2.1. Modeling Soil Moisture Dynamics
Soil moisture dynamics is described by the following non-linear differential equation of water balance, written for a hydrologically effective soil profile of thickness
Z [
26,
27]:
where
t is time (in day),
n is the vertically averaged soil porosity, and
S (0 ≤
S ≤1) is relative soil moisture content (expressed as
S =
θ/
n, i.e., the volumetric soil-water content,
θ, normalized by soil porosity,
n) averaged over the entire depth
Z (in mm) of the soil control volume. We attach a more functional than physical meaning to the soil control volume of depth
Z as it represents the hydrologically active, uniform soil profile where the evapotranspiration process plays a dominant role. Equation (1) is a stochastic model of soil moisture dynamics at a point since we treat precipitation (
P) as a Poisson stationary random process characterized by the inter-arrival time,
τ (in days), between independent precipitation events, and the daily precipitation depth,
p (in mm), and duration,
tp (in day) [
8].
The right-hand side of Equation (1) comprises the following incoming and outgoing fluxes:
where
P is precipitation,
CI is canopy interception,
Q is surface runoff,
E and
T are actual evaporation and transpiration, respectively, and
L is leakage (i.e., the drainage losses) from the lower boundary of the soil profile. The flux in Equation (1) has dimension LT
−1 and, unless otherwise specified, the linear dimension has units of “mm”, whereas the time dimension has units of “days”.
Actual rates of soil evaporation, E, actual plant transpiration, T, and leakage, L, are considered only as a function of the spatial average soil saturation (S) and season of the year, whereas surface runoff, Q, is generated according to the saturation-excess or infiltration-excess mechanisms depending on the soil condition at time t, which is influenced by previous fire occurrence under the assumption that the effect of fire on soil properties (such as a reduction in soil infiltration capacity due to the loss of soil wettability after the formation of a water repellent uppermost soil layer) lasts one year. Actually, variable π[S(t), t] of Equation (2) includes the portion of precipitation (P) that infiltrates into the control volume through the soil surface, after the subtraction of canopy interception (CI) and surface runoff (Q) if they both occur.
In the absence of a fire event, the interception by vegetation canopy is modeled rather simplistically such that
ci(
t) = min{Δ,
p(
t)}, where Δ (in cm) is a threshold value of precipitation depth below which no water reaches the soil surface [
9,
28]. Crown fire reduces the interception capacity of the overstory, also in accordance with Caylor et al. [
29] who suggested that the amount of interception is proportional to the leaf area index and the number of canopies present. The threshold is calculated as follows:
where
Bu and
Bl are the biomass density of the upper and lower vegetation layers, respectively. Therefore, the depth of net rainfall,
r′(
t), that reaches the soil surface is as follows:
Obviously, when no vegetation interception occurs and/or just after a fire event (i.e., when Δ=0), the net rainfall (i.e., throughfall) is
r′ =
r =
p (see
Table A1).
The infiltration capacity of soil,
f(
t), undergoes a substantial reduction after a fire event [
30,
31]. The Hortonian process for overland flow generation occurs when rainfall intensity
j(
t) =
r′(
t)/
tp(
t) exceeds soil infiltration capacity,
f(
t), with
tp(
t) being the duration of the rainfall event. Therefore we use the following relation:
where
S0 is the relative soil moisture in the control volume at the beginning of a rainfall event.
Consequently, overland flow (
q) can be generated by either saturation-excess or infiltration-excess mechanisms and is computed as follows:
Severe fires, which typically ignite at the end of summer, can alter the soil structure and hence increase the imperviousness of recently burned soils. This will limit the amount of rainfall that infiltrates over the subsequent rainy period (autumn season), and therefore replenishes the soil profile and later on becomes available for vegetation [
17,
31,
32,
33,
34,
35,
36].
The biomass density of the upper and lower vegetation layers, namely Bu and Bl, is limited by the local environmental conditions other than soil moisture availability and fire to the carrying capacity ku and kl, respectively. The carrying capacities of the two layers are used to derive dimensionless biomass density B*u=Bu/ku and B*u=Bl/kl.
The impact of a fire on soil properties lasts for about one year after fire [
37,
38,
39,
40,
41]. With
R*u and
R*l being the dimensionless burning biomass density, i.e., the maximum amount of biomass burned normalized to the carrying capacity according to the living biomass density (see [
24]), our modeling approach considers that, any time the during the one-year time lag (
t–1yr;
t), the partitioning of precipitation into overland flow is affected by a reduced soil infiltration capacity according to the following expression:
where
f0 is soil infiltration capacity in the absence of fire and:
The parameter
Kϕ depends on the soil composition and vegetation cover, and accounts for the degree of imperviousness induced by the fire event. Ursino and Rulli [
23] presented an extensive sensitivity analysis of the fire regime for
Kϕ ranging from 0 to
Kϕ>>1 (actually, this parameter is set at 1 in the present study; see
Table A1 in the
Appendix A).
With a view to the modeling objectives of this study, namely to evaluate the impact of hydrological processes on fire regime, we follow Guswa et al. [
26] who suggested separating the actual evaporation at the soil surface,
E(
S), from the actual transpiration by plants,
T(
S). These two variables are computed as follows:
In Equation (10),
S** is a soil moisture threshold below which a reduction in evaporation rate occurs [
42], whereas
Sh is the so-called hygroscopic moisture content, namely the average soil moisture content when soil suction head at the soil-atmosphere interface (|
ψs-a|) is low enough for evaporation from the soil surface to cease. Soil suction head |
ψs-a| is often set at a value ranging from 150⋅10
3 to 500⋅10
3 cm, and we posit |
ψs-a|=500⋅10
3 cm in the present study. In Equation (11),
Swp is soil moisture at permanent wilting condition (i.e., the wilting point) and
S* is soil moisture at incipient stomatal closure, and we assume that both these parameters take on different values for overstory (e.g., olive or chestnut trees) or understory (e.g., shrubs). The exponents
e1 and
t1 featuring in Equations (10)–(11) account for possible nonlinearity in these relationships and here are both set equal to 1.0 [
19]. Values for these parameters are reported in
Table A3 and
Table A4 in the
Appendix A.
Evapotranspiration of the two vegetation types is computed as follows:
where
E(
S) is actual evaporation flux estimated according to Equation (10), but note that this variable takes on different values during the wet or dry season because vegetative activity is assumed to take place during the dry season [
20]. See
Table A5 in the
Appendix A for the relevant parameter values used in these equations.
The vertically lumped bucket model assumes that the drainage rates occur under the condition of the unit gradient of the total hydraulic potential and are expressed as a function of the soil saturated hydraulic conductivity,
Ks, as follows:
where γ is the soil-pore/connectivity parameter and
Sfc is the relative soil moisture at “field capacity” [
20]. The determination of the
Sfc value deserves some comments. Since the model control volume is not an actual (mostly, layered) soil profile, but rather an equivalent uniform soil, in this study we take advantage of the recent findings made by Nasta and Romano [
43], who set up a functional evaluation and an analytical procedure to identify the effective value of soil moisture at field capacity in the case of an actual layered soil profile.
2.2. The Modified Predator–prey Model for Fire Dynamics
Predator–prey interactions are often used to interpret density-dependent limiting factors occurring in a certain environment, and the related analyses have proved to be quite successful especially in behavioral ecology studies to describe the patterns of time variations of the investigated variables or species [
44,
45]. Following suggestions made by some researchers, predator–prey models rapidly attracted the attention of researchers, managers, and professionals who had to deal with vegetation fires and their dynamics [
46]. One feature of a predator–prey model that interested us was its potential to serve as a stochastic tool for a system of two competing attributes, which makes it very suitable to be coupled with the stochastic description of soil moisture dynamics offered by Equation (1).
Previous predator–prey models developed to investigate fire regimes were based on average annual precipitation and therefore simulated annual water balance [
23,
24]. Instead, given the chief aims of the present study, the stochastic precipitation variables, i.e.,
p,
tp, and
τ, are independent and exponentially distributed with season-dependent averages. During the wet or dry season, the average precipitation amount, the reciprocal of precipitation duration, and inter-arrival time are denoted by the symbols
ζ,
δ, and
λ, respectively, with the subscript “
wet” or “
dry” that refers to the specific season considered (see
Table 1). As the wet seasons of Mediterranean climates are typically out-of-phase with vegetative growing periods, the amount of precipitation potentially involved in the water balance is restricted approximately to the amount of water that is stored in the soil profile at the beginning of the growing season plus the rainfall during the dry season. Note that the values for variable
I can undergo a reduction over the first few years following a fire event because a burning event usually creates a nearly impermeable soil layer that reduces the amount of water entering the soil control volume.
Frequency and magnitude of fire are dictated by fuel availability and environmental dryness, hence ultimately by climate indirectly through its influence on vegetation growth and directly under conditions of the high flammability of fuel. To reduce the number of dependent variables to only two variables per species (living and burning biomass density), the burning biomass is the “predator” and the living biomass is the “prey”, even though it is the dry matter that becomes fuel for a fire at the end of the dry season. The amount of fuel available every year is assumed to be proportional to the living biomass.
The four dimensionless balance equations that determine the predator–prey dynamics of the living and burning trees and shrubs (upper and lower layer or overstory and understory, respectively) are the following:
During the dry season,
Gu and
Gl are biomass logistic growth functions. At very low values of
S, corresponding to prolonged conditions of water scarcity and droughts in the ecosystem, vegetation does not grow and does not produce fuel. At higher
S values, instead, it is not flammable. The two situations mentioned above are synthesized through the following analytical expressions:
The net primary productivity (NPP) of Mediterranean forest ranges between 0.5 and 1.5 kg·m
2·yr
−1, whereas the NPP of Mediterranean scrubland ranges between 0.3 and 0.6 kg·m
2·yr
−1 [
47]. The parameters
ru and
ku, as well as
rl and
kl, characterize the vegetation growth rate (
ru and
rl) and carrying capacity (
ku and
kl) over their area of occupancy, according to the referred literature data. The biomass densities
and
represent each species’ abundance, whereas the burning biomass density (
Ru and
Rl) is responsible for the impact of fire on hydrological processes.
In Equation (15), ru is proportional to yield and the ratio Tu/Tmax,u, whereas rl to species-specific yield and Tl/Tmax,l. Actual transpiration is calculated as a function of relative soil moisture soil saturation through Equation (11) and accounts for the lack of biomass production due to the water stress occurring during very dry years.
When enough fuel is available and the environment is sufficiently dry, fire develops as soon as
Fu,l<
Du,l. The burning biomass of each layer attacks the living biomass of both layers, igniting a fire with severity that is inversely proportional to the soil moisture:
where:
Soil moisture content (
S) is interpreted as a proxy of the moisture content of the plant biological tissues that inhibit fire development [
48,
49,
50,
51,
52]. High parameter
χ could be used for less drought-tolerant species, drying out quickly when soil moisture availability decreases. Even moderately low soil moisture values favor the development of fires. Low parameter
χ restricts the development of fire only to very dry conditions, representing the behavior of more drought-tolerant and less flammable species. When soil moisture exceeds the threshold
Sfire, then no fire can develop.
Burning plants of the two layers become extinct at a given rate:
The symbols
δu and
δl are the fire extinction rates in the tree and shrub layers, respectively, that are typical of the Mediterranean ecosystem under consideration. The dynamics of burning species are much faster than those of living species. Parameter values are chosen according to previous literature contributions and specified in
Table A5 of the
Appendix A. The vegetation is dormant during the rainy season and
Gu =
Gl = 0 and fire does not ignite spontaneously, namely
Fu =
Fl =
Du =
Dl = 0.
2.3. Scenarios for Sensitivity Analysis
Equation (1) requires as input information the daily precipitation that indirectly drives biomass growth and likely fire occurrence through dryness and fuel abundancy. Within Campania, reference was made to the weather stations of Salerno and Gioi Cilento. Salerno is a city by the sea and its station has long time-series of daily rainfall data that are employed here as suitable information for the Amalfi Coast and the Vesuvius National Park. Instead, the village of Gioi Cilento is situated in the Cilento, Vallo di Diano and Alburni National Park.
The weather station of Salerno (X-UTM: 479,039 m; Y-UTM: 4,503,239 m) is located at 13 m above sea level (a.s.l.), whereas the weather station in the village of Gioi Cilento (X-UTM: 518,534 m; Y-UTM: 4,460,028 m) is located at 668 m a.s.l. Therefore, the two stations can be viewed as representative of precipitation regimes occurring near the coastal areas and in hilly zones, respectively, of the region in question. These stations have values of the seasonality index (SI; [
53]) in the range 0.40–0.59, meaning that the precipitation regime is rather seasonal with a short dry season.
To provide the reader with a clear understanding and a less biased perspective of observed dry spells in a typical zone of Mediterranean Europe, we computed the Standardized Precipitation Index (SPI; [
54]) for several weather stations located in the southern Italian region of Campania. Computing SPI is highly recommended by the World Meteorological Organization to characterize the meteorological drought, and the use of a standardized indicator helps compare the outcomes from various stations. SPI values quantify the precipitation deficit (negative values) or surplus (positive values) with respect to the median value in the observed period. According to the SPI classification of drought conditions, a period is severely dry for SPI values ranging from −1.5 to −1.99, whereas it is extremely dry when SPI values are lower than −2.00.
Figure 1 and
Figure 2 depict the three-month standardized precipitation index (SPI-3) for the two weather stations of Salerno and Gioi Cilento by using the daily rainfall data recorded from 1920 to 2018 at these points. No rainfall data were recorded at these stations during the Second World War and for a few years after its conclusion. We selected an accumulation period of three months for SPI since this time scale seems to reflect more medium-term soil moisture conditions and takes seasonal precipitation regime into due account. In both bottom panels of
Figure 1 and
Figure 2, the line segment in magenta connects the median values of SPI-3 and highlights the occurrence of more frequent precipitation anomalies slightly after the year 1990 (in a few cases close to -1.0 for Gioi Cilento and even greater than -1.0 for Salerno). Moreover, in the recording period from 1990 to 2018, the median dry anomalies are greater than the wet ones for both Salerno and Gioi Cilento weather stations.
The impact that the typical seasonality of a Mediterranean precipitation regime may exert on the time evolution of the occurrence of wildfires is evaluated by identifying two different precipitation scenarios, referred to as S1 and S2. It is worth noting that the individual parameter values attached to these seasonal precipitation regimes rely on actual long-term rainfall records available from about 250 weather stations located throughout Campania and can be conveniently viewed as representative of rainfall situations occurring in differently located zones of the study region (see
Table 1).
To account for intra-annual rainfall seasonality, both scenarios S1 and S2 in
Table 1 refer to a conventional hydrological year, starting on April 1st. Based on datasets available in the literature (e.g., [
7,
20]), precipitation scenario S1-A has a mean annual precipitation (
Pyear) of more than about 1150 mm/yr, which can be considered as representative of average conditions occurring in some Mediterranean hill zones. Instead, precipitation scenario S2-A has a mean annual precipitation of approximately 550 mm/yr which occurs more frequently in southern, coastal zones of this region.
Table A2 reports the Poisson parameters pertaining to these two precipitation scenarios. Both scenarios S1-A and S2-A refer to a hydrological year that is split into a dry season, lasting six months (namely 182 days from April to September of a certain year), typically characterized by fewer precipitation events and vegetation re-growth, and a wet season, lasting the other six months (namely 183 days from October to March of the subsequent year), when vegetation is virtually dormant and typically characterized by more precipitation events. Scenarios S1-B and S2-B use the same Poisson parameters as the previous cases, but refer only to an arbitrary (albeit realistic) increase in the number of days (NDdry) from 183 to 245 over a dry period (i.e., a dry period lasting nearly 8 months). It should be pointed out, however, that we made the simplistic assumption that the same Poisson parameter values are held in the cases considered.
To overcome the limits of simulations based on only one realization of the rainfall process over the observation time of 50 years, we further address the probability of achieving a certain forest composition under prescribed stochastic climate conditions by analyzing much longer simulation runs (e.g., 10,000 runs) within a Monte Carlo approach. We addressed the frequency histograms of the main dependent variables for both scenarios S1 and S2 as well as for the two soil types (loam and sandy clay loam soil, respectively).