan (2007), combining semi-empirical modeling and information theory, analyzed the dependency of tree cover on soil texture, fire, domestic livestock, human population density and cultivation intensity, and found that in the semiarid and mesic savanna the maximum fire frequency is f E0.5 yr  1. Sankaran's data demonstrate that the fire frequency may be fE 1 yr  1 in pastoral and protected area when 400 o MAR o800 mm yr  1 . The model can be used to further investigate if the low fire frequency at higher MAR can be attributed to the physiological 1.0

1.0

Bt kt

0.8

traits of mesic savanna and in particular to the limited interspecific interaction between burning biomass and standing biomass that follows the condition γ t ¼ γ g 5 βt ; β g . The impact of grass fires on tree recruitment and establishment may be strengthened by increasing γt to investigate the extent to which tree vulnerability to fire attack may affect the frequency of wild-fires in savannas. Tree density and fire frequency evaluated with γ t ¼ 1 (as before), 2, and 5 kg  1 m2 yr  1 have been plotted as a function of MAR in Fig. 7. When increasing tree vulnerability to grass fire, the transition between grass-dominated and tree-dominated savannas switches at higher MAR values. Correspondingly, fire frequency progressively increases with γt, in the wet regime.

6. Discussion and conclusion Different mechanisms could promote the tree–grass coexistence at a local scale, since there may be different savannas and each savanna could be sustained by different feedback mechanisms (Favier et al., 2012). The minimal model for the ecohydrology-driven wildfire regime in savanna that was presented here is based on the following assumptions: (i) grass represents the major source of fuel in savanna (Casagrandi and Rinaldi, 1999 and references therein), (ii) water is a fundamental driver of vegetation growth (e.g., Rodriguez-Iturbe and Porporato, 2005) and fuel production, (iii) grass is more tolerant of water stress than trees (Schenk and Jackson, 2002; Scholes et al., 2002; Williams et al., 1996), and thus grass fuel production is higher at low MAR, as compared to tree fuel production and (iv) fire develops if enough fuel is present (Casagrandi and Rinaldi, 1999), either in the grass or in the tree layer, and although fire attacks both layers, grasses are much more flammable. Points (i)–(iv) concur in creating a positive grass–fire feedback, where grass feeds fire and fire reduces tree cover to the advantage of grass. Feedbacks are crucial determinants of fire regime (Staver et al., 2011a; Staver and Levin, 2012). The model demonstrates that climate and fire shape savannas and that the environment, which changes with climate, may lead the savanna ecosystem to different fire regimes. The study of the dependence of fire frequency on climate indicates that two distinct eco-hydrological and fire regimes may be modeled with a two-predator–two-prey approach: (i) a dry regime in which grass is abundant (grass density is between 0:3kg and 0:6kg ), the impact of fire on the environment is enhanced by large fuel availability and the fire frequency is low and (ii) a wetter regime in which the tree density approaches its upper limit below tree carrying capacity, the fire regime is regulated by the low grass availability and fire occurs at higher frequency. In between these

f

Sankaran (2005) data Pastoral area

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Protected area 0.6

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0.0

Wildfire zone

0.0 0

200

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200

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MAR

Fig. 6. Average density of trees and fire frequency estimated for αt ¼ 5 and αg ¼ 0:1. Left: comparison between predicted average biomass density of trees and Sankaran's dataset (I ¼ 0; 6 MAR). Continuous line: dimensionless tree density Bt =kt ; dashed line: dimensionless tree carrying capacity kt S=kt ¼ S, corresponding to the steady state solution in the absence of fire. Right: comparison between predicted fire frequency (continuous line) and Sankaran's dataset (I ¼ 0; 6MAR).

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1.0

75

0.4

0.8 0.3 0.6 0.2 0.4 t

0.1 0.2

t

0.0

0.0 200

400

600

800

1000

200

400

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1000

Fig. 7. Left: dimensionless average density of trees (continuous line) and grass (gray line) versus MAR, estimated for αt ¼ 5 and αg ¼ 0:1, and different values of γt. Namely γ t ¼ 1 (same as in Fig. 6), 2, and 5 yr  1. Right: corresponding fire frequency as a function of MAR.

two eco-hydrological and fire regimes the calculated tree cover increases with water availability according to local and global experimental evidence (Sankaran et al., 2007; Staver et al., 2011b), semi-empirical (Buccini and Hanan, 2007) and theoretical (Accatino et al., 2010) models. The transition between the two states along a MAR gradient becomes less gradual at higher MAR values, as the negative impact of grass fire on tree density increases. The model originally predicts that the transition between dry and mesic savanna is characterized by a shift in wildfire frequency driven by major differences in soil moisture available for plants and vegetation composition. Although, the two-predator–two prey model demonstrates not only the link between two distinct ecohydrological regimes and the corresponding fire regime. The model evidences that (i) niche competition plays a role in determining the existence of the two fire regimes and (ii) the fire restriction of tree recruitment affects the transition between the two regimes. The model better predicted fire frequency at low MAR, suggesting that in the wet regime, the eco-hydrology-driven fire frequency may be dictated by a more complex interaction between climate, vegetation growth, fuel production and fire, most likely establishing at time scales shorter than 1 year. Several factors that are not evaluated here may increase the forecasting capability of the model and deserve further investigation: including the wide ranges of ecological conditions that characterize the data (Sankaran et al., 2007), a site specific model parameterizations, increasing the time resolution in the description of relevant processes involved in the soil moisture balance (RodriguezIturbe and Porporato, 2005; Fernandez-Illescas and RodriguezIturbe, 2004) and vegetation growth (Scholes and Archer, 1997). The variety of species in the community (Favier et al., 2012) (here reduced to two functional types), the change of physiological vegetation traits along the MAR gradient (Ratnam et al., 2011), and management practices (Buccini and Hanan, 2007) that have not been modeled here, certainly have a non-negligible impact on both tree density and the fire regime, which the model cannot capture in the present form. Accounting for these could most likely improve the match between the model outcome and experimental data. This study introduces a new conceptual framework for modeling wildfire in savannas, originally demonstrates that an eco-hydrologydriven predator–prey model may predict fire as a result of the savanna composition and represents a new contribution to the debate on the eco-hydrological dynamics of flammable savannas. The model was applied to MAR o 1000 mm yr  1 where climate

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