Temporal fire disturbance patterns on a forest landscape

ELSEVIER

Temporal fire disturbance patterns on a forest landscape

Ecological Modelling 99 (1997) I37- 150

Chao Li *, Michael Ter-Mikaelian, Ajith Perera

Ontario Forest Research Institute, P.O. Box 969, 1235 Queen Street East, Sault Ste. Marie, Ontario P5A .iV5. Canada

Accepted 12 November 1996

Abstract

Potential temporal fire disturbance patterns on a forest landscape were investigated using a fire regime model with four different fire probability functions: (1) forest age-independent; (2) hyperbolic increase with forest age; (3) sigmoid increase with age; and (4) linear increase with age. Different combinations of parameter values for a logistic equation were used to approximate different fire probability functions. An extensive mode1 behavior study suggested that fire regimes similar to the observations in Ontario could result from any of the fire probability functions, but with different parameter values. Simulation results on the case study area indicated that when the fire rotation period was fixed to 200 years (corresponding to fire regimes in the southern part of northwestern Ontario), the predicted temporal disturbance patterns (average interval between two successive fires) were similar for small and intermediate fires, but different for large and severe fires. The results from the fire probability function with a sigmoid shape appeared the most appropriate among the four tested fire probability functions. The average interval between two successive fires for each size group is: small fires every 5.8 years, intermediate fires every 34.4 years, and large fires every 151.6 years. Better prediction of a temporal disturbance pattern, especially for large and severe fires, will require an explicit understanding of the quantitative relationship between fire probability and forest age. 0 1997 Elsevier Science B.V.

Keywords: Fire regime; Ontario; Landscape structure; Disturbance pattern; Simulation mode1

1. Introduction

Forest dynamics are profoundly influenced by their associated disturbance regimes or the tempo- ral and spatial pattern of the creation of open or altered patches (Pickett and White, 1985). Tempo-

* Corresponding author. Tel.: + 1 705 9462981; fax: + I 705 9462030; e-mail: [email protected]

ral patterns of disturbance regimes may be crucial in determining the demographic characteristics of landscape-level dynamics. For example, in a sys- tematic exploration of the effect of varying the rate of disturbance, the size of individual distur- bances, and the temporal and spatial autocorrela- tion among individual disturbances on the distribution and abundance of competing plant species, Moloney and Levin (1996) found that the

0304-3800/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO304-3800(96)01944-S

138 C. Li et al. / Ecolqical Mode&g 99 (1997) 137-15-O

biggest impact on species abundance of a serpen- tine grassland occurred when the temporal auto- correlation structure of the disturbance regime changes.

The demographic dynamics of forest ecosys- tems may also be largely influenced by the tempo- ral disturbance patterns that set up timers for forest succession processes and produce spatial- temporal mosaics of stands within ecosystems. Since a newly burned stand can be less likely to burn than a non-burned stand, the forest may passively ‘respond’ to fire disturbances through their effects on the possible fire processes of fol- lowing years. Forests may then recover their ‘nor- mal’ susceptibilities to fire disturbances in subsequent successional processes. The speed of this recovery may vary with different types of forests.

Based on these processes, different fire proba- bility functions (indicating forests passive ‘re- sponse’ patterns to fire disturbances) have been assumed in various fire models. For example, fire probability was assumed to be forest age-depen- dent, that is: (1) to increase with forest succes- sional stage in a hyperbolic manner to an asymptote (Antonovski et al., 1992); (2) to in- crease with forest age in a sigmoid way to its maximum value (Peterson, 1994); (3) to increase linearly with forest age (Baker et al., 1991); (4) to increase quadratically (Ratz, 1995); (5) to increase first to its maximum value and then decrease (Gardner et al., 1996); and (6) to decrease first to its minimum value and then increase in a sigmoid pattern (Peterson, 1994). An age-independent fire probability was also assumed in several fire mod- els (e.g. Van Wagner, 1978). The comparison of simulation results under different fire probability functions, however, was never carried out using a single fire model. This raised the question of whether the simulated landscape dynamics and associated fire regimes would be changed using different fire probability functions.

In this paper, we test the hypothesis that there is no significant difference among temporal distur- bance patterns predicted using different fire prob- ability functions. This is tested using a developed fire regime model that is able to simulate fire regime and landscape dynamics under different

fire probability functions, without modifying the model’s structure. Four different fire probability functions were evaluated: (1) age-independent fire probability; (2) hyperbolic increase age-dependent fire probability; (3) sigmoid increase age-depen- dent fire probability; and (4) linear increase age- dependent fire probability. For simulating a fire regime on a case study area, the model uses GIS information for the area as input. If the hypothe- sis is true, then simulated temporal disturbance patterns on the case study area under different fire probability functions should not be significantly different. The fire regime model will be presented first, followed by a sensitivity analysis showing the model’s behavior. The hypothesis will then be tested based on a fixed fire rotation period (i.e. the time required to burn a total area that is equiva- lent to the study area) by comparing the simula- tion results under the four assumed fire probability functions.

2. Model development

2.1. General model structure

The model represents a forest landscape using a grid of equal-sized rectangular cells with m rows and IZ columns. In the case study presented in this paper, each cell represents an area of 1 ha that could be covered by forest, lake, or other land forms. The number of rows and columns could be equal or unequal, depending on the study area being investigated, and both are user-defined: they are limited only by the computer hardware’s abil- ity to complete a simulation in an acceptable period of time.

Fire hazard is represented by fire probability. A baseline map of fire hazard is the spatial distribu- tion of fire probabilities across the forest land- scape. Fire probabilities are assumed to be a function of forest stand age (i.e. time since last burn), which is related to fuel accumulation. The stand ages 1 year after a yearly time step, unless it is burned and then the age is reset to zero.

To consider the interactions among fire, forest, topography, and weather, two components are essential: fire hazard maps that indicate fire sus-

C. Li et ul. /Ecological Modelling 99 (1997) 137- 150 139

ceptibilities across the landscape and a fire-spread algorithm. Using this algorithm, the final fire shapes and sizes will be simulated in a natural way. Therefore, model development concentrated on: (1) constructing baseline fire hazard maps; (2) modifying the maps by fuel types, topography, weather etc.; and (3) implementing fire initiation and spread processes based on the modified maps.

2.2. Construction of baseline fire hazard maps

The probabilities of fire initiation and spread were assumed to be functions of cell age (if the cell is covered by forest), which accounts for changes in fuel quantity. This age-dependent fire probability has been described by different func- tions in the literature. For example, Baker et al. (1991) assumed a linear fire probability function in their DISPATCH model; Antonovski et al. (1992) used a discrete form of hyperbolic fire probability function in their FORLAND model; various shapes of fire probability functions, including the discrete form of sigmoid shape and discrete form of nega- tive exponential followed by a sigmoid shape, were used in Peterson (1994) prototype and fuel models; in EMBYR (Gardner et al., 1996) the fire probability was represented by discrete forms of either hyperbolic or domed shapes; age-indepen- dent fire probability was used in the Van Wagner (1978) model and Peterson (1994) Null model as a constant probability regardless of forest age. All of the parameter values were estimated based on qualitative knowledge of fire spread processes.

In our model, the age-dependent fire probabil- ity function was described using a logistic equa- tion (mathematically, it can approximate different function shapes by varying combinations of parameter values, see examples in Fig. l), as follows:

P=k/(l +exp(a-b x Age)) (1)

where a, b and k are parameters. The parameter a determines the age where the slope starts to in- crease; b controls the speed of slope increase; and k defines the asymptote of the function curve. The assumption behind this equation is that the amount of accumulated fuel is proportional to the yield of forest (usually described by an S-shaped

asymptotic curve, also called a sigmoid curve, Vanclay, 1994), i.e. an old stand is more likely to be burned than a young stand in the presence of a source of initiation. This equation may also be better suited for fire probability because the prob- ability is constrained between 0 and 1.

For a given cell, the probability of fire spread (i.e. due to its proximity to a burning cell) was assumed to be higher than the probability of fire initiation (e.g. due to a lightning strike). This assumption relies on a comparison of the chances of a given cell burning under two scenarios: (1) a single tree within the cell burns due to a lightning strike, and the initial amount of heat for burning the whole cell is only available from this single tree; (2) for a cell next to a burning cell, the initial amount of heat arises from an entire burning cell, not just a single tree. In scenario 2, the chance that the cell burns is likely higher. Therefore, the model uses two sets of parameter values for Eq. (1) to characterize the relationship between the probabilities of fire and cell age, called the base- line fire initiation probability, PBasrinitiate, and the baseline fire spread probability PBasespread (Fig. 2a).

0.7

0.0 0 50 100 150 300

Age

Fig. 1. Examples of various age-dependent fire probability function shapes approximated by Eq. (1). Curve approximates an almost constant situation with parameter values of 0.6. 4. and 2 for k, a, and b, respectively; hyperbolic shape: 0.6, 2, and 0.1 for k, a, and b, respectively; linear shape: 0.65, 2.5, and 0.025 for k, a. and b, respectively; and finally, sigmoid shape: 0.6, 2.5, and 0.05 for k, a, and b, respectively.

140 C. Li et al. /Ecological Modelling 99 (1997) 137-150

(a) (W

Fig. 2. Baseline fire probability curves and their modifiers used in the fire regime model: (a) baseline fire probability curves for ignition and spread; (b) influence of slope modifier; (c) influence of primary soil moisture types; and (d) weather influence indicated by dryness of fuel.

2.3. Mod@ation of baseline fire hazard maps

The actual fire initiation and spread probabili- ties (Plnitiation and PSpread, respectively) are not merely determined by forest age, but also influ- enced by several factors as follows:

PInitiation = p Basemitiate F Fueltype F- F Moisture Weather (2)

PSpread = P F F F- F Basespread Fueltype Slope Molsturc Weather

(3)

where FFurltype, FSlopeY FMOIStUTe and Fweathcr are the modifiers representing the effects of fuel type, slope, soil moisture regime, and weather condi- tions.

Fuel types were assumed to correspond to forest canopy cover types. The information about canopy cover types was obtained from the land cover GIS database of the Forest Landscape

Ecology Program of the Ontario Forest Research Institute. There are 16 land cover types in the database (Spectranalysis Inc., 1992). Different fuel types influence the rate of fire spread (ROS), which is a measurement of the speed of fire spread under equilibrium conditions (Forestry Canada Fire Danger Group, 1992). High ROS, therefore, may indicate a high probability of fire. If the fire probability could be assumed as proportional to the ROS, then the effects of fuel type on baseline fire probabilities can be estimated under a fixed initial spread index (ISI, which is a function of weather conditions) for various canopy cover types. These effects are the measurements of rela- tive potential in fire initiation and spread under similar weather conditions. Therefore, they can serve as modifiers for the base fire probabilities. Table 1 shows the values of the modifiers corre- sponding to the canopy cover types used in the simulations. Assuming a ‘black box’ type of suc-

C. Li et al. ‘Ecological Modelling 99 (1997) 137-- 150 141

cessional pathway (which implies that under a stable climate, succession will result in the same forest cover as pre disturbance) (Suffling, 1983; Ratz, 1995), land cover type for each cell was assumed to not change after a fire disturbance.

2.3.2. Fs,,,,, Effect of topography on the fire probability was

directly from Van Wagner (1977, 1988) results. Van Wagner investigated the effect of slope, both uphill and downhill, on fire spread processes. In the uphill cases, he obtained a spread factor equa- tion in which the relative spread rate was calcu- lated based on a factor of 1 for level ground:

F s,ope = exp(3.533(tan S)‘.2) (4)

where S is the radian angle between slope and horizontal. Van Wagner (1988) also obtained a spread factor equation for downhill situations:

F Slope = (1 - 0.033A + 0.000749/12) x cos(S) (5)

where A is the slope angle. Fig. 2(b) shows the slope effects used in the simulations.

2.3.3. I;Moisture The index of primary soil moisture from the

Ontario Land Inventory was used to imply the general moisture content of fuel near the soil surface. Primary soil moisture is an indication of the average amount of water available to the rooting zone over the growing season, i.e. the

Table 1 Modifiers of fire probability due to different canopy cover types used in the simulations

Canopy cover type Modifier of fire probability

Dense jack pine 0.9665 Dense black spruce 0.9165 Sparse coniferous 0.5649 Mixed coniferous 0.9415 White pine 0.3034 Red pine 0.3034 White and red pine 0.3034 Mixed deciduous 0.2202 Dense deciduous 0.2202 Sparse deciduous 0.1321 Poorly vegetated area 0.0793 Recent cut and burn 0.0793

long-term average of moisture regime (Ontario Ministry of Natural Resources, unpublished data). There are five categories in the moisture regime classification: (1) dry, deficiency of soil water over the growing season; (2) fresh, optimum soil water for most forest (tree) species; (3) moist, slight excess of soil water; (4) wet, large excess of soil water over the growing season; and (5) satu- rated. Since the quantitative description of the effects of primary soil moisture on fuel moisture (hence the fire probability) were not found in literature, a hypothetical relationship based on qualitative knowledge was used in the simula- tions. The model assumed that the baseline fire probability curves would be modified by the pri- mary soil moisture regime in an approximate lin- ear manner as shown in Fig. 2(c).

Using the primary soil moisture index to indi- cate fuel moisture is probably suitable for long- term simulations (i.e. decades to centuries), rather than short-term (e.g. within a fire season) fire regime dynamics. For example, if a cell within the landscape is classified as dry, then it would indi- cate that not enough water content will be avail- able for most fire seasons. Thus, the moisture content of fuel would be lower, and the probabil- ities of fire initiation and spread would be higher during most fire seasons. On the contrary, if a cell is classified as moist, wet, or even saturated, then higher fuel moisture content would likely result from enough water content in the soil and the probabilities of fire initiation and spread would be lower.

2.3.4. F,,,,<>urhrr The model assumes that the fire initiation and

spread probabilities are inversely proportional to relative humidity and time since the last rainfall. This assumption was partially based on Swetnam (1993) results that regionally synchronous fire oc- currence was inversely related to yearly fluctua- tions in precipitation. A stable climate scenario was assumed at this stage of model development. The annual index of weather conditions, repre- sented by the average dryness of fuel in a given fire season (year), which depends directly on the weather, was simulated as a random event that follows a normal probability distribution. For a

142 C. Li et al. /Ecological Modelling 99 (1997) 137-150

stable climate scenario, therefore, the mean value of the distribution was set to 1. Simulated yearly indices of weather conditions served as modifiers of the baseline fire probability functions (Fig. 2(d)).

The normal distribution may be a reasonable choice for simulating fluctuations in fuel moisture and hence fire probability. For example, Renkin and Despain (1992) showed that the relative fre- quency of thousand-hour timelag fuel moisture (THRFM, a measure of long-term moisture con- ditions) values approximated a normal distribu- tion during the fire seasons of 1965- 1988 in Yellowstone National Park.

By incorporating the weather effect, the overall fire probabilities will rise when weather favors fire occurrences, which reduces the importance of spa- tial fuel configuration in controlling the fire spread processes. This treatment was to account for the fire occurrence patterns observed in Yel- lowstone National Park (Turner and Romme, 1994), i.e. although spatial fuel configuration may control the spread of fires, this constraint could be overridden when weather in a particular year (or month) extremely favors fire processes.

2.4. Implementation of fire initiation and spread processes

A fire process can be divided into two stages: initiation and spread (Li and Apps, 1996). At the initiation stage, a fire source, such as a lightning strike, ignites a fire and a certain area of forest burns. This area corresponds to the grain of the landscape model. Whether an ignition initiates a fire depends on the cell’s condition. Most of the forest in the cell will burn once a fire is initiated; in other words, it would not be referred to as an initiation if most of the forest in the cell remained intact.

Once a fire initiates, it may spread to neighbor- ing cells, including four cells with adjacent com- mon edges and four cells with adjacent common corners. The fire-spread stage extends from the time a fire is initiated until it stops expanding in any direction or reaches the boundaries of the defined landscape.

Fig. 3. Mean annual fire sources in Northwestern Ontario for an area of 40 x 40 km’. Black circle with solid line, lightning fires; open square with dashed line, fires caused by human activities; and black triangle with dashed line, total fire source. Data were provided by the Fire, Flood and Aviation Branch of the Ontario Ministry of Natural Resources.

2.4.1. Fire initiation Three conditions must be met for a fire initia-

tion to occur: appropriate quantity and quality of fuel, and a source of fire (e.g. Li and Apps, 1995).

The main source of fire in Ontario is lightning strikes. More than 90% of the area burned in boreal Ontario is the result of fires caused by lightning (Johnson, 1992). A data set of historical fire occurrence in northwestern Ontario (1976- 1994), provided by the Fire, Flood, and Aviation Branch of the Ontario Ministry of Natural Re- sources, was used to analyze the dynamics of the annual number of fire sources. The data set con- tains 11295 records that occurred across a total land area of 144960 km2. The fire records were then grouped into 32 areas, each 40 x 40 km2. For each area, the records were subgrouped by causes: lightning or human activity. Fig. 3 summarizes the mean annual number of fire sources. A time series analysis (Venables and Ripley, 1994) was applied to the mean number of fire sources over the 32 areas in three data sets (lightning, human and total fires). A 3.6-year cycle was the most significant fluctuation in all three data sets. The second significant cycle was about 2.3-2.6 years. A 9-18-year cycle may exist, but the data sets only covered 19 years; not long enough to confirm the cycle. Based on this analysis, a stochastic simulated time series with similar cyclic pattern of fluctuation and amplitude (Venables and Ripley,

C. Li et al. 1 Ecologicul Modding 99 (1997) 137-150 143

1994) was used in the simulations to represent the potential annual number of fire sources.

The model assumes that the locations of fire sources (mainly lightning strikes) are randomly distributed within the landscape. The actual lo- cations of fire initiations are thus determined primarily by the fire initiation probabilities of the cells, Prnic,ation. Consequently, simulated numbers of fire initiation were usually less than the numbers of fire sources, which influenced mean fire size estimates.

2.4.2. Fire spread The model uses a percolation algorithm to

simulate fire spread. The percolation algorithm has been widely used and discussed in fire dis- turbance models (e.g. Albinet et al., 1986; Von Niessen and Blumen, 1986; Gardner et al., 1987; Gardner and O’Neill, 1991; Ratz, 1995). Based on a hypothetical landscape with a single cover type, Ratz (1995) compared simulation results under two variants of the percolation algorithm: fire spreads to its four nearest neighbors or to its eight adjacent cells. The results were almost identical for fire structure and system behavior, but the parameter values for the two variants differ.

The algorithm used in our model was that fire will spread to all eight neighboring cells. This helped to account for the effect of slope on spread processes, because fire probabilities differ when fires come from various directions with different elevations (Eqs. (4) and (5) Fig. 2(b)). Whether an unburned cell would be burned due to its proximity to a burning cell, in theory, should be determined by the fire spread po- tential of the burning cell and the susceptibility of the unburned cell. We assumed that once a fire initiates in a cell, the potential of fire spread from this cell is the same regardless of cell age. By this simplification, fire spread to an adjacent cell will be determined by the susceptibility of the adjacent cell (if it is cover- ed with forest), called the probability of fire spread.

The model was written in FORTRAN 77 and run on a Dncstation 5000. The execution of a

single stochastic simulation on a 100 x 100 cell landscape (1200 time steps, i.e. years) took about 3.5 h of CPU time. The landscape’s forest age structure was randomly initialized at the be- ginning of each simulation run. The results of the first 200 years were discarded to eliminate the possible effects of initial conditions. The cal- culation of mean fire sizes and rotation periods were the average of the next 1000 years’ simula- tion results.

3. Case study area and fire regimes in Northwestern Ontario

Simulations were carried out based on a real forest landscape in Northwestern Ontario. The case study area is 10 x 10 km2 with a spatial resolution of 1 ha, located in UTM Zone 15, between 5 580 900 and 5 590 900 in northing and between 424000 and 434 000 in easting. Fig. 4(a) shows the map of the land cover type for the study area. From a landscape analysis of the study area, 77.11% of the area is identified as covered by forest, of which 82.99% are conifers. The rest of the study area is covered by water (15.18%) areas where forests were recently cut or burned (4.37%) etc.

Fig. 4(b) is an elevation map for the case study area. Elevation data assembled from digi- tal terrain elevation data (DTED) produced by the U.S. Defense Mapping Agency (Spectranaly- sis Inc., 1994) indicated a range from 320 to 428 m above sea level.

Fig. 4(c) is a map of soil moisture regime for the case study area. Only three moisture regime categories (dry, fresh, and moist) existed in this particular area.

Fire regimes can be described by mean fire size and fire rotation period (Heinselman, 1981; Johnson, 1992; Romme, 1982). Estimated mean fire size for the southern part of northwestern Ontario, where the case study area is located, is about 95.4 ha and the mean fire rotation period is about 198.7 years (unpublished data from the Fire, Flood, and Aviation Branch of the On- tario Ministry of Natural Resources).

144 C. Li et al. /Ecological Modeiiing 99 (1997) 137-150

Elevation

320

428

Soil Moisture

Fig. 4. The maps of vegetation cover (a), elevation (b), and soil moisture (c) for the study area, which were used as a model input.

C. Li et ul. /Ecological Modelling 99 (1997) 137-150 145

4. Sensitivity analysis

Behavior of the developed model was exten- sively examined through comparisons between simulation results and observed fire regime in northwestern Ontario. The characteristics of ob- served fire regimes were used as criteria for justi- fying model behavior. Many characteristics of fire models, such as spatial pattern formation and dissolution, and self-organization, have been ex- amined and discussed (e.g. Baker, 1992, 1995; Wissel, 1992; Peterson, 1994; Ratz, 1995; Holling et al., 1996). In this sensitivity analysis, we exam- ined the influences of different fire probability function shapes, soil moisture regime and varia- tion in annual weather conditions on the simu- lated mean fire size and rotation period.

4.1. Shape of jive probability function

Model behavior in terms of mean fire size and rotation period was examined under four fire probability functions (Fig. 1). These fire probabil- ity functions can be approximated by varying combinations of parameter values of the logistic Eq. (1). Forest landscapes in northwestern On- tario are usually separated by a large number of small lakes, so that different fire regimes could be expected from various lake distributions. In this sensitivity analysis, the model was run with the GIS information for the case study area as input, but without land cover type. This treatment al- lowed the investigation of fire regimes on non- fragmented forest landscape, which is comparable to theoretical studies reported in literature. Al- though the exact k values could be different under various types of landscape fragmentation, the qualitative conclusions would not change.

A systematic approach was used for exploring model behavior under the four different fire prob- ability functions. Since the combination of parameter a and b in Eq. (1) determined the fundamental shape of the functions, the sensitivity analysis was carried out based on ranges of k value that lead to results comparable to observed fire regimes in Ontario. Five replications for each of the k values under each of the four fire proba- bility functions were used to obtain estimations of mean fire size and rotation period.

Model behavior could be changed by varying the k values in each of the four fire probability functions. Observed fire regimes in Ontario can be simulated by any of the four fire probability func- tions, but with different ranges of k. The results are summarized as follows.

The first fire probability function that is invari- ant with forest age came from the Van Wagner (1978) assumption. Under this assumption, only a narrow range of k (around 0.2) for Eq. (l), could result in the mean fire sizes and rotation periods that are comparable to those of observed fire regimes (Fig. 5(a)). When k was less than 0.17, the resulting fires were too small to rejuvenate the forests; and once k increased to approach 0.4, the resulting fires were so large that the model could not complete a simulation run. For example, there were eight fires during the first 27 years and each of them burned more than 93”/0 of the study area, which was outside the capability of the system. This is similar to the situation observed by Ratz (1995) except for slight differences in parameter values.

The second fire probability function increases with forest age at earlier successional stage(s) until approaching an upper asymptote. This kind of fire probability function shape characterizes the fire regime in western Siberia (Antonovski et al., 1992). The reasonable range of k that could lead to mean fire sizes and rotation periods compara- ble to those from observed fire regimes in Ontario was narrow, about 0.2 (Fig. 5(b)). However, in- creased k values were not outside the bounds of the system’s capabilities.

When fire probability increases with forest age in a sigmoid manner as assumed in Peterson’s model (1994), or in a linear manner as assumed in the Baker et al. (1991) model, a wide range of k values for Eq. (1) may lead to fire regimes that are comparable to those observed in Ontario (Fig. 5(c) and (d)). The sigmoid pattern of fire proba- bility assumes that the probability is proportional to the accumulation of standing biomass in the forest; and the linear increase pattern assumes a steady increase in probability with increasing forest age. Differences between the simulation results from these two fire probability functions are the consequent ranges of mean fire size and

146 C. Li et al. /Ecological Mode&g 99 (1997) 137-150

3 c

mo-

0.25 0.

k Value k Value

Fig. 5. Mean fire size and fire rotation period under (a) constant, (b) hyperbolic, (c) sigmoid and (d) linear forest’s response patterns to fire disturbances.

rotation period within the tested ranges of k. As an example, Table 2 shows selected simulation results for mean fire size and rotation period. For the same range of k, fire sizes for the sigmoid fire probability function were from 47 to 498 ha, and for the linear fire probability function were from 31 to 828 ha. Fire rotation periods were also different. Furthermore, different model behavior could result from various slope values under the assumption of a linear increase in fire probabil- ity.With the increasing slope value, however,

Table 2 Simulation results under two hypothetical response patterns of forests to fire disturbances

Scenario

k value Fire Size (ha) Rotation period (Year)

Sigmoid Linear

0.2 0.8 0.2 0.8 47 498 31 828

548 41 263 78

model behavior would be closer to those of the sigmoid, hyperbolic, and constant fire probability functions. This is because the fire probability has an upper limit of 1 regardless the actual slope value. For the same reason, Ratz (1995) quadratic fire probability function could also be approxi- mated by a sigmoid fire probability function.

The simulation results suggested that, theoreti- cally, similar fire regimes could result from differ- ent fire probability functions, but with different parameter values. More specifically, if a forest could quickly recover to its ‘normal’ fire suscepti- bility after a disturbance, such as in the assump- tions of constant and hyperbolic fire increase probabilities, then the k value needs to be more precisely estimated. In other cases where the re- covery of normal fire susceptibility requires a longer period after being burned, a small error in estimating the k value would not lead to signifi- cantly different system behavior.

C. Li et ul. /Ecological Modelling 99 (1997) 137-150 147

Table 3 Modifiers used in simulations to modify the baseline fire probability function

Soil moisture category Small effect Large effect

Dry I.1 1.2 Fresh 1 I Moist 0.91 0.83 Wet 0.83 0.69 Saturated 0.75 0.58

For the rest of the model behavior investiga- tion, the assumption that fire probability increases in a sigmoid manner with forest age was used. Furthermore, a k value of 0.6, intermediate within the range of potential k values, was used except where indicated.

4.2. Soil moisture

Relative effects of soil moisture on fire proba- bility were tested by comparing the simulation results from two sets of modifiers: one assumed a small effect and the other assumed a large effect. Table 3 shows the values of modifiers used in the simulations.

The assumption of large relative effects of pri- mary soil moisture on fire probability resulted in larger fire size (410.0 f 3.0 ha) than the assump- tion of small relative effects (388.1 f 4.7 ha). Sim- ulated mean fire rotation period was shorter (49.6 & 0.4 years) than the small relative effects (52.4 + 0.6 years) scenario.

Changes in relative effects of soil moisture regime on fuel moisture may influence the simu- lated fire regimes. When the influence of soil moisture regime on fuel moisture is small, the simulated fire regimes have smaller mean fire size and longer fire rotation period. When the influ- ence of soil moisture regime on fuel moisture is larger, the fire probabilities within areas with dry soil moisture regime (e.g. 71% of the study area in Fig. 4(c)) generally increase. The general increase in fire probability would likely result in a condi- tion that favors the spread of fires. Thus, a larger mean fire size results.

4.3. Weather

To evaluate the effect of weather on the mod- el’s behavior, two scenarios were assumed under a hypothetical stable climate condition. In the first scenario, the year-to-year variation in weather was small, represented by a normal distribution with a variance of 0.08. This distribution covers a range of dryness indices from 0.7 to 1.3. A larger year-to-year variation was assumed in the second scenario, represented by a normal distribution with a variance of 0.25, which covers a range of dryness indices from 0.1 to 1.9. Both scenarios were based on the observed relationship between ROS and ISI, where all the observed IS1 were less than 70 m/min (Forestry Canada Fire Danger Group, 1992). If we assume 35 m/min of IS1 represents a normal situation, then the calculated maximum dryness index would range from 1.40 to 1.75. The two scenarios, therefore. represent the extremes. Effects of spatial fuel configuration on controlling the spread of fires in the second sce- nario would be less than that in the first scenario, because the effects would have a higher probabil- ity of being overridden by weather that extremely favors fire processes.

Simulation results indicated larger mean fire sizes (459.4 + 4.2 ha) in the second scenario than the first (388.1 _+ 4.7 ha). Fire rotation periods were longer in the first scenario (52.4 + 0.6 years) than in the second scenario (44.3 f 0.4 years).

In the simulations, however, the number of fire initiations did not show a significant change when the variation in the relative effects increased. This may be the result of the assumption that the fire probability for initiation is lower than that for spread (Fig. 2(a)).

Weather conditions may have a profound im- pact on simulated fire regimes. In the first sce- nario, the largest increase in fire probability was limited to 1.3 times the normal situation. In such stable year-to-year weather conditions, the chance of controlling fire spread by spatial fuel configura- tion would likely be much larger than by weather, in most years. Consequently, resulting fire sizes were mainly determined by the age-mosaic of the forest landscape. When the largest increase in fire probability in years that extremely favor fire pro-

148 C. Li et al. /Ecological Modelling 99 (1997) 137-1.50

cesses is increased (e.g. 1.9 times the normal situa- tion as assumed in the second scenario) there would be increased chance for the effects of spa- tial fuel configuration to be overridden by the effects of weather conditions. In years when weather conditions extremely favor fire processes, the burned area would likely be larger than that under normal weather conditions. As a result, a larger mean fire size and shorter fire rotation period appeared in the simulation results.

5. Simulated temporal disturbance patterns and discussion

The developed model was then used to test the hypothesis that there is no significant difference among temporal disturbance patterns predicted using different fire probability functions. If the hypothesis is true, then there may not be a strong need to identify the exact shape of the fire proba- bility function to predict the temporal fire distur- bance pattern. If the hypothesis is not true, however, it is necessary to identify the exact shape of the fire probability function.

To compare the simulated temporal disturbance patterns, the fire rotation period was fixed at about 200 years, by the justification of k values for each of the four fire probability functions. The 200-year fire rotation period corresponded to fire regimes with a mean fire size of about 140 ha. This mean fire size appeared consistent with the situations in northwestern Ontario where the case study area is located.

Simulated fires were categorized into five groups according to the percentage of area burned by each fire. Group 1 was for fires that burned less than 1 ha, i.e. a fire ignition did not result in a fire initiation; group two included fires that burned less than 5% of the forested area within the landscape called light fire; group three contained the fires that burned 5-15% of the forested area called intermediate fire; group four were the fires that burned 15-30% of the forested land called large fire; and group five included severe fires that burned more than 30% of the forested land.

Temporal disturbance pattern was described as the mean interval between successive fires in each of the fire size groups. Fig. 6 shows the average situation for five replications for each of the fire probability functions. The results indicated an almost identical interval between two small fires for all four fire probability functions ranging from 5.2 to 6.2 years with an average of 5.8 years. A small variance existed in the intermediate fire group from 27.8 to 45.6 years with an average of 34.4 years. For large fires, the predictions differed among response patterns: ranging from 76.1 to 246.7 years with an average of 151.6 years. For severe fires (not shown in Fig. 6), predictions from three fire probability functions (constant, sigmoid and linear) were similar: no more than one fire in 1000 years. Since the analysis was based on simu- lation results of lOOO-year periods, no estimate of the mean interval between successive severe fires could be made. However, the hyperbolic fire probability function would result in three to four severe fires during a lOOO-year period, indicating a significant difference in predicting temporal dis- turbance pattern from the other fire probability function assumptions.

In the database of historical fire records, only three small fires had occurred in the case study area since 1976. The predictions of temporal dis- turbance pattern from all four fire probability functions were consistent with this observation. However, predictions for other fire size groups cannot be confirmed with the current 19-year database.

1 T

100

50

Fig. 6. Simulated average intervals for each of the forest’s response patterns to fire disturbances for each fire size group.

C. Li et al. /Ecological Modelling 99 (1997) 137-150 149

One-way ANOVA (Chambers and Hastie, 1993) was used to determine the mean intervals between two successive fires in the small, interme- diate and large fire size groups. This identified whether differences in the predicted intervals were caused by the different fire probability functions. In each size group, mean intervals of five replica- tions for each of the fire probability functions were used in the ANOVA.

For small fires, age-independent fire probability resulted in a shorter fire interval (P < 0.01) than the age-dependent fire probability functions. There were no significant differences among the predictions from three age-dependent fire proba- bility functions.

For intermediate fires, linear fire probability resulted in a longer fire interval (P < 0.01) than the other three fire probability functions. No sig- nificant differences were found among predictions from constant, hyperbolic and sigmoid fire proba- bility function assumptions.

Significant differences (P < 0.01) in predicting temporal disturbance pattern existed for large fires, except for hyperbolic and sigmoid fire prob- ability functions.

The hypothesis that there is no significant dif- ference among temporal disturbance patterns pre- dicted using different fire probability functions should generally be rejected, especially when the prediction of large fires is emphasized. When pre- dicting fires that burn less than 15% of the total study area, however, variations caused by using different fire probability functions would be small. Better prediction of a temporal disturbance pat- tern, especially for large and severe fires, will require quantitative knowledge of the fire proba- bility function.

Results from the model behavior study also suggest that more attention should be paid to identifying the potential for the role of spatial fuel configuration in controlling fire spread processes to be overridden by weather conditions. Further- more, to improve the prediction of temporal fire disturbance patterns on a real landscape, the in- teraction between fire and insect pest disturbances and human activity such as fire management, may need to be incorporated into the model.

6. Conclusions

Temporal patterns of small and intermediate fire disturbances over a long period could be predicted using different fire probability func- tions, with different asymptotes. To obtain a bet- ter prediction of temporal patterns for large and severe fire disturbances, however, an explicit un- derstanding of the quantitative relationship be- tween fire probability and forest age may be required.

Acknowledgements

We thank Dave Baldwin and Ryan Bae for assistance in GIS data analysis; Paul Ward, Al Tithecott, Paul McBay, Rob Janser, and Jim Ca- puto of the Aviation, Flood and Fire Manage- ment Branch of the Ontario Ministry of Natural Resources for providing historical fire data; Lisa Buse for critical reading of an earlier version of this paper. Model development benefited from discussions with Buzz Holling, Garry Peterson, and Paul Marples of the University of Florida, Bob Gardner of the Appalachian Environmental Laboratory and Bob O’Neill of the Oak Ridge National Laboratory. This research was funded by the Sustainable Forestry Initiative and Envi- ronmental Assessment Condition # 107(a), On- tario Ministry of Natural Resources.

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